Peak-CNNv2: A Deformable FPN Architecture with YOLO-style Modules for Accurate Particle Localization

Peak-CNNv2: A Deformable FPN Architecture with YOLO-style Modules for Accurate Particle Localization | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Peak-CNNv2: A Deformable FPN Architecture with YOLO-style Modules for Accurate Particle Localization Haoyu Wu, Yaoming Liu, Liangting He, Yan Pan, Yitong Shi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7848113/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Accurate detection of particle clusters in high-density 2D images is crucial for high-energy physics, medical imaging, and astrophysics. Traditional clustering methods, such as K-means and DBSCAN, often fail with overlapping or closely spaced particles, causing false detections and errors in centroid-of-gravity (COG) measurements. To address these challenges, we propose Peak-CNNv2, a lightweight convolutional neural network that directly predicts cluster peak positions for precise localization. The network incorporates YOLO-style convolutional blocks—drawing on YOLO’s (You Only Look Once) design principle of efficient local feature extraction to enhance responsiveness to dense particle distributions—alongside deformable convolutions and a multi-scale Feature Pyramid Network (FPN) to improve handling of overlapping and irregular clusters. Compared to heavier models, Peak-CNNv2 achieves high accuracy with manageable computational cost. Experimental results demonstrate that Peak-CNNv2 achieves an F1-score exceeding 93% for 1–60 clusters and robust performance under overlapping conditions. It also reduces centroid errors and minimizes false positives. By integrating deep learning with multi-scale feature fusion and YOLO-inspired efficient convolution design, Peak-CNNv2 provides a scalable and accurate solution for detecting particles in complex and noisy data. This approach advances image analysis in physics, medical imaging, and astrophysics, enabling reliable processing of dense datasets. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics Particle cluster detection Convolutional neural network Centroid localization Deformable Convolution YOLO Feature Pyramid Network (FPN) Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 1. Introduction Cluster detection in high-density 2D images is foundational to high-energy physics, medical imaging, and astrophysics. In high-energy physics, detectors like the SPHENIX TPC and LHCb generate data with ~ 1 particle per square centimeter, where accurate cluster identification links collision signals to particles and underpins momentum/energy calculations [ 1 , 4 ]; in medical imaging, it supports tumor cell analysis, and in astrophysics, it enables galaxy cluster mapping. However, several challenges remain, including false positives (FP), false negatives (FN), and centroid-of-gravity (COG) errors—for example, a 0.1-pixel COG error can distort physics momentum calculations by 5%. Traditional methods fail to address these issues: K-means requires predefined cluster counts, DBSCAN collapses overlapping high-density regions [ 1 ], local maxima detection misidentifies noise as clusters (causing FP), and thresholding may miss true clusters (leading to FN) or merge clusters (resulting in COG errors), showing their inability to handle real-world data complexity. Despite the advancements brought by deep learning, significant limitations persist in cluster detection tasks. Early CNNs like LeNet-5 and AlexNet lack sufficient depth for overlapping clusters [ 5 ], ResNet focuses more on classification than localization, U-Net excels in segmentation but not sub-pixel COG precision [ 3 ], CenterNet's fixed kernels struggle with irregular clusters, the Particle Flow Network (PFN) has physics-specific optimizations but relies on error-prone traditional preprocessing [ 2 ], and the original Peak-CNN performs poorly when cluster counts exceed 30 due to weak noise suppression. To tackle these gaps, this study proposes Peak-CNNv2, an end-to-end CNN that reduces FN by capturing weak signals, minimizes FP via noise suppression, and eliminates COG errors. The proposed pipeline comprises several key components: adaptive normalization to ensure consistent feature representation; deformable convolutions that adapt to irregular cluster shapes; a multi-scale FPN architecture with auxiliary supervision to enhance cross-scale feature representation and localization accuracy; and a weighted MSE heatmap regression with a multi-scale loss function, which enables sub-pixel centroid-of-gravity (COG) localization and prioritizes challenging samples. This research offers practical value: it can integrate into LHCb's particle-identification (PID) ecosystem to build a fully machine-learning-driven PID framework [ 4 ] and provide a user-friendly tool for medical imaging and astrophysics researchers. The study evaluates Peak-CNNv2 against SOTA models (CenterNet, PFN, U-Net) using metrics like F1-score and MSE on datasets with 1 to 90 clusters. The paper is divided into six main sections. The introduction, presenting the challenges of particle cluster detection in high-density 2D images and the limitations of traditional clustering methods, is followed by a literature review that outlines the evolution of relevant convolutional neural networks and specialized particle detection models, enabling the definition of the gap in existing literature that this paper addresses. The methodology section details the data source, preprocessing steps, model architecture (including deformable convolutions and FPN structure), loss function, optimization strategies, and evaluation metrics. It is followed by the experiments section, which covers experiment settings, hyperparameter optimization, in-depth analysis of key hyperparameters, ablation study, and comparison with state-of-the-art methods to assess the model's performance. In the results section, the main findings are discussed through comprehensive testing on the test dataset, including batch prediction, quantitative metric calculation, grouped visualization analysis, and the variation of detection metrics with different cluster counts, allowing for a clear understanding of the model's performance. Finally, conclusions summarize the key achievements, limitations of the proposed Peak-CNNv2, and directions for future research, aiming to further improve the algorithm's performance in real-world scenarios. 2. literature review The detection of overlapping particles in high-density environments is a challenging computer vision task that demands robust feature extraction and precise sub-pixel localization, with the evolution of Convolutional Neural Networks (CNNs) being pivotal in addressing this. Early architectures like LeNet-5 and AlexNet enabled automatic feature learning, though their limited depth and receptive fields restricted their effectiveness for dense particle localization [ 5 ], a limitation subsequently addressed by deeper models such as VGGNet and ResNet, which advanced hierarchical multi-scale feature learning through mechanisms like residual connections, thereby improving the detection of subtle patterns in crowded backgrounds. A significant paradigm shift occurred with the introduction of one-stage detectors, notably YOLO [ 6 ], which reframed object detection as a single regression problem, thereby enabling fast, real-time performance. Subsequent iterations in the YOLO series, such as YOLOv4 [ 9 ], further enhanced this framework by incorporating an effective balance of speed and accuracy through enhancements like the CSPDarknet53 backbone, Spatial Pyramid Pooling (SPP), and the Path Aggregation Network (PANet), which collectively improve feature extraction and multi-scale fusion for complex scenes [ 9 ]. Handling multi-scale objects is critical in dense particle fields, as deep CNNs naturally produce multi-scale features where high-level features are semantically strong yet spatially coarse, while low-level ones are detailed but lack semantics [ 7 ], a problem directly solved by the Feature Pyramid Network (FPN) [ 7 ] through its use of top-down pathways with lateral connections to build a feature pyramid where every level is rich in both semantics and spatial detail, making it a standard for multi-scale detection. To further address the challenge of irregular particle shapes and dense overlaps, deformable convolutional networks [ 11 ] were introduced, augmenting standard convolutions with learnable offsets to allow the receptive field to adapt to particle geometry, a concept that has evolved through versions like DCNv3, which integrates an attention mechanism [ 12 ], and more recently DCNv4, which was proposed to enhance expressive power and optimize memory access, resulting in faster convergence and superior efficiency for vision tasks [ 12 ]. This progression represents a continuous effort to combine the adaptability of attention with the efficiency of convolutions—a principle also explored in architectures like the deformable embedding vision transformer (DEViT), which adaptively extracts multi-scale features for targets of varying sizes [ 10 ]. Inspired by these advancements, our work builds upon keypoint detectors like CenterNet to avoid anchor-based complexities, focusing instead on direct cluster peak prediction and, unlike specialized tools like PFN that prioritize traditional clustering, unifying feature extraction and localization within a deep learning framework; consequently, we propose Peak-CNNv2, which integrates multi-scale learning via FPN [ 7 ] and enhances geometric flexibility with deformable convolutions [ 11 , 12 ], while also incorporating efficient convolutional blocks inspired by YOLOv4 [ 9 ] and employing multi-scale auxiliary supervision to stabilize training and refine localization, with experiments demonstrating that our model achieves high accuracy in dense particle detection with computational efficiency, effectively addressing the key challenges of overlap and shape variation. 3. Methodology 3.1 Data source and Acquisition observation. Data from this dataset was stored in a text file and read to generate high-resolution 64×64 grayscale maps with variable numbers of clusters, where the precise locations of these clusters (represented by local maxima) are embedded as high-resolution coordinates in the images. The 2D Gaussian clusters have key parameters including a width of 1.0 pixel units in both x and y directions, and 3000 samples per cluster. These parameters result in a resolution of the COG of about 0.02 pixel units. The cluster width and resolution were set to match that of the clusters seen in the SPHENIX TPC. 3.2 Preprocessing The raw data is stored in the text file and a piece of python code is used to read this data and transform it into 64*64 pixel images with clusters shown. The images are visualized using grayscale colormaps. Each number is corresponded to one pixel. From zero to bigger numbers, the brightness of pixels increases. Thus, clusters can be clearly shown in the images. With the initial high-resolution coordinates given, we used them to compare with the calculated coordinates of COG points. The coordinate differences (dx, dy) between the initial and calculated centers of gravity were computed. These differences were used to plot a normal distribution, from which the systematic bias and standard deviation were derived. Subsequently, the Peak-CNN model was trained on the dataset to produce cluster images with significantly improved centroid-of-gravity recognition accuracy. PeakCNN is particularly well-suited for integration into the LHCb particle-identification (PID) ecosystem (LHCb Collaboration, 2008) [ 4 ]. In this system, charged hadrons (2–100 GeV/c) are identified by dual Ring-Imaging Cherenkov detectors, muons by dedicated muon chambers, and electrons/photons by calorimeters. The CNN-derived peak coordinates can be seamlessly incorporated into downstream PID pipelines, either through likelihood-based combination of subdetector responses or via advanced neural-network classifiers. This integration results in a fully machine-learning-driven global PID framework. Thus, PeakCNN not only represents a significant advancement in high-density particle imaging but also highlights the growing synergy between deep learning and experimental high-energy physics. A canonical CNN consists of four key components: convolutional layers that apply learnable kernel filters to extract hierarchical features, pooling layers that reduce spatial resolution while preserving important information, fully connected layers that handle high-level reasoning, and non-linear activation layers (e.g., ReLU, Leaky ReLU, Parametric ReLU) that introduce non-linearity and enable gradient-based optimization [ 5 ]. Building upon this foundation, our Peak-CNNv2 employs a Feature Pyramid Network (FPN) with deformable convolutions to enhance multi-scale feature extraction and geometric adaptability. The model incorporates YOLO-style convolutional blocks and multi-scale auxiliary supervision, enabling precise localization of overlapping particles through heatmap regression. This architecture achieves robust performance in dense particle detection while maintaining computational efficiency. 3.4 Loss Function and Optimization The model employs a multi-strategy weighted MSE loss for particle detection, with dynamic weight map W adjusted based on ground truth heatmap Y and spatial features. It prioritizes peak regions via \(\:{\text{W}}_{\text{base}}\text{=1+12}\text{⋅}\text{Y}\text{(}\text{i}\text{,}\text{j}\text{)}\) , penalizes false positives \(\:\text{(}{\text{W}}_{\text{fp}}\text{=4.0}\text{}\) for high predictions with low truth) and adaptively boosts challenging targets— \(\:\text{1.5×}\) for weak signals, \(\:\text{2.0×}\) for single particles, \(\:\text{2}\text{.5×}\) for overlaps, and \(\:\text{2.0×}\) or dense samples (> 30 clusters). This focuses learning on critical regions while mitigating noise. For multi-scale supervision, the total loss combines the main output loss with auxiliary losses from intermediate feature maps: \(\:\text{Ltotal}\text{=}\text{Lmain}\text{+0.3×(}\text{Laux}\text{8}\text{x}\text{+}\text{Laux}\text{16}\text{x}\text{+}\text{Laux}\text{32}\text{x}\text{)}\) where each auxiliary loss is computed using the same weighted MSE strategy, ensuring consistent optimization across scales. Optimization employs AdamW \(\:\text{(}\text{lr}\text{=3×1}{\text{0}}^{\text{-4}}\text{,}\text{weig}\text{h}\text{t}\text{}\text{decay}\text{=1×1}{\text{0}}^{\text{-4}}\text{)}\) with a Cosine Annealing scheduler \(\:\text{(}\text{Tmax}\text{=70}\text{epoc}\text{h}\text{s}\text{,}{\eta}_{\text{min}}\text{=1×1}{\text{0}}^{\text{-7}}\text{).\:}\) Training strategies include gradient accumulation (2 steps, effective batch size 32), early stopping (15 epochs patience), and checkpointing. HDF5 preprocessing and stratified sampling ensure balanced training across 1–90 cluster densities, enabling robust convergence despite the model's complexity from deformable convolutions and multi-scale FPN architecture. 3.5 Evaluation metric Multiple metrics have been used to measure the model fit and accuracy of the model incorporated. MSE -Measure the average pixel squared difference between the predicted heatmap and the true heatmap \(\:MSE=\frac{1}{H\times\:W}\sum\:_{i=1}^{H}\sum\:_{j=1}^{W}({Y}_{i,j}-{\widehat{Y}}_{i,j}{)}^{2}\) (a) $$\:{\widehat{Y}}_{i,j}:\text{P}\text{r}\text{e}\text{d}\text{i}\text{c}\text{t}\text{e}\text{d}\:\text{h}\text{e}\text{a}\text{t}\text{m}\text{a}\text{p}\:\text{p}\text{o}\text{s}\text{i}\text{t}\text{i}\text{o}\text{n}\:\left(\text{p}\text{i}\text{x}\text{e}\text{l}\:\text{v}\text{a}\text{l}\text{u}\text{e}\:\text{a}\text{t}\:\right(\text{i},\text{j}\left)\right)$$ $$\:{Y}_{i,j}:the\:pixel\:value\:of\:the\:true\:heatmap\:position\left(\right(i,j\left)\right)$$ $$\:H\times\:W=64\times\:64:height\times\:width$$ PSNR - Image quality assessment metric based on MSE \(\:PSNR=20\times\:{\text{log}}_{10}\left(\frac{MA{X}_{Y}}{\sqrt{MSE}}\right)\) (b) True Positives, TP -The distance between the predicted centroid and the true centroid is less than the set threshold $$\:distance(\widehat{p},p)=\sqrt{({x}_{\widehat{p}}-{x}_{p}{)}^{2}+({y}_{\widehat{p}}-{y}_{p}{)}^{2}}$$ \(\:\text{DISTANCE\_THRESHOLD=}\text{0}\text{.}\) 1 (c) \(\:FP={N}_{pred}-TP\) (d) \(\:{Bias}_{x}={N}_{1}\sum\:_{i=1}^{N}({x}_{{i}_{found}}-{x}_{{i}_{true}})\) (e) \(\:{Bias}_{y}={N}_{1}\sum\:_{i=1}^{N}({y}_{{i}_{found}}-{y}_{{i}_{true}})\) (f) False Negatives, FN - The number of true centroids that are not matched by the predicted centroids (particles missed by the model). \(\:FP={N}_{true}-TP\) (g) Fake rate -The proportion of false positives in the total number of detections reflects the "false alarm" tendency of the model. $$\:Fake\:Rate=\frac{FP}{TP+FP}\:\left(h\right)$$ Precision - Measures the proportion of true positives among the predicted centroids, reflecting the model's ability to control "false detections". \(\:Precision=\frac{TP}{TP+FP}\) (i) Recall -Measure the proportion of true centroids that are successfully detected, reflecting the model's ability to control "missed detections". \(\:Recall=\frac{TP}{TP+FN}\) (j) Mean Coordinate Error -The average Euclidean distance between successfully matched centroid pairs, which measures the positioning accuracy. \(\:Mean\:Coordinate\:Error=\frac{1}{TP}\sum\:_{k=1}^{TP}\sqrt{({y}_{k}-{\widehat{y}}_{k}{)}^{2}+({x}_{k}-{\widehat{x}}_{k}{)}^{2}}\) (k) Efficiency -The proportion of accurate matches within the error tolerance range, which reflects the high-precision positioning ability. \(\:Efficiency=\frac{Accurate\:Matches}{TP}\) (l) Count Error - Predict the absolute difference between the number of predicted centroids and the number of true centroids to measure the accuracy of particle counting. \(\:Count\:Error=\left|{N}_{true}-{N}_{pred}\right|\) (m) F1 score - the harmonic mean of precision and recall, evaluating a model's performance by balancing false positives and false negatives. \(\:F1=2\times\:\frac{Precision\times\:Recall}{Precision+Recall}\) (n) 4. Experiments 4.1 Experiments settings Peak-CNNv2 is a deep learning model for particle detection and dense object localization that generates target-position heatmaps from 64×64 single-channel images, employing a Feature Pyramid Network (FPN) architecture integrated with deformable convolutions and multi-scale supervision, as well as YOLO-style convolutional blocks (equipped with LeakyReLU activations and batch normalization) to enhance feature representation. Specifically, deformable convolutions adapt to target deformations via predicted (x,y) offsets, strengthening the capture of complex features from irregular particle shapes; the FPN structure consists of a bottom-up pathway for feature extraction and a top-down pathway with lateral connections for multi-scale feature fusion, generating feature maps at multiple scales (4×4, 8×8, 16×16, 32×32, and 64×64) to effectively handle varying particle sizes and densities. Additionally, the network incorporates auxiliary supervision at intermediate scales (8×, 16×, and 32× downsampling) to stabilize training and improve localization precision, while its final output layer produces 64×64 heatmaps through transposed convolution and sigmoid activation. With strict size constraints, a multi-scale weighted MSE loss, and balanced sampling, Peak-CNNv2 can accurately localize dense, low-intensity, or irregular targets—making it suitable for particle analysis and cell detection—and the overall FPN architecture of Peak-CNNv2 (integrating deformable convolutions and multi-scale supervision) is illustrated in Fig. 1 . 4.2 Hyperparameter Optimization After establishing a robust model architecture, we further refined its hyperparameters. This process involved selecting the top 10 candidate architectures from a model architecture search, wherein hyperparameter sets were randomly sampled from a broad search space and evaluated to identify the optimal configuration. As shown in Fig. 2 , a weight decay of 1e − 5 was selected for its ability to minimize validation loss. Similarly, a Gaussian sigma value of 0.3 was chosen based on Fig. 3, as it yielded the most stable training process and the lowest loss. A batch size of 8, presented in Fig. 4 , was adopted to introduce beneficial gradient noise, thereby promoting more fine-grained feature learning. Finally, as indicated in Fig. 5, a learning rate of 1e − 4 was identified as optimal, achieving both the lowest loss and robust convergence. 4.3 In-depth Analysis of Hyperparameters A deeper understanding of the impact of hyperparameters on model performance was pursued through a series of manual experiments. These experiments were conducted using the 10 optimal models identified during the model architecture search phase. In each experiment, only one hyperparameter was adjusted while keeping others fixed, with the best validation loss serving as the performance evaluation metric. Among the tested hyperparameters, batch size, learning rate, weight decay, and Gaussian sigma emerged as key influential factors: As quantitatively evidenced in the Fig. 4 , a smaller batch size (e.g 8) enhanced the model's ability to learn fine-grained features by introducing moderate gradient noise, which prevents convergence to overly smooth local optima—this configuration is particularly compatible with the model's architectural demands for capturing subtle patterns. In contrast, larger batch sizes (32 and above) reduced the model's sensitivity to feature variations due to excessively stable gradient estimates, leading to increased validation loss. For learning rate, the analysis in the Fig. 5 reveals that a value of 1e-4 not only achieved the lowest validation loss but also ensured robust convergence, as an excessively small learning rate (e.g 1e-5) caused slow convergence with suboptimal loss, while an intermediate value (e.g. 5e-5) failed to attain the same level of performance as 1e-4. Regarding weight decay, the Fig. 2 plot shows that a small value (1e-5) minimized validation loss by effectively regularizing the model without undercutting feature learning capacity; in contrast, larger weight decay values (e.g., 1e-4,5e-4) increased loss due to over-regularization. For Gaussian sigma, the Fig. 3 plot indicates that a value of 0.3 delivered both the most stable training process and the lowest loss, whereas larger values (e.g 0.5 0.7) either elevated loss or compromised training stability. The optimal values of these parameters (batch size = 8, learning rate = 1e-4, weight decay = 1e-5, Gaussian sigma = 0.3) are highly aligned with the model's architectural characteristics, as they synergistically leverage the model's capacity for adaptive feature extraction while ensuring balanced regularization and convergence, thereby achieving the minimum validation loss across experiments. 4.4 Ablation Study To rigorously evaluate the contribution of each key component in the proposed PeakCNNv2 architecture, a comprehensive ablation study was conducted. The performance of the full model was compared against several ablated variants on the test dataset. The specific modifications applied to each variant, along with their corresponding performance differences, are summarized in Table 1 . Results from this ablation study confirm the necessity and effectiveness of each core component within the architectural design. Table 1 Model Modification Comparison Table Ablation Variant Modification Baseline Base Model None(baseline) Deformable Conv No Deformable Conv Remove Deformable Conv Pyramid Structure No Pyramid Remove Pyramid Structure Activation ReLU Instead of Leaky Replace Leaky ReLU with ReLU Auxiliary Loss No Auxiliary Loss Remove Auxiliary Loss This study report the training loss, validation loss, and validation MSE of models with different architectural variants, all using the same loss function (weighted MSE loss with multi-scale supervision for applicable models) and evaluation metric (MSE) to ensure fair comparison: the Base Model, equipped with deformable convolutions, a pyramid feature fusion structure, LeakyReLU activation, and auxiliary loss, performs remarkably in training efficiency and final performance — its training loss drops rapidly in early epochs and stabilizes at an extremely low value, the validation loss follows a steep trajectory to plummet and converge to the minimum, and the validation MSE also reaches the lowest among all variants, demonstrating strong precise peak prediction capability; in contrast, the "No Deformable Conv" variant (replacing deformable convolutions with standard ones) exhibits significantly slower reduction in training and validation losses, with higher final loss values and elevated validation MSE due to the lack of adaptive spatial sampling ability of standard convolutions, the "No Pyramid" variant (abandoning pyramid fusion and relying only on simple upsampling) shows even slower loss decay, with training loss decreasing sluggishly, validation loss plateauing at a relatively high level, and validation MSE failing to reach the low range of the Base Model (reflecting the importance of the pyramid structure in fusing multi-scale features), the "ReLU Instead of Leaky" variant (swapping LeakyReLU with ReLU) suffers from slower training convergence, higher final validation loss, and increased validation MSE as ReLU’s hard zeroing of negative values causes premature gradient vanishing while LeakyReLU preserves gradient flow for negative inputs, and the "No Auxiliary Loss" variant (omitting auxiliary supervision) has a deceptively low final training loss (even lower than the Base Model) but much worse validation loss and MSE since auxiliary loss regularizes training via multi-scale intermediate supervision, prevents overfitting, and enhances generalization; in summary, each architectural component of the Base Model contributes synergistically to efficient training and strong peak prediction, and removing any component leads to measurable degradation in convergence speed, final loss, or generalization, as visually summarized in the loss and MSE convergence curves of Fig. 6 . Additionally, this ablation study systematically evaluates the impact of various architectural components on model complexity, measured in terms of both parameter count (in millions, “M”) and computational demand in FLOPs (billions, “B”). As shown in Fig. 7 and Fig. 8, which illustrate model parameters and FLOPs respectively, the Base model serves as the reference point with 6.266M parameters and 0.3370B FLOPs. Removing the deformable convolution module reduces parameters slightly to 6.034M, as depicted in Fig. 7 , but increases FLOPs to 0.5348B, which is shown in Fig. 8. This indicates that deformable convolutions introduce a substantial computational footprint despite a minor drop in parameter count. Omitting the pyramid structure results in the largest reduction across both metrics, bringing parameters down to 5.131M and FLOPs to 0.1200B, as visible in Fig. 7 and Fig. 8 respectively. This highlights the dominant role of the pyramid structure in driving model complexity. Substituting LeakyReLU with ReLU keeps both parameters and FLOPs identical to the Base model, at 6.266M and 0.3370B, which suggests that LeakyReLU does not incur extra computational overhead relative to ReLU in this architecture. Finally, excluding the auxiliary loss leaves both parameters and FLOPs unchanged from the Base model, implying that auxiliary loss has a negligible effect on overall computational requirements and parameter count. 4.5 Comparison with State-of-the-Art (SOTA) Methods To thoroughly evaluate the performance of PeakCNNv2, a comparative analysis was conducted against four representative models under identical training data, preprocessing, and evaluation protocols: CenterNet, recognized for its efficient anchor-free structure; PFN, a network specifically designed for particle detection; U-Net, a key architecture widely used in biomedical image segmentation; and LightweightYOLO, a newly introduced lightweight detector emphasizing inference efficiency. This comparison covered a range of cluster densities from 1 to 80 clusters to rigorously assess how well the models handle both isolated clusters and heavy overlap. As detailed in Fig. 9 and Fig. 10 , which respectively illustrate the metrics comparison at a 0.1 threshold and the precision, recall, and F1 score across models, the results conclusively demonstrate that PeakCNNv2 achieves state-of-the-art accuracy. Figure 10 shows that PeakCNNv2 outperforms or matches the detection performance, particularly in F1-score, of PFN, U-Net, and LightweightYOLO across all densities, with notable advantages in precision—maintaining the highest precision values among all models. Meanwhile, Fig. 9 indicates that PeakCNNv2 delivers superior localization precision compared to CenterNet, reflected in a lower false rate and strong efficiency at the 0.1 threshold, and achieves accuracy levels far exceeding those of LightweightYOLO, which prioritizes inference speed at the cost of precision in dense scenarios. This performance validates our architectural design, as the synergistic integration of deformable convolutions and attention mechanisms effectively addresses the core challenges of overlap and shape variation where traditional methods often struggle. Regarding computational costs, as summarized in Table 2 which compares model performance in terms of parameters and inference efficiency, PeakCNNv2 exhibits a favorable trade-off. Although it incorporates more parameters, at 6.034 million, than CenterNet, PFN, and LightweightYOLO, its computational demand measured in FLOPs—533.7 million—is substantially lower than that of PFN with 973.7 MF and U-Net with 1720.6 MF, indicating a more efficient use of model capacity. Furthermore, while the inference time of PeakCNNv2 is slightly longer than that of CenterNet, at 1.514 milliseconds compared to 1.449 ms, and markedly higher than LightweightYOLO's 0.736 ms, this modest increase is justified by significant gains in accuracy. This is especially evident in dense and overlapping scenarios, where PeakCNNv2 maintains robust performance with far less degradation than other models. Table 2 Model Performance Comparison (Parameters and Inference Efficiency) Model Parameters (M) FLOPs (MF) Inference Time (ms) Throughput (samples/sec) Total Evaluation Time (sec) CenterNet 0.917 146.2 1.449 ± 0.013 690.3 ± 6.4 119.68 ± 0.30 PFN 1.506 973.7 0.743 ± 0.004 1345.6 ± 7.6 118.89 ± 0.70 UNet 3.704 1720.6 1.265 ± 0.013 790.6 ± 8.1 119.30 ± 0.38 PeakCNNv2 6.034 533.7 1.514 ± 0.036 660.8 ± 15.7 121.11 ± 0.30 LightweightYOLO 0.119 29.6 0.736 ± 0.002 1357.9 ± 4.5 122.01 ± 0.62 In conclusion, PeakCNNv2 establishes a new state-of-the-art for dense particle detection. It delivers top-tier accuracy through advanced architectural features, achieving a competitive balance between precision and computational efficiency. Although LightweightYOLO excels in pure inference speed, PeakCNNv2 is better suited for applications that prioritize precision, such as offline data analysis and high-precision measurements, where its substantial accuracy advantages outweigh the moderate increase in inference time. 4.5.1 Statistical Significance Testing The performance and statistical significance of the PeakCNNv2 model for peak detection were evaluated through a comparison with four competing architectures—CenterNet, PFN, LightweightYOLO, and UNet. Robustness was further assessed using bootstrap analysis with 100 random seeds. PeakCNNv2 integrated deformable convolutions (for adaptive spatial feature capture) and a Feature Pyramid Network (FPN) for multi-scale fusion, while other models followed canonical designs; peak extraction from heatmaps relied on thresholding with non-maximum suppression, and key metrics (Recall, F1 score) were used for quantification. As shown in the bootstrap analysis plots for Recall in Fig. 11 and for F1 score in Fig. 12 , PeakCNNv2 exhibited the highest mean Recall (~ 0.920) and F1 score (~ 0.927) with narrow 95% confidence intervals (indicating robust initialization stability), whereas other models (e.g., CenterNet, PFN) had lower mean values and wider intervals. Statistical significance via independent-samples t-tests showed: PeakCNNv2 significantly outperformed LightweightYOLO in F1 score and precision (p = 0.0219, *), with no significant recall difference (p = 0.1136); against UNet, all metrics were significant—F1 score and precision (p = 0.0022, **), recall (p = 0.0267, *)—with PeakCNNv2 achieving ~ 3.21% higher F1, ~ 3.18% higher precision, and ~ 3.23% higher recall. Collectively, PeakCNNv2 demonstrates statistically significant and superior peak detection performance versus key competitors, validating its effectiveness and robustness. Table 3. Significance Test Results of Detection Performance Metrics Between Models Pair Metric p-value Sig. % Difference (Neg = Model1 Modl2) LightweightYOLO vs PeakCNNv2 f1 0.021917 * -2.19 precision 0.021917 * -2.16 recall 0.113636 ns -2.19 (No significant difference) PeakCNNv2 vs UNet f1 0.002255 ** 3.21 precision 0.002255 ** 3.18 recall 0.026729 * 3.23 Note: *p < 0.05, **p < 0.01, ***p < 0.001, ns = p ≥ 0.05 (No significant difference) 5. Experimental Results A comprehensive evaluation of model performance was carried out on the test dataset, incorporating batch prediction, quantitative metric calculation, and grouped visualization analysis. The comparative analysis of heatmaps and error metrics across different cases, as illustrated in Fig. 13 , offers intuitive visual evidence of the model’s performance under varied conditions. Key metrics across all samples are summarized in Table 4 . Further analysis categorized performance into best, average, and worst-performing samples by sorting results according to F1-score. Directional errors—dx for the x-direction and dy for the y-direction—were also computed to evaluate localization accuracy along each axis. The distributions of these errors, visualized in the histograms in Fig. 14 and Fig. 15, show concentrations near zero with minimal bias, indicating unbiased estimation capabilities and providing insight into the error patterns of the model. The evaluation framework also incorporates grouped statistics for the best, average, and worst-performing samples, comparing metrics such as average precision, recall, F1-score, MSE, and count error across groups. Representative samples from each group are visualized with their original images, true heatmaps, predicted heatmaps, and key metrics, as exemplified in Fig. 13 , allowing observation of how the model handles different peak distribution scenarios. Table 4 Average of Particle Detection Metrics on 80,000 Samples Cluster Count MSE Precision Recall F1 Score Efficiency Fake Rate 10 0.000009 ± 0.000000 0.995576 ± 0.000443 0.980540 ± 0.000997 0.987624 ± 0.000570 0.980540 ± 0.000997 0.004424 ± 0.000443 20 0.000022 ± 0.000000 0.988253 ± 0.000749 0.959160 ± 0.001103 0.973111 ± 0.000895 0.959160 ± 0.001103 0.011747 ± 0.000749 30 0.000041 ± 0.000000 0.980657 ± 0.000601 0.935267 ± 0.000893 0.957051 ± 0.000669 0.935267 ± 0.000893 0.019343 ± 0.000601 40 0.000070 ± 0.000001 0.976939 ± 0.000550 0.923709 ± 0.000774 0.949214 ± 0.000608 0.923709 ± 0.000774 0.023061 ± 0.000550 50 0.000068 ± 0.000001 0.972774 ± 0.000906 0.914150 ± 0.001131 0.942201 ± 0.000873 0.914150 ± 0.001131 0.027226 ± 0.000906 60 0.000110 ± 0.000001 0.970057 ± 0.000559 0.901889 ± 0.000965 0.934364 ± 0.000652 0.901889 ± 0.000965 0.029943 ± 0.000559 70 0.000131 ± 0.000002 0.967474 ± 0.001060 0.894140 ± 0.000801 0.928997 ± 0.000743 0.894140 ± 0.000801 0.032526 ± 0.001060 80 0.004339 ± 0.000015 0.944050 ± 0.000549 0.902234 ± 0.000684 0.922461 ± 0.000617 0.904031 ± 0.000636 0.057719 ± 0.000636 To further validate the robustness of PeakCNNv2 in real-world complex scenarios, experiments were conducted on the EMPS dataset—a collection of 465 electron microscopy images with pixel-level semantic instance segmentations stored as 32-bit integer maps [ 8 ]. This dataset contains images of various cells and crystals, making it well-suited for testing the model's performance in medical and related fields, as it can simulate the practical demands of cell localization in clinical diagnostics and crystal particle analysis in material science. Beyond this, the dataset presents significant challenges for peak detection due to strong noise, heavy object overlaps, and highly irregular instance shapes. As demonstrated in Fig. 16 , which showcases detection results on this dataset, PeakCNNv2 exhibits strong adaptability in such demanding conditions. The model's deformable convolutions enable adaptive feature extraction for irregularly shaped targets (e.g., irregular cell edges or non-uniform crystal morphologies), while its Feature Pyramid Network facilitates multi-scale feature fusion to address overlaps and noise. Additionally, the sub-pixel fitting strategy achieves fine-grained centroid localization. Both visual assessments in Fig. 16 and quantitative evaluations confirm that PeakCNNv2 effectively filters noise-induced false positives, disentangles heavily overlapping instances (such as clustered cells or aggregated crystal particles), and accurately locates centroids even for highly irregular objects, thereby validating its reliability in complex real-world contexts—including medical and material science scenarios targeted by the EMPS dataset. 6. Conclusion This study tackles the persistent challenge of accurately detecting and localizing overlapping particle clusters in high-density 2D images—a critical task in high-energy physics, medical imaging, and astrophysics. Traditional methods like K-means and DBSCAN often fail under high-density conditions, resulting in false detections and centroid errors. To address this, we proposed Peak-CNNv2, an end-to-end convolutional neural network that incorporates deformable convolutions, YOLO-style convolutional blocks, and a multi-scale Feature Pyramid Network (FPN) architecture with auxiliary supervision to adaptively capture irregular cluster shapes and suppress background noise, enabling precise sub-pixel localization of cluster peaks. Experimental results demonstrate that Peak-CNNv2 achieves state-of-the-art performance, with an F1-score exceeding 93% across 1–60 clusters and significant reductions in centroid localization error compared to models such as CenterNet, U-Net, and PFN. These improvements validate the effectiveness of our architectural innovations—particularly the deformable convolution modules that enhance geometric transformation modeling, the YOLO-style blocks that provide efficient feature extraction with LeakyReLU activations and batch normalization, and the multi-scale FPN structure that improves feature representation across varying particle sizes and densities—in handling dense and overlapping particles. The model holds practical value for integration into experimental pipelines such as the LHCb particle identification system, offering a scalable, deep learning–based solution for precision-sensitive applications. Despite its high accuracy, the model exhibits several limitations: First, the integration of deformable convolutions, YOLO-style blocks, and multi-scale FPN architecture increases computational complexity, resulting in higher GPU memory usage and longer inference times compared to lightweight models. Second, performance remains limited under extreme crowding or real experimental noise, occasionally leading to cluster merging. Third, as observed in the corresponding Fig. 16 , there is a failure to detect edge clusters—this is primarily attributed to the lack of edge cluster samples in the training dataset, which prevented the model from learning the feature patterns of clusters in edge regions through its deformable convolution and multi-scale fusion mechanisms. Future work will focus on improving computational efficiency through model compression and optimization of the deformable convolution operations, incorporating real experimental data (including edge cluster samples) to enhance generalization, and extending the framework to 3D detection for broader scientific utility. Declarations Funding Declaration The authors declare that this research received no specific financial support from any finding agency in the public, commercial, or not-for-profit sectors. None of the authors have received grants or financial contributions from organizations that could influence the design, execution, analysis, or interpretation of this study. Competing interests The authors declare no competing interests. Author Contribution Author Contributions StatementH.W. designed and debugged the model, conducted model training and testing, and solely performed the experiments and analysis for SOTA comparison, statistical significance testing, and ablation studies. H.W. also created all figures and tables and participated in the writing and final revision of the entire manuscript, specifically for sections 3.4 ,4.2, 4.3, 4.4, 4.5, 4.5.1, 5, and 6. Y.L. contributed to literature collection, paper outline development, and wrote sections 2 (Literature Review) and 3.5 (Evaluation Metric). Y.L. also reviewed the entire manuscript. Y.P. contributed to literature collection, paper outline development, and wrote sections 1 (Introduction) and 2 (Literature Review). Y.S. contributed to literature collection, paper outline development, and wrote section 1 (Introduction). L.H. participated in model design and partial debugging, and wrote sections 3.1 (Data source and Acquisition), 3.2 (Preprocessing), and 3.4 (Loss Function and Optimization).All authors reviewed and approved the final manuscript. Data Availability The electron microscopy image data used in this study are from the EMPS dataset, which was originally published by B. Yildirim and J. M. Cole in "Bayesian Particle Instance Segmentation for Electron Microscopy Image Quantification", Journal of Chemical Information and Modeling, 2021 (https://doi.org/10.1021/acs.jcim.0c01455). The dataset used for training and evaluation in our work is available from the corresponding author of the original publication upon reasonable request. References Fröhlich, B. et al. Track reconstruction in high-density environments. J. Instrum. 8 (05), P05013 (2013). LeCun, Y., Bengio, Y. & Hinton, G. Deep learning, Nature, vol. 521, no. 7553, pp. 436–444, (2015). Krizhevsky, A., Sutskever, I. & Hinton, G. E. ImageNet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems, vol. 25, pp. 1097–1105, (2012). LHCb & Collaboration The LHCb detector at the LHC. J. Instrum. 3 (08), S08005 (2008). LeCun, Y., Bottou, L., Bengio, Y. & Haffner, P. Gradient-based learning applied to document recognition, Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, (1998). Redmon, J., Divvala, S., Girshick, R. & Farhadi, A. You only look once: Unified, real-time object detection, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 779–788, (2016). Lin, T. Y. et al. Feature pyramid networks for object detection, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2117–2125, (2017). Yildirim, B. & Cole, J. M. Bayesian particle instance segmentation for electron microscopy image quantification. J. Chem. Inf. Model. 61 (4), 1810–1818 (2021). Bochkovskiy, A., Wang, C. Y. & Liao, H. Y. M. YOLOv4: Optimal Speed and Accuracy of Object Detection, arXiv preprint arXiv:2004.10934, (2020). Wu, Y. & Li, J. YOLOv4 with Deformable-Embedding-Transformer Feature Extractor for Exact Object Detection in Aerial Imagery, Sensors, 23 , 5, p. 2522, (2023). Dai, J. et al. Deformable Convolutional Networks, in Proceedings of the IEEE International Conference on Computer Vision, pp. 764–773, (2017). Xiong, Y. et al. Efficient Deformable ConvNets: Rethinking Dynamic and Sparse Operator for Vision Applications, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5652–5661, (2024). Wu, H., Liu, Y., He, L., Shi, Y. & Pan, Y. Peak-CNNv2, GitHub, [Online]. Available: haoyuwu2006615/Peakcnn-V2: Official PyTorch implementation of PeakCNNv2 — a deep learning model with deformable convolutions and Feature Pyramid Network (FPN) for high-accuracy, sub-pixel particle detection in dense and overlapping environments. (2025). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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07:40:28","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":107572,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eModel Parameters (Millions)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/35714af68d1ad402fe18c701.png"},{"id":96158966,"identity":"7d839f43-139c-4fd2-98fd-43d0b4d7f812","added_by":"auto","created_at":"2025-11-18 08:39:38","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":116984,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFLOPs (Billions)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/af756935b51c7f58edb72fbc.png"},{"id":96158996,"identity":"a31ac2cf-d038-497c-ab45-c0ca2819f2c9","added_by":"auto","created_at":"2025-11-18 08:39:40","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":102787,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePrecision，Efficiency，Fake rate Comparison at 0.1 Threshold Across Models\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/db4f2bb144c8a1bdfc791916.png"},{"id":96158880,"identity":"ee2695bc-a316-4a78-96b7-11576fb34e62","added_by":"auto","created_at":"2025-11-18 08:39:29","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":116067,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePrecision, Recall, and F1 Score Comparison Across Models\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/3e3a38f636d2e350528b1046.png"},{"id":96159031,"identity":"a11e1630-6401-4efc-8dc5-73755a18ac1f","added_by":"auto","created_at":"2025-11-18 08:39:46","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":109027,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eF1 Across Seeds (Bootstrap Analysis)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/74c3d67b013028cbfcf9db38.png"},{"id":96159003,"identity":"d8998f02-691d-4eb5-86fe-2a246e7714be","added_by":"auto","created_at":"2025-11-18 08:39:41","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":118285,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRecall Across Seeds (Bootstrap Analysis)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/6b162572d504b31d19c5ba45.png"},{"id":96158995,"identity":"fcfc1e4c-c474-4da5-8120-9558b33d4caa","added_by":"auto","created_at":"2025-11-18 08:39:40","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":382015,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eComparison of Heat Maps and Error Metrics for Different Cases\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/f94124e45fdcfc4c286d65dd.png"},{"id":96159020,"identity":"617e9966-1ee0-4757-b557-db23d359e5f6","added_by":"auto","created_at":"2025-11-18 08:39:46","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":46308,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eX-direction Error Distribution (dx = predicted x - true x)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/99f377ed16f94bde671ebf0f.png"},{"id":96158950,"identity":"10b31ad4-0dce-447a-ab9c-60064efd1966","added_by":"auto","created_at":"2025-11-18 08:39:36","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":45203,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eY-direction Error Distribution (dy = predicted y - true y)\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/bcc77883b6f6700fc806b38a.png"},{"id":96159044,"identity":"cb8894ad-9603-4be8-8365-71e1bba92eca","added_by":"auto","created_at":"2025-11-18 08:39:47","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":519329,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDetection Results on EMPS Dataset\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/d55dc1c8e6deb0cb039df981.png"},{"id":96709697,"identity":"3c1d31a0-806d-4ec4-8bce-ead6ae10a986","added_by":"auto","created_at":"2025-11-25 10:09:32","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3241097,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7848113/v1/c5097457-3e2b-4fac-8ae6-074adf28c7e7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Peak-CNNv2: A Deformable FPN Architecture with YOLO-style Modules for Accurate Particle Localization","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eCluster detection in high-density 2D images is foundational to high-energy physics, medical imaging, and astrophysics. In high-energy physics, detectors like the SPHENIX TPC and LHCb generate data with ~\u0026thinsp;1 particle per square centimeter, where accurate cluster identification links collision signals to particles and underpins momentum/energy calculations [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]; in medical imaging, it supports tumor cell analysis, and in astrophysics, it enables galaxy cluster mapping. However, several challenges remain, including false positives (FP), false negatives (FN), and centroid-of-gravity (COG) errors\u0026mdash;for example, a 0.1-pixel COG error can distort physics momentum calculations by 5%. Traditional methods fail to address these issues: K-means requires predefined cluster counts, DBSCAN collapses overlapping high-density regions [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], local maxima detection misidentifies noise as clusters (causing FP), and thresholding may miss true clusters (leading to FN) or merge clusters (resulting in COG errors), showing their inability to handle real-world data complexity.\u003c/p\u003e\u003cp\u003eDespite the advancements brought by deep learning, significant limitations persist in cluster detection tasks. Early CNNs like LeNet-5 and AlexNet lack sufficient depth for overlapping clusters [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], ResNet focuses more on classification than localization, U-Net excels in segmentation but not sub-pixel COG precision [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], CenterNet's fixed kernels struggle with irregular clusters, the Particle Flow Network (PFN) has physics-specific optimizations but relies on error-prone traditional preprocessing [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], and the original Peak-CNN performs poorly when cluster counts exceed 30 due to weak noise suppression. To tackle these gaps, this study proposes Peak-CNNv2, an end-to-end CNN that reduces FN by capturing weak signals, minimizes FP via noise suppression, and eliminates COG errors. The proposed pipeline comprises several key components: adaptive normalization to ensure consistent feature representation; deformable convolutions that adapt to irregular cluster shapes; a multi-scale FPN architecture with auxiliary supervision to enhance cross-scale feature representation and localization accuracy; and a weighted MSE heatmap regression with a multi-scale loss function, which enables sub-pixel centroid-of-gravity (COG) localization and prioritizes challenging samples.\u003c/p\u003e\u003cp\u003eThis research offers practical value: it can integrate into LHCb's particle-identification (PID) ecosystem to build a fully machine-learning-driven PID framework [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] and provide a user-friendly tool for medical imaging and astrophysics researchers. The study evaluates Peak-CNNv2 against SOTA models (CenterNet, PFN, U-Net) using metrics like F1-score and MSE on datasets with 1 to 90 clusters.\u003c/p\u003e\u003cp\u003eThe paper is divided into six main sections. The introduction, presenting the challenges of particle cluster detection in high-density 2D images and the limitations of traditional clustering methods, is followed by a literature review that outlines the evolution of relevant convolutional neural networks and specialized particle detection models, enabling the definition of the gap in existing literature that this paper addresses. The methodology section details the data source, preprocessing steps, model architecture (including deformable convolutions and FPN structure), loss function, optimization strategies, and evaluation metrics. It is followed by the experiments section, which covers experiment settings, hyperparameter optimization, in-depth analysis of key hyperparameters, ablation study, and comparison with state-of-the-art methods to assess the model's performance. In the results section, the main findings are discussed through comprehensive testing on the test dataset, including batch prediction, quantitative metric calculation, grouped visualization analysis, and the variation of detection metrics with different cluster counts, allowing for a clear understanding of the model's performance. Finally, conclusions summarize the key achievements, limitations of the proposed Peak-CNNv2, and directions for future research, aiming to further improve the algorithm's performance in real-world scenarios.\u003c/p\u003e"},{"header":"2. literature review","content":"\u003cp\u003eThe detection of overlapping particles in high-density environments is a challenging computer vision task that demands robust feature extraction and precise sub-pixel localization, with the evolution of Convolutional Neural Networks (CNNs) being pivotal in addressing this. Early architectures like LeNet-5 and AlexNet enabled automatic feature learning, though their limited depth and receptive fields restricted their effectiveness for dense particle localization [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], a limitation subsequently addressed by deeper models such as VGGNet and ResNet, which advanced hierarchical multi-scale feature learning through mechanisms like residual connections, thereby improving the detection of subtle patterns in crowded backgrounds. A significant paradigm shift occurred with the introduction of one-stage detectors, notably YOLO [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], which reframed object detection as a single regression problem, thereby enabling fast, real-time performance. Subsequent iterations in the YOLO series, such as YOLOv4 [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], further enhanced this framework by incorporating an effective balance of speed and accuracy through enhancements like the CSPDarknet53 backbone, Spatial Pyramid Pooling (SPP), and the Path Aggregation Network (PANet), which collectively improve feature extraction and multi-scale fusion for complex scenes [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eHandling multi-scale objects is critical in dense particle fields, as deep CNNs naturally produce multi-scale features where high-level features are semantically strong yet spatially coarse, while low-level ones are detailed but lack semantics [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], a problem directly solved by the Feature Pyramid Network (FPN) [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] through its use of top-down pathways with lateral connections to build a feature pyramid where every level is rich in both semantics and spatial detail, making it a standard for multi-scale detection. To further address the challenge of irregular particle shapes and dense overlaps, deformable convolutional networks [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] were introduced, augmenting standard convolutions with learnable offsets to allow the receptive field to adapt to particle geometry, a concept that has evolved through versions like DCNv3, which integrates an attention mechanism [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], and more recently DCNv4, which was proposed to enhance expressive power and optimize memory access, resulting in faster convergence and superior efficiency for vision tasks [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. This progression represents a continuous effort to combine the adaptability of attention with the efficiency of convolutions\u0026mdash;a principle also explored in architectures like the deformable embedding vision transformer (DEViT), which adaptively extracts multi-scale features for targets of varying sizes [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Inspired by these advancements, our work builds upon keypoint detectors like CenterNet to avoid anchor-based complexities, focusing instead on direct cluster peak prediction and, unlike specialized tools like PFN that prioritize traditional clustering, unifying feature extraction and localization within a deep learning framework; consequently, we propose Peak-CNNv2, which integrates multi-scale learning via FPN [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] and enhances geometric flexibility with deformable convolutions [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], while also incorporating efficient convolutional blocks inspired by YOLOv4 [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] and employing multi-scale auxiliary supervision to stabilize training and refine localization, with experiments demonstrating that our model achieves high accuracy in dense particle detection with computational efficiency, effectively addressing the key challenges of overlap and shape variation.\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Data source and Acquisition\u003c/h2\u003e\u003cp\u003eobservation. Data from this dataset was stored in a text file and read to generate high-resolution 64\u0026times;64 grayscale maps with variable numbers of clusters, where the precise locations of these clusters (represented by local maxima) are embedded as high-resolution coordinates in the images.\u003c/p\u003e\u003cp\u003eThe 2D Gaussian clusters have key parameters including a width of 1.0 pixel units in both x and y directions, and 3000 samples per cluster. These parameters result in a resolution of the COG of about 0.02 pixel units. The cluster width and resolution were set to match that of the clusters seen in the SPHENIX TPC.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Preprocessing\u003c/h2\u003e\u003cp\u003eThe raw data is stored in the text file and a piece of python code is used to read this data and transform it into 64*64 pixel images with clusters shown. The images are visualized using grayscale colormaps. Each number is corresponded to one pixel. From zero to bigger numbers, the brightness of pixels increases. Thus, clusters can be clearly shown in the images. With the initial high-resolution coordinates given, we used them to compare with the calculated coordinates of COG points. The coordinate differences (dx, dy) between the initial and calculated centers of gravity were computed. These differences were used to plot a normal distribution, from which the systematic bias and standard deviation were derived. Subsequently, the Peak-CNN model was trained on the dataset to produce cluster images with significantly improved centroid-of-gravity recognition accuracy.\u003c/p\u003e\u003cp\u003ePeakCNN is particularly well-suited for integration into the LHCb particle-identification (PID) ecosystem (LHCb Collaboration, 2008) [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. In this system, charged hadrons (2\u0026ndash;100 GeV/c) are identified by dual Ring-Imaging Cherenkov detectors, muons by dedicated muon chambers, and electrons/photons by calorimeters. The CNN-derived peak coordinates can be seamlessly incorporated into downstream PID pipelines, either through likelihood-based combination of subdetector responses or via advanced neural-network classifiers. This integration results in a fully machine-learning-driven global PID framework. Thus, PeakCNN not only represents a significant advancement in high-density particle imaging but also highlights the growing synergy between deep learning and experimental high-energy physics.\u003c/p\u003e\u003cp\u003eA canonical CNN consists of four key components: convolutional layers that apply learnable kernel filters to extract hierarchical features, pooling layers that reduce spatial resolution while preserving important information, fully connected layers that handle high-level reasoning, and non-linear activation layers (e.g., ReLU, Leaky ReLU, Parametric ReLU) that introduce non-linearity and enable gradient-based optimization [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eBuilding upon this foundation, our Peak-CNNv2 employs a Feature Pyramid Network (FPN) with deformable convolutions to enhance multi-scale feature extraction and geometric adaptability. The model incorporates YOLO-style convolutional blocks and multi-scale auxiliary supervision, enabling precise localization of overlapping particles through heatmap regression. This architecture achieves robust performance in dense particle detection while maintaining computational efficiency.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e3.4 Loss Function and Optimization\u003c/h2\u003e\u003cp\u003eThe model employs a multi-strategy weighted MSE loss for particle detection, with dynamic weight map W adjusted based on ground truth heatmap Y and spatial features. It prioritizes peak regions via \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{W}}_{\\text{base}}\\text{=1+12}\\text{\u0026sdot;}\\text{Y}\\text{(}\\text{i}\\text{,}\\text{j}\\text{)}\\)\u003c/span\u003e\u003c/span\u003e, penalizes false positives \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{(}{\\text{W}}_{\\text{fp}}\\text{=4.0}\\text{}\\)\u003c/span\u003e\u003c/span\u003efor high predictions with low truth) and adaptively boosts challenging targets\u0026mdash;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{1.5\u0026times;}\\)\u003c/span\u003e\u003c/span\u003e for weak signals, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{2.0\u0026times;}\\)\u003c/span\u003e\u003c/span\u003e for single particles, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{2}\\text{.5\u0026times;}\\)\u003c/span\u003e\u003c/span\u003e for overlaps, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{2.0\u0026times;}\\)\u003c/span\u003e\u003c/span\u003e or dense samples (\u0026gt;\u0026thinsp;30 clusters). This focuses learning on critical regions while mitigating noise.\u003c/p\u003e\u003cp\u003eFor multi-scale supervision, the total loss combines the main output loss with auxiliary losses from intermediate feature maps: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{Ltotal}\\text{=}\\text{Lmain}\\text{+0.3\u0026times;(}\\text{Laux}\\text{8}\\text{x}\\text{+}\\text{Laux}\\text{16}\\text{x}\\text{+}\\text{Laux}\\text{32}\\text{x}\\text{)}\\)\u003c/span\u003e\u003c/span\u003e where each auxiliary loss is computed using the same weighted MSE strategy, ensuring consistent optimization across scales.\u003c/p\u003e\u003cp\u003eOptimization employs AdamW \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{(}\\text{lr}\\text{=3\u0026times;1}{\\text{0}}^{\\text{-4}}\\text{,}\\text{weig}\\text{h}\\text{t}\\text{}\\text{decay}\\text{=1\u0026times;1}{\\text{0}}^{\\text{-4}}\\text{)}\\)\u003c/span\u003e\u003c/span\u003e with a Cosine Annealing scheduler \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{(}\\text{Tmax}\\text{=70}\\text{epoc}\\text{h}\\text{s}\\text{,}{\\eta}_{\\text{min}}\\text{=1\u0026times;1}{\\text{0}}^{\\text{-7}}\\text{).\\:}\\)\u003c/span\u003e\u003c/span\u003eTraining strategies include gradient accumulation (2 steps, effective batch size 32), early stopping (15 epochs patience), and checkpointing. HDF5 preprocessing and stratified sampling ensure balanced training across 1\u0026ndash;90 cluster densities, enabling robust convergence despite the model's complexity from deformable convolutions and multi-scale FPN architecture.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e3.5 Evaluation metric\u003c/h2\u003e\u003cp\u003eMultiple metrics have been used to measure the model fit and accuracy of the model incorporated.\u003c/p\u003e\u003cp\u003eMSE -Measure the average pixel squared difference between the predicted heatmap and the true heatmap\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:MSE=\\frac{1}{H\\times\\:W}\\sum\\:_{i=1}^{H}\\sum\\:_{j=1}^{W}({Y}_{i,j}-{\\widehat{Y}}_{i,j}{)}^{2}\\)\u003c/span\u003e\u003c/span\u003e (a)\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{\\widehat{Y}}_{i,j}:\\text{P}\\text{r}\\text{e}\\text{d}\\text{i}\\text{c}\\text{t}\\text{e}\\text{d}\\:\\text{h}\\text{e}\\text{a}\\text{t}\\text{m}\\text{a}\\text{p}\\:\\text{p}\\text{o}\\text{s}\\text{i}\\text{t}\\text{i}\\text{o}\\text{n}\\:\\left(\\text{p}\\text{i}\\text{x}\\text{e}\\text{l}\\:\\text{v}\\text{a}\\text{l}\\text{u}\\text{e}\\:\\text{a}\\text{t}\\:\\right(\\text{i},\\text{j}\\left)\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:{Y}_{i,j}:the\\:pixel\\:value\\:of\\:the\\:true\\:heatmap\\:position\\left(\\right(i,j\\left)\\right)$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:H\\times\\:W=64\\times\\:64:height\\times\\:width$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ePSNR - Image quality assessment metric based on MSE\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:PSNR=20\\times\\:{\\text{log}}_{10}\\left(\\frac{MA{X}_{Y}}{\\sqrt{MSE}}\\right)\\)\u003c/span\u003e\u003c/span\u003e (b)\u003c/p\u003e\u003cp\u003eTrue Positives, TP -The distance between the predicted centroid and the true centroid is less than the set threshold\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:distance(\\widehat{p},p)=\\sqrt{({x}_{\\widehat{p}}-{x}_{p}{)}^{2}+({y}_{\\widehat{p}}-{y}_{p}{)}^{2}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{DISTANCE\\_THRESHOLD=}\\text{0}\\text{.}\\)\u003c/span\u003e\u003c/span\u003e1 (c)\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:FP={N}_{pred}-TP\\)\u003c/span\u003e\u003c/span\u003e (d)\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Bias}_{x}={N}_{1}\\sum\\:_{i=1}^{N}({x}_{{i}_{found}}-{x}_{{i}_{true}})\\)\u003c/span\u003e\u003c/span\u003e (e)\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{Bias}_{y}={N}_{1}\\sum\\:_{i=1}^{N}({y}_{{i}_{found}}-{y}_{{i}_{true}})\\)\u003c/span\u003e\u003c/span\u003e (f)\u003c/p\u003e\u003cp\u003eFalse Negatives, FN - The number of true centroids that are not matched by the predicted centroids (particles missed by the model).\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:FP={N}_{true}-TP\\)\u003c/span\u003e\u003c/span\u003e (g)\u003c/p\u003e\u003cp\u003eFake rate -The proportion of false positives in the total number of detections reflects the \"false alarm\" tendency of the model.\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:Fake\\:Rate=\\frac{FP}{TP+FP}\\:\\left(h\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003ePrecision - Measures the proportion of true positives among the predicted centroids, reflecting the model's ability to control \"false detections\".\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Precision=\\frac{TP}{TP+FP}\\)\u003c/span\u003e\u003c/span\u003e (i)\u003c/p\u003e\u003cp\u003eRecall -Measure the proportion of true centroids that are successfully detected, reflecting the model's ability to control \"missed detections\".\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Recall=\\frac{TP}{TP+FN}\\)\u003c/span\u003e\u003c/span\u003e (j)\u003c/p\u003e\u003cp\u003eMean Coordinate Error -The average Euclidean distance between successfully matched centroid pairs, which measures the positioning accuracy.\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Mean\\:Coordinate\\:Error=\\frac{1}{TP}\\sum\\:_{k=1}^{TP}\\sqrt{({y}_{k}-{\\widehat{y}}_{k}{)}^{2}+({x}_{k}-{\\widehat{x}}_{k}{)}^{2}}\\)\u003c/span\u003e\u003c/span\u003e (k)\u003c/p\u003e\u003cp\u003eEfficiency -The proportion of accurate matches within the error tolerance range, which reflects the high-precision positioning ability.\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Efficiency=\\frac{Accurate\\:Matches}{TP}\\)\u003c/span\u003e\u003c/span\u003e (l)\u003c/p\u003e\u003cp\u003eCount Error - Predict the absolute difference between the number of predicted centroids and the number of true centroids to measure the accuracy of particle counting.\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Count\\:Error=\\left|{N}_{true}-{N}_{pred}\\right|\\)\u003c/span\u003e\u003c/span\u003e (m)\u003c/p\u003e\u003cp\u003eF1 score - the harmonic mean of precision and recall, evaluating a model's performance by balancing false positives and false negatives.\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:F1=2\\times\\:\\frac{Precision\\times\\:Recall}{Precision+Recall}\\)\u003c/span\u003e\u003c/span\u003e (n)\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Experiments","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e4.1 Experiments settings\u003c/h2\u003e\u003cp\u003ePeak-CNNv2 is a deep learning model for particle detection and dense object localization that generates target-position heatmaps from 64\u0026times;64 single-channel images, employing a Feature Pyramid Network (FPN) architecture integrated with deformable convolutions and multi-scale supervision, as well as YOLO-style convolutional blocks (equipped with LeakyReLU activations and batch normalization) to enhance feature representation. Specifically, deformable convolutions adapt to target deformations via predicted (x,y) offsets, strengthening the capture of complex features from irregular particle shapes; the FPN structure consists of a bottom-up pathway for feature extraction and a top-down pathway with lateral connections for multi-scale feature fusion, generating feature maps at multiple scales (4\u0026times;4, 8\u0026times;8, 16\u0026times;16, 32\u0026times;32, and 64\u0026times;64) to effectively handle varying particle sizes and densities. Additionally, the network incorporates auxiliary supervision at intermediate scales (8\u0026times;, 16\u0026times;, and 32\u0026times; downsampling) to stabilize training and improve localization precision, while its final output layer produces 64\u0026times;64 heatmaps through transposed convolution and sigmoid activation. With strict size constraints, a multi-scale weighted MSE loss, and balanced sampling, Peak-CNNv2 can accurately localize dense, low-intensity, or irregular targets\u0026mdash;making it suitable for particle analysis and cell detection\u0026mdash;and the overall FPN architecture of Peak-CNNv2 (integrating deformable convolutions and multi-scale supervision) is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003e4.2 Hyperparameter Optimization\u003c/h2\u003e\u003cp\u003eAfter establishing a robust model architecture, we further refined its hyperparameters. This process involved selecting the top 10 candidate architectures from a model architecture search, wherein hyperparameter sets were randomly sampled from a broad search space and evaluated to identify the optimal configuration. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, a weight decay of 1e\u0026thinsp;\u0026minus;\u0026thinsp;5 was selected for its ability to minimize validation loss. Similarly, a Gaussian sigma value of 0.3 was chosen based on Fig.\u0026nbsp;3, as it yielded the most stable training process and the lowest loss. A batch size of 8, presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003e, was adopted to introduce beneficial gradient noise, thereby promoting more fine-grained feature learning. Finally, as indicated in Fig.\u0026nbsp;5, a learning rate of 1e\u0026thinsp;\u0026minus;\u0026thinsp;4 was identified as optimal, achieving both the lowest loss and robust convergence.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003e4.3 In-depth Analysis of Hyperparameters\u003c/h2\u003e\u003cp\u003eA deeper understanding of the impact of hyperparameters on model performance was pursued through a series of manual experiments. These experiments were conducted using the 10 optimal models identified during the model architecture search phase. In each experiment, only one hyperparameter was adjusted while keeping others fixed, with the best validation loss serving as the performance evaluation metric. Among the tested hyperparameters, batch size, learning rate, weight decay, and Gaussian sigma emerged as key influential factors: As quantitatively evidenced in the Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003e, a smaller batch size (e.g 8) enhanced the model's ability to learn fine-grained features by introducing moderate gradient noise, which prevents convergence to overly smooth local optima\u0026mdash;this configuration is particularly compatible with the model's architectural demands for capturing subtle patterns. In contrast, larger batch sizes (32 and above) reduced the model's sensitivity to feature variations due to excessively stable gradient estimates, leading to increased validation loss. For learning rate, the analysis in the Fig.\u0026nbsp;5 reveals that a value of 1e-4 not only achieved the lowest validation loss but also ensured robust convergence, as an excessively small learning rate (e.g 1e-5) caused slow convergence with suboptimal loss, while an intermediate value (e.g. 5e-5) failed to attain the same level of performance as 1e-4. Regarding weight decay, the Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e plot shows that a small value (1e-5) minimized validation loss by effectively regularizing the model without undercutting feature learning capacity; in contrast, larger weight decay values (e.g., 1e-4,5e-4) increased loss due to over-regularization. For Gaussian sigma, the Fig.\u0026nbsp;3 plot indicates that a value of 0.3 delivered both the most stable training process and the lowest loss, whereas larger values (e.g 0.5 0.7) either elevated loss or compromised training stability. The optimal values of these parameters (batch size\u0026thinsp;=\u0026thinsp;8, learning rate\u0026thinsp;=\u0026thinsp;1e-4, weight decay\u0026thinsp;=\u0026thinsp;1e-5, Gaussian sigma\u0026thinsp;=\u0026thinsp;0.3) are highly aligned with the model's architectural characteristics, as they synergistically leverage the model's capacity for adaptive feature extraction while ensuring balanced regularization and convergence, thereby achieving the minimum validation loss across experiments.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\u003ch2\u003e4.4 Ablation Study\u003c/h2\u003e\u003cp\u003eTo rigorously evaluate the contribution of each key component in the proposed PeakCNNv2 architecture, a comprehensive ablation study was conducted. The performance of the full model was compared against several ablated variants on the test dataset. The specific modifications applied to each variant, along with their corresponding performance differences, are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Results from this ablation study confirm the necessity and effectiveness of each core component within the architectural design.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eModel Modification Comparison Table\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAblation\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eVariant\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eModification\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eBaseline\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eBase Model\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eNone(baseline)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eDeformable Conv\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNo Deformable Conv\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eRemove Deformable Conv\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePyramid Structure\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNo Pyramid\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eRemove Pyramid Structure\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eActivation\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eReLU Instead of Leaky\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eReplace Leaky ReLU with ReLU\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eAuxiliary Loss\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNo Auxiliary Loss\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eRemove Auxiliary Loss\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eThis study report the training loss, validation loss, and validation MSE of models with different architectural variants, all using the same loss function (weighted MSE loss with multi-scale supervision for applicable models) and evaluation metric (MSE) to ensure fair comparison: the Base Model, equipped with deformable convolutions, a pyramid feature fusion structure, LeakyReLU activation, and auxiliary loss, performs remarkably in training efficiency and final performance \u0026mdash; its training loss drops rapidly in early epochs and stabilizes at an extremely low value, the validation loss follows a steep trajectory to plummet and converge to the minimum, and the validation MSE also reaches the lowest among all variants, demonstrating strong precise peak prediction capability; in contrast, the \"No Deformable Conv\" variant (replacing deformable convolutions with standard ones) exhibits significantly slower reduction in training and validation losses, with higher final loss values and elevated validation MSE due to the lack of adaptive spatial sampling ability of standard convolutions, the \"No Pyramid\" variant (abandoning pyramid fusion and relying only on simple upsampling) shows even slower loss decay, with training loss decreasing sluggishly, validation loss plateauing at a relatively high level, and validation MSE failing to reach the low range of the Base Model (reflecting the importance of the pyramid structure in fusing multi-scale features), the \"ReLU Instead of Leaky\" variant (swapping LeakyReLU with ReLU) suffers from slower training convergence, higher final validation loss, and increased validation MSE as ReLU\u0026rsquo;s hard zeroing of negative values causes premature gradient vanishing while LeakyReLU preserves gradient flow for negative inputs, and the \"No Auxiliary Loss\" variant (omitting auxiliary supervision) has a deceptively low final training loss (even lower than the Base Model) but much worse validation loss and MSE since auxiliary loss regularizes training via multi-scale intermediate supervision, prevents overfitting, and enhances generalization; in summary, each architectural component of the Base Model contributes synergistically to efficient training and strong peak prediction, and removing any component leads to measurable degradation in convergence speed, final loss, or generalization, as visually summarized in the loss and MSE convergence curves of Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAdditionally, this ablation study systematically evaluates the impact of various architectural components on model complexity, measured in terms of both parameter count (in millions, \u0026ldquo;M\u0026rdquo;) and computational demand in FLOPs (billions, \u0026ldquo;B\u0026rdquo;). As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Fig.\u0026nbsp;8, which illustrate model parameters and FLOPs respectively, the Base model serves as the reference point with 6.266M parameters and 0.3370B FLOPs. Removing the deformable convolution module reduces parameters slightly to 6.034M, as depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e7\u003c/span\u003e, but increases FLOPs to 0.5348B, which is shown in Fig.\u0026nbsp;8. This indicates that deformable convolutions introduce a substantial computational footprint despite a minor drop in parameter count. Omitting the pyramid structure results in the largest reduction across both metrics, bringing parameters down to 5.131M and FLOPs to 0.1200B, as visible in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Fig.\u0026nbsp;8 respectively. This highlights the dominant role of the pyramid structure in driving model complexity. Substituting LeakyReLU with ReLU keeps both parameters and FLOPs identical to the Base model, at 6.266M and 0.3370B, which suggests that LeakyReLU does not incur extra computational overhead relative to ReLU in this architecture. Finally, excluding the auxiliary loss leaves both parameters and FLOPs unchanged from the Base model, implying that auxiliary loss has a negligible effect on overall computational requirements and parameter count.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003e4.5 Comparison with State-of-the-Art (SOTA) Methods\u003c/h2\u003e\u003cp\u003eTo thoroughly evaluate the performance of PeakCNNv2, a comparative analysis was conducted against four representative models under identical training data, preprocessing, and evaluation protocols: CenterNet, recognized for its efficient anchor-free structure; PFN, a network specifically designed for particle detection; U-Net, a key architecture widely used in biomedical image segmentation; and LightweightYOLO, a newly introduced lightweight detector emphasizing inference efficiency. This comparison covered a range of cluster densities from 1 to 80 clusters to rigorously assess how well the models handle both isolated clusters and heavy overlap.\u003c/p\u003e\u003cp\u003eAs detailed in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e9\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e10\u003c/span\u003e, which respectively illustrate the metrics comparison at a 0.1 threshold and the precision, recall, and F1 score across models, the results conclusively demonstrate that PeakCNNv2 achieves state-of-the-art accuracy. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e10\u003c/span\u003e shows that PeakCNNv2 outperforms or matches the detection performance, particularly in F1-score, of PFN, U-Net, and LightweightYOLO across all densities, with notable advantages in precision\u0026mdash;maintaining the highest precision values among all models. Meanwhile, Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e9\u003c/span\u003e indicates that PeakCNNv2 delivers superior localization precision compared to CenterNet, reflected in a lower false rate and strong efficiency at the 0.1 threshold, and achieves accuracy levels far exceeding those of LightweightYOLO, which prioritizes inference speed at the cost of precision in dense scenarios. This performance validates our architectural design, as the synergistic integration of deformable convolutions and attention mechanisms effectively addresses the core challenges of overlap and shape variation where traditional methods often struggle.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eRegarding computational costs, as summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e which compares model performance in terms of parameters and inference efficiency, PeakCNNv2 exhibits a favorable trade-off. Although it incorporates more parameters, at 6.034\u0026nbsp;million, than CenterNet, PFN, and LightweightYOLO, its computational demand measured in FLOPs\u0026mdash;533.7 million\u0026mdash;is substantially lower than that of PFN with 973.7 MF and U-Net with 1720.6 MF, indicating a more efficient use of model capacity. Furthermore, while the inference time of PeakCNNv2 is slightly longer than that of CenterNet, at 1.514 milliseconds compared to 1.449 ms, and markedly higher than LightweightYOLO's 0.736 ms, this modest increase is justified by significant gains in accuracy. This is especially evident in dense and overlapping scenarios, where PeakCNNv2 maintains robust performance with far less degradation than other models.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eModel Performance Comparison (Parameters and Inference Efficiency)\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"6\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eParameters (M)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eFLOPs (MF)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eInference Time (ms)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eThroughput (samples/sec)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eTotal Evaluation Time (sec)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCenterNet\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.917\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e146.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e1.449\u0026thinsp;\u0026plusmn;\u0026thinsp;0.013\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e690.3\u0026thinsp;\u0026plusmn;\u0026thinsp;6.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e119.68\u0026thinsp;\u0026plusmn;\u0026thinsp;0.30\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePFN\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e1.506\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e973.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.743\u0026thinsp;\u0026plusmn;\u0026thinsp;0.004\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e1345.6\u0026thinsp;\u0026plusmn;\u0026thinsp;7.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e118.89\u0026thinsp;\u0026plusmn;\u0026thinsp;0.70\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eUNet\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e3.704\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e1720.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e1.265\u0026thinsp;\u0026plusmn;\u0026thinsp;0.013\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e790.6\u0026thinsp;\u0026plusmn;\u0026thinsp;8.1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e119.30\u0026thinsp;\u0026plusmn;\u0026thinsp;0.38\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePeakCNNv2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e6.034\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e533.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e1.514\u0026thinsp;\u0026plusmn;\u0026thinsp;0.036\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e660.8\u0026thinsp;\u0026plusmn;\u0026thinsp;15.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e121.11\u0026thinsp;\u0026plusmn;\u0026thinsp;0.30\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLightweightYOLO\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.119\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e29.6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.736\u0026thinsp;\u0026plusmn;\u0026thinsp;0.002\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e1357.9\u0026thinsp;\u0026plusmn;\u0026thinsp;4.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e122.01\u0026thinsp;\u0026plusmn;\u0026thinsp;0.62\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eIn conclusion, PeakCNNv2 establishes a new state-of-the-art for dense particle detection. It delivers top-tier accuracy through advanced architectural features, achieving a competitive balance between precision and computational efficiency. Although LightweightYOLO excels in pure inference speed, PeakCNNv2 is better suited for applications that prioritize precision, such as offline data analysis and high-precision measurements, where its substantial accuracy advantages outweigh the moderate increase in inference time.\u003c/p\u003e\u003cdiv id=\"Sec14\" class=\"Section3\"\u003e\u003ch2\u003e4.5.1 Statistical Significance Testing\u003c/h2\u003e\u003cp\u003eThe performance and statistical significance of the PeakCNNv2 model for peak detection were evaluated through a comparison with four competing architectures\u0026mdash;CenterNet, PFN, LightweightYOLO, and UNet. Robustness was further assessed using bootstrap analysis with 100 random seeds. PeakCNNv2 integrated deformable convolutions (for adaptive spatial feature capture) and a Feature Pyramid Network (FPN) for multi-scale fusion, while other models followed canonical designs; peak extraction from heatmaps relied on thresholding with non-maximum suppression, and key metrics (Recall, F1 score) were used for quantification.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs shown in the bootstrap analysis plots for Recall in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e11\u003c/span\u003e and for F1 score in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e12\u003c/span\u003e, PeakCNNv2 exhibited the highest mean Recall (~\u0026thinsp;0.920) and F1 score (~\u0026thinsp;0.927) with narrow 95% confidence intervals (indicating robust initialization stability), whereas other models (e.g., CenterNet, PFN) had lower mean values and wider intervals. Statistical significance via independent-samples t-tests showed: PeakCNNv2 significantly outperformed LightweightYOLO in F1 score and precision (p\u0026thinsp;=\u0026thinsp;0.0219, *), with no significant recall difference (p\u0026thinsp;=\u0026thinsp;0.1136); against UNet, all metrics were significant\u0026mdash;F1 score and precision (p\u0026thinsp;=\u0026thinsp;0.0022, **), recall (p\u0026thinsp;=\u0026thinsp;0.0267, *)\u0026mdash;with PeakCNNv2 achieving\u0026thinsp;~\u0026thinsp;3.21% higher F1, ~\u0026thinsp;3.18% higher precision, and ~\u0026thinsp;3.23% higher recall. Collectively, PeakCNNv2 demonstrates statistically significant and superior peak detection performance versus key competitors, validating its effectiveness and robustness.\u003c/p\u003e\u003cp\u003e\u003cb\u003eTable\u0026nbsp;3. Significance Test Results of Detection Performance Metrics Between Models\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e\u003ccolgroup cols=\"5\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePair\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMetric\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003ep-value\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eSig.\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003e% Difference (Neg\u0026thinsp;=\u0026thinsp;Model1\u0026thinsp;\u0026lt;\u0026thinsp;Model2; Pos\u0026thinsp;=\u0026thinsp;Model1\u0026thinsp;\u0026gt;\u0026thinsp;Modl2)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLightweightYOLO vs PeakCNNv2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003ef1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.021917\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e*\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-2.19\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eprecision\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.021917\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e*\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-2.16\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003erecall\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.113636\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003ens\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e-2.19 (No significant difference)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003ePeakCNNv2 vs UNet\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003ef1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.002255\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e**\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.21\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eprecision\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.002255\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e**\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.18\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003erecall\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0.026729\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e*\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.23\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003ctfoot\u003e\u003ctr\u003e\u003ctd colspan=\"5\"\u003e\u003cb\u003eNote: *p\u0026thinsp;\u0026lt;\u0026thinsp;0.05, **p\u0026thinsp;\u0026lt;\u0026thinsp;0.01, ***p\u0026thinsp;\u0026lt;\u0026thinsp;0.001, ns\u0026thinsp;=\u0026thinsp;p\u0026thinsp;\u0026ge;\u0026thinsp;0.05 (No significant difference)\u003c/b\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tfoot\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e"},{"header":"5. Experimental Results","content":"\u003cp\u003eA comprehensive evaluation of model performance was carried out on the test dataset, incorporating batch prediction, quantitative metric calculation, and grouped visualization analysis. The comparative analysis of heatmaps and error metrics across different cases, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e13\u003c/span\u003e, offers intuitive visual evidence of the model\u0026rsquo;s performance under varied conditions.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eKey metrics across all samples are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Further analysis categorized performance into best, average, and worst-performing samples by sorting results according to F1-score. Directional errors\u0026mdash;dx for the x-direction and dy for the y-direction\u0026mdash;were also computed to evaluate localization accuracy along each axis. The distributions of these errors, visualized in the histograms in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e14\u003c/span\u003e and Fig.\u0026nbsp;15, show concentrations near zero with minimal bias, indicating unbiased estimation capabilities and providing insight into the error patterns of the model.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe evaluation framework also incorporates grouped statistics for the best, average, and worst-performing samples, comparing metrics such as average precision, recall, F1-score, MSE, and count error across groups. Representative samples from each group are visualized with their original images, true heatmaps, predicted heatmaps, and key metrics, as exemplified in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e13\u003c/span\u003e, allowing observation of how the model handles different peak distribution scenarios.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eAverage of Particle Detection Metrics on 80,000 Samples\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\"\u0026plusmn;\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCluster Count\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMSE\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003ePrecision\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eRecall\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eF1 Score\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eEfficiency\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003eFake Rate\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000009\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.995576\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000443\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.980540\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000997\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.987624\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000570\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.980540\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000997\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.004424\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000443\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000022\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.988253 \u0026plusmn;\u003c/p\u003e\u003cp\u003e0.000749\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.959160\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001103\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.973111\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000895\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.959160\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001103\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.011747\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000749\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000041\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000000\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.980657\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000601\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.935267\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000893\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.957051\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000669\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.935267\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000893\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.019343\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000601\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000070\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000001\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.976939\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000550\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.923709\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000774\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.949214\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000608\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.923709\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000774\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.023061\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000550\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000068\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000001\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.972774\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000906\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.914150\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001131\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.942201\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000873\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.914150\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001131\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.027226\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000906\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000110\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000001\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.970057\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000559\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.901889\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000965\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.934364\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000652\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.901889\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000965\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.029943\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000559\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e70\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.000131\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000002\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.967474\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001060\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.894140\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000801\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.928997\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000743\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.894140\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000801\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.032526\u0026thinsp;\u0026plusmn;\u0026thinsp;0.001060\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e80\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c2\"\u003e\u003cp\u003e0.004339\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000015\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c3\"\u003e\u003cp\u003e0.944050\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000549\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c4\"\u003e\u003cp\u003e0.902234\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000684\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c5\"\u003e\u003cp\u003e0.922461\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000617\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c6\"\u003e\u003cp\u003e0.904031\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000636\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\"\u0026plusmn;\" colname=\"c7\"\u003e\u003cp\u003e0.057719\u0026thinsp;\u0026plusmn;\u0026thinsp;0.000636\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eTo further validate the robustness of PeakCNNv2 in real-world complex scenarios, experiments were conducted on the EMPS dataset\u0026mdash;a collection of 465 electron microscopy images with pixel-level semantic instance segmentations stored as 32-bit integer maps [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. This dataset contains images of various cells and crystals, making it well-suited for testing the model's performance in medical and related fields, as it can simulate the practical demands of cell localization in clinical diagnostics and crystal particle analysis in material science. Beyond this, the dataset presents significant challenges for peak detection due to strong noise, heavy object overlaps, and highly irregular instance shapes. As demonstrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e16\u003c/span\u003e, which showcases detection results on this dataset, PeakCNNv2 exhibits strong adaptability in such demanding conditions. The model's deformable convolutions enable adaptive feature extraction for irregularly shaped targets (e.g., irregular cell edges or non-uniform crystal morphologies), while its Feature Pyramid Network facilitates multi-scale feature fusion to address overlaps and noise. Additionally, the sub-pixel fitting strategy achieves fine-grained centroid localization. Both visual assessments in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e16\u003c/span\u003e and quantitative evaluations confirm that PeakCNNv2 effectively filters noise-induced false positives, disentangles heavily overlapping instances (such as clustered cells or aggregated crystal particles), and accurately locates centroids even for highly irregular objects, thereby validating its reliability in complex real-world contexts\u0026mdash;including medical and material science scenarios targeted by the EMPS dataset.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eThis study tackles the persistent challenge of accurately detecting and localizing overlapping particle clusters in high-density 2D images\u0026mdash;a critical task in high-energy physics, medical imaging, and astrophysics. Traditional methods like K-means and DBSCAN often fail under high-density conditions, resulting in false detections and centroid errors. To address this, we proposed Peak-CNNv2, an end-to-end convolutional neural network that incorporates deformable convolutions, YOLO-style convolutional blocks, and a multi-scale Feature Pyramid Network (FPN) architecture with auxiliary supervision to adaptively capture irregular cluster shapes and suppress background noise, enabling precise sub-pixel localization of cluster peaks.\u003c/p\u003e\u003cp\u003eExperimental results demonstrate that Peak-CNNv2 achieves state-of-the-art performance, with an F1-score exceeding 93% across 1\u0026ndash;60 clusters and significant reductions in centroid localization error compared to models such as CenterNet, U-Net, and PFN. These improvements validate the effectiveness of our architectural innovations\u0026mdash;particularly the deformable convolution modules that enhance geometric transformation modeling, the YOLO-style blocks that provide efficient feature extraction with LeakyReLU activations and batch normalization, and the multi-scale FPN structure that improves feature representation across varying particle sizes and densities\u0026mdash;in handling dense and overlapping particles. The model holds practical value for integration into experimental pipelines such as the LHCb particle identification system, offering a scalable, deep learning\u0026ndash;based solution for precision-sensitive applications.\u003c/p\u003e\u003cp\u003eDespite its high accuracy, the model exhibits several limitations: First, the integration of deformable convolutions, YOLO-style blocks, and multi-scale FPN architecture increases computational complexity, resulting in higher GPU memory usage and longer inference times compared to lightweight models. Second, performance remains limited under extreme crowding or real experimental noise, occasionally leading to cluster merging. Third, as observed in the corresponding Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e16\u003c/span\u003e, there is a failure to detect edge clusters\u0026mdash;this is primarily attributed to the lack of edge cluster samples in the training dataset, which prevented the model from learning the feature patterns of clusters in edge regions through its deformable convolution and multi-scale fusion mechanisms. Future work will focus on improving computational efficiency through model compression and optimization of the deformable convolution operations, incorporating real experimental data (including edge cluster samples) to enhance generalization, and extending the framework to 3D detection for broader scientific utility.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cb\u003eFunding Declaration\u003c/b\u003e\u003c/p\u003e\u003cp\u003eThe authors declare that this research received no specific financial support from any finding agency in the public, commercial, or not-for-profit sectors. None of the authors have received grants or financial contributions from organizations that could influence the design, execution, analysis, or interpretation of this study.\u003c/p\u003e\u003cp\u003e\u003ch2\u003eCompeting interests\u003c/h2\u003e\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAuthor Contributions StatementH.W. designed and debugged the model, conducted model training and testing, and solely performed the experiments and analysis for SOTA comparison, statistical significance testing, and ablation studies. H.W. also created all figures and tables and participated in the writing and final revision of the entire manuscript, specifically for sections 3.4 ,4.2, 4.3, 4.4, 4.5, 4.5.1, 5, and 6. Y.L. contributed to literature collection, paper outline development, and wrote sections 2 (Literature Review) and 3.5 (Evaluation Metric). Y.L. also reviewed the entire manuscript. Y.P. contributed to literature collection, paper outline development, and wrote sections 1 (Introduction) and 2 (Literature Review). Y.S. contributed to literature collection, paper outline development, and wrote section 1 (Introduction). L.H. participated in model design and partial debugging, and wrote sections 3.1 (Data source and Acquisition), 3.2 (Preprocessing), and 3.4 (Loss Function and Optimization).All authors reviewed and approved the final manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe electron microscopy image data used in this study are from the EMPS dataset, which was originally published by B. Yildirim and J. M. Cole in \"Bayesian Particle Instance Segmentation for Electron Microscopy Image Quantification\", Journal of Chemical Information and Modeling, 2021 (https://doi.org/10.1021/acs.jcim.0c01455). The dataset used for training and evaluation in our work is available from the corresponding author of the original publication upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eFr\u0026ouml;hlich, B. et al. Track reconstruction in high-density environments. \u003cem\u003eJ. Instrum.\u003c/em\u003e \u003cb\u003e8\u003c/b\u003e (05), P05013 (2013).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLeCun, Y., Bengio, Y. \u0026amp; Hinton, G. Deep learning, Nature, vol. 521, no. 7553, pp. 436\u0026ndash;444, (2015).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKrizhevsky, A., Sutskever, I. \u0026amp; Hinton, G. E. ImageNet classification with deep convolutional neural networks, in Advances in Neural Information Processing Systems, vol. 25, pp. 1097\u0026ndash;1105, (2012).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLHCb \u0026amp; Collaboration The LHCb detector at the LHC. \u003cem\u003eJ. Instrum.\u003c/em\u003e \u003cb\u003e3\u003c/b\u003e (08), S08005 (2008).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLeCun, Y., Bottou, L., Bengio, Y. \u0026amp; Haffner, P. Gradient-based learning applied to document recognition, Proceedings of the IEEE, vol. 86, no. 11, pp. 2278\u0026ndash;2324, (1998).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eRedmon, J., Divvala, S., Girshick, R. \u0026amp; Farhadi, A. You only look once: Unified, real-time object detection, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 779\u0026ndash;788, (2016).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLin, T. Y. et al. Feature pyramid networks for object detection, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2117\u0026ndash;2125, (2017).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eYildirim, B. \u0026amp; Cole, J. M. Bayesian particle instance segmentation for electron microscopy image quantification. \u003cem\u003eJ. Chem. Inf. Model.\u003c/em\u003e \u003cb\u003e61\u003c/b\u003e (4), 1810\u0026ndash;1818 (2021).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBochkovskiy, A., Wang, C. Y. \u0026amp; Liao, H. Y. M. YOLOv4: Optimal Speed and Accuracy of Object Detection, arXiv preprint arXiv:2004.10934, (2020).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWu, Y. \u0026amp; Li, J. YOLOv4 with Deformable-Embedding-Transformer Feature Extractor for Exact Object Detection in Aerial Imagery, Sensors, \u003cb\u003e23\u003c/b\u003e, 5, p. 2522, (2023).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eDai, J. et al. Deformable Convolutional Networks, in Proceedings of the IEEE International Conference on Computer Vision, pp. 764\u0026ndash;773, (2017).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eXiong, Y. et al. Efficient Deformable ConvNets: Rethinking Dynamic and Sparse Operator for Vision Applications, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5652\u0026ndash;5661, (2024).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eWu, H., Liu, Y., He, L., Shi, Y. \u0026amp; Pan, Y. Peak-CNNv2, GitHub, [Online]. Available: haoyuwu2006615/Peakcnn-V2: Official PyTorch implementation of PeakCNNv2 \u0026mdash; a deep learning model with deformable convolutions and Feature Pyramid Network (FPN) for high-accuracy, sub-pixel particle detection in dense and overlapping environments. (2025).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Particle cluster detection, Convolutional neural network, Centroid localization, Deformable Convolution, YOLO, Feature Pyramid Network (FPN)","lastPublishedDoi":"10.21203/rs.3.rs-7848113/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7848113/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate detection of particle clusters in high-density 2D images is crucial for high-energy physics, medical imaging, and astrophysics. Traditional clustering methods, such as K-means and DBSCAN, often fail with overlapping or closely spaced particles, causing false detections and errors in centroid-of-gravity (COG) measurements. To address these challenges, we propose Peak-CNNv2, a lightweight convolutional neural network that directly predicts cluster peak positions for precise localization. The network incorporates YOLO-style convolutional blocks\u0026mdash;drawing on YOLO\u0026rsquo;s (You Only Look Once) design principle of efficient local feature extraction to enhance responsiveness to dense particle distributions\u0026mdash;alongside deformable convolutions and a multi-scale Feature Pyramid Network (FPN) to improve handling of overlapping and irregular clusters. Compared to heavier models, Peak-CNNv2 achieves high accuracy with manageable computational cost. Experimental results demonstrate that Peak-CNNv2 achieves an F1-score exceeding 93% for 1\u0026ndash;60 clusters and robust performance under overlapping conditions. It also reduces centroid errors and minimizes false positives. By integrating deep learning with multi-scale feature fusion and YOLO-inspired efficient convolution design, Peak-CNNv2 provides a scalable and accurate solution for detecting particles in complex and noisy data. This approach advances image analysis in physics, medical imaging, and astrophysics, enabling reliable processing of dense datasets.\u003c/p\u003e","manuscriptTitle":"Peak-CNNv2: A Deformable FPN Architecture with YOLO-style Modules for Accurate Particle Localization","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-18 08:38:52","doi":"10.21203/rs.3.rs-7848113/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"886b6000-2206-4e1e-b0c9-9d729efd9a8a","owner":[],"postedDate":"November 18th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":57684644,"name":"Physical sciences/Engineering"},{"id":57684645,"name":"Physical sciences/Mathematics and computing"},{"id":57684646,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2025-11-24T17:53:25+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-18 08:38:52","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7848113","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7848113","identity":"rs-7848113","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}