A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images

A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images | 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 Research Article A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images Dan Wang, Xiu-bin Zhu, Witold Pedrycz, Zhenhua Yu, Zhiwu Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1219938/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Aug, 2023 Read the published version in Soft Computing → Version 1 posted 4 You are reading this latest preprint version Abstract Image compression techniques realized in various ways have become an indispensable part in the practical storage and transmission of digital images. In this study, we present a novel method of lossy compression based on sampling and fuzzy encoding for grayscale images and discuss the problem of their reconstruction. First, an image is divided into a number of non-overlapping blocks of pixels. Next, we perform multiple rounds of random sampling. In each round, a number of pixels are selected as prototypes for the representing the corresponding block. Each pixel in the block is reconstructed based on the gray-levels of the prototypes and membership degrees computed with respect to the distances of each pixel to the prototypes. The reconstruction abilities delivered by the prototypes are quantified by a certain objective fidelity criterion and the prototypes leading to lowest reconstruction error are determined as representatives of current block. Finally, once the representatives in each block have been determined, we reconstruct the whole image based on these prototypes. Experimental studies as well as visual evaluations show that the proposed algorithm is able to achieve high compression ratios while preserving the overall fidelity in the decompressed images. Image compression grayscale image fuzzy encoding coordinated-based Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 I. Introduction Digital images play an import role in multimedia and information intercommunication by carrying a wealth of information. Images may represent objects in the form of picture-like bitmaps, vector-based drawings or 3D rendering. With the rapid development of information technology, a large amount of digital image data is generated at high pace. Although in recent years we have witnessed the increase of bandwidth of the communication networks and the decrease of the price of storage devices, digital images produced by various technologies and applications have established higher demands with regard to the bandwidth and storage capacity. In order to overcome the existing limitations on the bandwidth of communication networks and storage capacity, the problem of digital image compression has been extensively researched. We have been witnessing a highly diversified plethora of approaches to image compressing. Image compression is a data compression technology which aims at reducing the cost of transmission and storage. Generally, there are two categories of compression algorithms: lossy compression and lossless compression [ 5 ]. Compared with lossless compression, lossy compression may lead to loss of fidelity, but it is especially useful when being used at low bit rates while loss of fidelity is acceptable in order to fulfill the bandwidth of transmission. Lossy compression algorithms usually tend to achieve a trade-off between the bit rate of the transmission system and the quality of the images being processed. Some lossy compression techniques may lead to an extremely high compression ratio. Existing image compression techniques usually make use of visual perception or exploit statistical properties of image data to reach the aim of compressing data. The motivation behind the proposed algorithm presented in this study is to use a certain number of neighboring prototypes to represent one pixel using fuzzy set-based encoding, given the fact that significant correlations or redundancy usually exist among consecutive or neighboring pixels. We present a novel image compression algorithm which makes use of the sampled pixels (prototypes) and membership degrees of each pixel to a group of prototypes with respect to the gray levels to reconstruct image. The membership values used for image reconstruction are implied by the distance of the pixels to be reconstructed to the selected prototypes. Given a digital image of fixed size, it could be represented by a two-dimensional plane where x and y are spatial coordinates, and any pair of coordinates ( x , y ) represents a specific pixel. When fuzzy encoding / decoding is involved, the membership degrees of the gray level of this pixel to the brightness of the prototypes become necessary. But since each pixel has its unique coordinate, the distances between different pixels are determined and could be calculated on the fly. Once a group of prototypes has been determined and membership degrees computed with respect to coordinates could approximate the membership grades between values of gray levels, the requirement for storage space could be reduced significantly, as we only need to store the index (coordinates) and values of gray-levels of the prototypes after compression is completed. Given the coordinates of the prototypes and their gray levels, it is enough to reconstruct the whole image, which means that we are able to achieve a very high compression ratio. It is worth noting that in current image compression techniques (no matter which compression algorithm has been adopted), the inner relationship between the gray levels of pixels and their coordinates has not been fully exploited. The method proposed in this study has not been explored in the existing literatures. The developed approach exhibits a tangible level of originality in the sense that a novel method which fully utilize the information implicit in the coordinates of each pixel to reconstruct the image. An overview of the overall framework highlighting the essence of the method is visualized in Fig. 1 . The completion of the compression process gives rise to a collection of prototypes whose coordinates and gray levels are stored for further decompression. In the decompression process, the reconstruction of the original image is based on these prototypes and the membership degrees of each pixel to these prototypes implied in their coordinates. In this study, we are interested in the compression of grayscale images. Grayscale images are usually stored in the form of two-dimensional arrays of gray-level values. The only colors of a grayscale image are the shades of gray and each pixel in the image is a scalar proportional to the intensity of gray level. This study is organized as follows. In Section II, we offer a brief introduction to the existing image compression algorithms with some focus on the techniques involving fuzzy sets. In Section III, we elaborate on the detailed image compression algorithm. Performance indexes used to evaluate the quality of reconstructed images are discussed in Section IV. A series of experiments and ensuing parametric analysis are reported in Section V while conclusions are offered in Section VI. Ii. Compression Of Images Image compression technologies have been utilized in many diverse applications, such as remote sensing, medical imaging, digital television, and video surveillance systems. The key to image compression technology is to remove redundant data from the original image, and later the compressed image is decompressed to exactly rebuild the original image or to reconstruct an approximation one. There are several different types of data redundancies, for example, spatial redundancy, temporal redundancy, coding redundancy, psychovisual redundancy and knowledge redundancy. General compression systems try to remove those redundancies to meet the objective of compressing images. Generally, image compression methods could be divided into two categories: lossless methods and lossy methods, depending on whether the compressed data supports error-free data reconstruction. A general compression system is usually constructed in the encoder-decoder architecture. The encoder removes redundancy information from the original image and enhances the immunity to noise while the decoder recovers/reconstructs the original image as faithfully as possible. Lossless compression is a necessary choice when degradation of original image is not acceptable. For example, in medical fields, the details of medical images are very important to diagnose illness and choose a right therapeutic scheme [ 1 ]. Other areas in which lossless compression are extensively adopted includes compression of thermal images taken by satellites [ 2 ], remote sensing images [ 3 ], and fingerprints images [ 4 ] etc. Many powerful lossless image compression algorithms have been developed. It is needless to say that there is no ideal lossless compression tool, which could efficiently compress all possible images [ 5 ] [ 6 ] [ 30 ][ 33 ]. For this reason, many different compression algorithms have been designed with specific assumptions about what kinds of redundancy the original images are likely to contain. Some of the most commonly used lossless compression algorithms include Huffman coding [ 7 ], arithmetic coding [ 8 ] and Run Length Coding [ 9 ]. In recent years, we also have witnessed a lot of such studies reported in the literature [ 11 ] [ 12 ] [ 13 ][ 27 ]. The main drawback of lossless image compression algorithms is that they cannot guarantee the compressed image being smaller in size than the original one. Compared with lossless compression algorithms, lossy image compression algorithms can reach higher compression rate at the expense of losing some information of the original images. While the degradation of the uncompressed image could be quantified through certain performance indexes, such as Mean Square Error (MSE) or Peak-Signal-To-Noise ratio (PSNR), the efficiency of the compression algorithm could also be assessed in terms of data compression ability and implementation complexity of the algorithm. Lossy compression is widely used, albeit imperfect, especially when the main goal is to reach a given compression rate to meet the bandwidth requirement or to overcome the restrictions on storage or to reduce encoding-decoding time. There are a number of approaches to lossy image compression, such as vector quantisation, predictive coding, and transform coding [ 5 ]. These algorithms are carried in the following aspects to achieve the goal of compressing image: i) quantification of the color space to reduce dimensionality of the problem, ii) implementing lower resolution for chroma information than for luma information when coding images, iii) fractal compression and iv) transform coding, which is also the most commonly used method. Fuzzy sets and Fuzzy relational equations also have been used in image compression/decompression [ 21 ] [ 22 ], which exploit a new field for the research of image processing. In these studies, images are represented as fuzzy relations since fuzzy sets are able to capture the essence of the problem from information granule point of view and come with a sound algorithmic framework. In the area of lossy image compression, we can refer to some recent studies presented in the literature [ 14 – 16 ] [ 25 ][ 29 ] [ 32 ]. In recent year, there also has been an interest in compressing images using neural networks, refer to some recent studies [ 19 ] [ 20 ] [ 18 ] [ 26 ] [ 31 ]. The performance of compression algorithms can be quantified in terms of the following criteria: image quality, compression ratio, and compression speed, which is affected by computational complexity and memory requirements. Iii. Fuzzy Compression-decompression Algorithm In what follows, we elaborate on the underlying compression-decompression mechanism based on sampling and fuzzy encoding completed on a basis of coordinated-based membership degrees. This will help us stress a systematic and coherent development process being proposed in this study as well as highlight the novelty along with the proposed mechanism. The algorithm is run for some specific values of setting-namely, the value of sampling ratio, the number of prototypes for reconstructing one pixel and the fuzzification coefficient. The effect of these values on the quality of the reconstruction will be studied later. A. Segmentation of image Before an image could be processed, it needs to be digitalized, encoded and stored. Grayscale images could be defined as a two-dimensional function, f ( x , y ), where x and y are spatial coordinate in horizontal and vertical directions, respectively. The gray level (brightness) of the image at any point ( x , y ) is denoted by its amplitude f . A sample of the image at a certain point is usually referred to as a pixel. An image consists of a finite number of pixels, each of which has a particular location and a gray-level value. There are several different ways of coding the values of gray-level at each point depending on the number of bits to represent one pixel. More bits are used to represent one pixel, more colors could be represented, and more memory and wider bandwidth are demanded when being processed. 8-bit color image is the most commonly one, which allows up to 256 colors (gray levels). A continuous digital representation of image is transformed into digital format and the values of gray-level of each square could be arranged in a two dimensional matrix. Suppose that the original image is represented by a matrix of pixels of size M × N , and the gray level of each pixel is stored as an 8-bit integer value which therefore allows for 256 different shades of brightness ranging from 0 to 255. We set value 0 to black, 255 to white and distribute the gray scale linearly in between. The first step is to partition an image into a number of non-overlapping blocks which are adjacent to each other. Each block is a rectangle with a fixed-size m × n where m << M and n << N , as shown in Fig. 2 . Note that the sizes of the blocks on the right or the bottom of the image may be smaller than m × n . These blocks are treated in the same way as others for uniformity. The choice of suitable values of m and n requires some attention. If the size of a block is too large, we may face the problem of explosion of search space. On the other hand, undersized block may hinder the visual quality of reconstructed image, especially in the areas near the boundaries of the blocks. B. Selection of prototypes In each block, we randomly pick up a sample of pixels, say g pixels; we will refer to them as prototypes and call them v 1 , v 2 , …, v g . The coordinates of these points are marked as ( x 1 , y 1 ), ( x 2 , y 2 ), …, ( x g , y g ) while the values of their gray levels are denoted as f ( x 1 , y 1 ), f ( x 2 , y 2 ), …, f ( x g , y g ), respectively. These sampling points are treated as “anchor” points based on which the block is reconstructed. The sampling ratio is described as the number of sampling points to the total number of pixels in this block and then forms the following expression p = g /( m × n ). The sampling percentage p influences the quality of the reconstruction and compression ratio. As there is an inherent factor of randomness, the whole process is repeated a certain number of iterations and the prototypes producing the lowest reconstruction error are treated as representatives of current block. As to the sampling ratio, one option is to adopt the same sampling ratio in each sub-block; another option worth considering is to choose different values of sampling ratio for different blocks depending upon the level of details presented in the given block and this ratio could be determined based on the reconstruction criterion. C. Determination of membership degrees Let us assume that for each block of pixels, we are provided a collection of prototypes v 1 , v 2 , …, v g , as shown in Fig. 3 . For any pixel s located at coordinates ( a , b ) in this block that is not designated as prototype, we calculate its membership degrees to the c nearest prototypes among v 1 , v 2 , …, v g , which are marked as v ’ 1 , v ’ 2 , …, v ’ c . The corresponding coordinates of these c prototypes are marked as ( x’ 1 , y’ 1 ), ( x’ 2 , y’ 2 ), …, ( x’ c , y’ c ). The choice of parameter c will influence the visual quality of the reconstructed image. The values of membership degrees are determined in the same manner as for Fuzzy C-Means (FCM) clustering method [ 23 ][ 24 ][ 28 ] $${u_i}(s)=\frac{1}{{\sum\limits_{{j=1}}^{c} {{{\left( {\frac{{||s - v{'_i}||}}{{||s - v{'_j}||}}} \right)}^{2/(m - 1)}}} }}$$ 1 where u i ( s ) stands for the value of membership degree of pixel s to prototype v ’ i . Here ||.|| denotes a certain distance function (we use a Manhattan distance in this study, any other form of the distance could also be considered), namely, \(||s - v{'_i}||={\text{|}}a - x{'_i}{\text{|}}+{\text{|}}b - y{'_i}{\text{|}}\) . In any image, each pixel of an image can be identified by its x and y coordinates. Given a number of prototypes v 1 , v 2 , …, v g and a specific pixel s ( a, b ), the vector of membership grades [ u 1 ( s ), u 2 ( s ),…, u c ( s )] of this pixel to the c prototypes that are closest to it do not need to be stored since their values could be calculated on the fly. The values of membership degree in this vector depend upon a number of parameters, such as m and c . The values of m (fuzzification coefficient in the FCM method) influence membership values. The most commonly used value of m is equal to 2 while higher values that 2 make the resulting membership functions become “spiky” and lower values of fuzzification coefficient produce Boolean-like (binary) membership regions [ 10 ]. D. Reconstruction of image through fuzzy decoding Having a collection of pixels which are considered as prototypes, we can formulate a reconstruction problem: determine the value of gray level of each pixel in the block given a group of prototypes v 1 , v 2 , …, v g . We identify two categories of pixels: (a) Pixels that have been identified as prototypes. Since the values of their gray levels have been stored in advance, the reconstruction doesn’t require any specialized handing. (b) Pixels that are not selected as prototypes, denoted as s , whose coordinate is represented as ( a , b ) and the gray level of the reconstructed pixel is denoted as F( a , b ). The reconstruction problem is solved in the same way as in the FCM thus the reconstruction result (value of gray level) is determined as: $$F(a,b)=\frac{{\sum\limits_{{i=1}}^{c} {{u_i}{{(s)}^m}f(x{'_i},y{'_i})} }}{{\sum\limits_{{i=1}}^{c} {{u_{^{i}}}{{(s)}^m}} }}$$ 2 where ( x ’ i , y ’ i ), i =1, 2, …, c , represents the coordinate of the prototype used for reconstructing s ( a , b ). The quality of the compression-decompression depends on a number of parameters and they can be optimized. In general, we could regard compression-decompression problem as an optimization of selection of prototypes guided by a minimization of some certain performance index which quantifies a departure of the values of gray level of the reconstructed pixels from their original ones, namely \((F({x_s},{y_s}) - f({x_s},{y_s}))\) where s denotes the pixel in current block. The performance index of the mean squared error (MSE) considered here comes in the following form. $${\text{MS}}{{\text{E}}_{{\text{block}}}}{\text{=}}\frac{1}{{m \times n}}{\sum\limits_{{s \in current{\text{ }}block}} {(F({x_s},{y_s}) - f({x_s},{y_s}))} ^2}$$ 3 E. Reconstruction of the overall image One the prototypes for each block have been determined, they form a backbone of the basis for the reconstruction of the overall image. The reconstruction the overall image is processed block by block. After the prototypes for each sub-block have been determined, for the pixels in each block that have not been identified as prototypes, the decompression (reconstruction) process consists of three steps: (a) determination of the c nearest prototypes v ’ 1 , v ’ 2 , …, v ’ c from the prototypes that represent current block according to the distances of this pixel to the prototypes, (b) calculation of membership degrees (matching levels) of this pixel to the selected c nearest prototypes and (c) aggregation of the gray levels of the selected prototypes weighted by the membership degrees. Iv. Performance Indexes The performance index of the Mean Square Error MSE all , which is used to evaluate the quality of the overall compression-decompression is expressed in the following form: $$MS{E_{all}}=\frac{1}{{M \times N}}{\sum\limits_{{s \in {\text{who}}le\_image}} {(F({x_s},{y_s}) - f({x_s},{y_s}))} ^2}$$ 4 where s represents the pixel that belongs to the whole image, f ( x , y ) denotes original gray level of pixel local at ( x , y ) while F ( x , y ) represented the value of the reconstructed gray level. Besides performance index MSE that we use to quantify the amount of resulting distortion in the reconstruction image, performance index Peak-Signal-to Noise- Ratio (PSNR) is defined as $$PSNR=10{\log _{10}}\frac{{x_{{peak}}^{2}}}{{\sigma _{q}^{2}}}=10{\log _{10}}\frac{{{{({2^{{\text{bits }}}} - {\text{1)}}}^{\text{2}}}}}{{\frac{1}{{MN}}\sum\limits_{{j=1}}^{N} {\sum\limits_{{i=1}}^{M} {{{(F(i,j) - f(i,j))}^2}} } }}$$ 5 where the value of bits is equal to 8. These two performance indexes are categorized as objective criteria. Compared with subjective criteria which rely on human’s visual perception to judge the quality of an image [ 17 ], objective criteria are much easier to generate and seemingly unbiased. The lower the value of MSE is, the closer the reconstructed image is to the original one. On the other hand, higher values of PSNR indicate better image quality. With regard to the compression of image, we are also interested in the compression ratio (CR) which is expressed as ratio of the original uncompressed input length to the sequence length of the compressed output. $$CR=\frac{{{\text{original image's file size}}}}{{{\text{compressed file size}}}}$$ 6 Another performance index worth considering is bits per pixel (BPP) which is determined as follows: $$BPP=\frac{{{\text{Number of bits}}}}{{{\text{Number of pixels}}}}$$ 7 Higher values of CR indicate better compression ability. Since each pixel has a range of values from 0 to 256, its values could be stored in a one byte array. Suppose the sampling ratio is equal to p (0<p<1), we need to allocate about p × M × N ×8 bits to store the values of gray-level of the sampled pixels in total. Beyond that, we also need to mark which points have been selected as prototypes. The most direct and simplest way is to use chromosome representation in which one bit is corresponding to one pixel in the image (0 means unselected while 1 means selected). Therefore, a storage space of M × N bits needs to be allocated. In addition, a few “tokens” to store the relevant information about current image, such as its size, sampling ratio and the values of relative parameters. Since the very limited number of tokens, they are almost negligible when calculating CR and BPP. On the basis of these considerations, the values of CR and BPP can be approximated as: $${\text{CR}} \approx \frac{{M \times N \times 8}}{{p \times M \times N \times 8+M \times N}}=\frac{8}{{8p+1}}$$ 8 $${\text{BPP}} \approx \frac{{p \times M \times N \times 8+M \times N}}{{M \times N}}=8p+1$$ 9 V. Experimental Studies In this section, we report on a series of experiments concerning the compression of grayscale images. Our goal is to quantify the performance of the compression mechanism, as well as evaluate the impact being brought by the sampling ratio p , the number of prototypes c used for representing / reconstructing each pixel, and the value of fuzzification coefficient m . The influence of sampling ratios on the reconstruction quality in terms of performance index MSE is obvious; by intuition, higher sampling ratio will improve the quality of the reconstructed image. Given a certain sampling ratio p , the effects of parameters m and c on the reconstruction quality require some careful attention. The optimal values of m and c are likely linked to the nature of the image. When the value of c equals to 1, it becomes to a Boolean encoding-decoding style compression. In this case, instead of using several nearest c ( c >1) prototypes to represent and reconstruct one pixel ( x , y ), we choose the prototype for which this pixel is the closet and use this prototype to represent/reconstruct pixel ( x , y ). At the same time, we determine the membership value with respect to the closest prototype as 1 while the membership grades of pixel ( x , y ) to other prototypes are determined as 0. The results of experiments show that suitable edge length of the sub-block ranges from 10 to 20. In the process of the experiments, when performing image segmentation, the size of each block is set as 15×15. The prototypes in each block are determined as the best family of randomly selected prototypes resulting in the best reconstruction quality of current block. For each block, the random selection is repeated 100 times. We sweep through the values of p starting from 0.05 and moving to the upper bound, which is 0.50 in this case. At the same time, the corresponding values of MSE and PSNR are recorded. Grayscale image-girl This image is from the network (URL: https://imgcache.cjmx.com/star/201605/20160518145710516.jpg ) and has been converted to grayscale. We perform the experiments on a grayscale image whose size is 400×300. Fig. 4 shows the original image and the reconstructed images when the value of c is set to 4, the value of fuzzification coefficient m is equal to 2.0, and sampling ratios are set to 0.15, 0.25 and 0.35, respectively. It is evident that the increase of the value of sampling ratio results in a visible improvement of the reconstructed image. The progression of the optimization of prototypes in each sub-block is quantified in terms of the MSE obtained in successive generations. The values of MSE obtained when p =0.35 for 312-th sub-block versus the successive generations are shown in Fig. 5 . The plots of MSE and PSNR (when c equals to 3) versus the fuzzification coefficient m for different values of sampling ratio are shown in Fig. 6. It is evident that the range of optimal values of m is positioned slightly beyond the typical value of 2.0. But as the value of m keeps increasing, the improvement in the quality of the reconstructed image is not overly critical when the value of m is larger than 3.0. The tendency is also evident that with the increase of the value of sampling ratio, the values of MSE decrease and the values of PSNR keep increasing, which indicates an improvement in the quality of the reconstructed images. The plots of the performance indexes MSE and PNSR versus c for different values of sampling ratios are shown in Fig. 7 (the value of fuzzification coefficient m is set to 2.0). The performance of using fuzzy encoding-decoding is superior over the Boolean style compression (the case when c = 1) in all cases. The improvements of performance expressed in terms of MSE are 28.6%, 30.4% and 34% when applying fuzzy encoding-decoding compared with Boolean encoding-decoding mechanism for p = 0.15, 0.25 and 0.35, respectively. The optimal value of c is positioned around 4 or 5 at which point the least reconstruction error is obtained. Beyond this region, we may see a slightly decrease in the quality of the reconstructed image. Image of Lena In the experiments on gray image of Lena with a size of 256×256, we follow the same setup as in the previous image and report the results. The original image and the reconstructed ones for different values of sampling ratio are shown in Fig. 8 . With the increase of the value of sampling ratio, the visual quality of the reconstructed keeps increasing. The values of MSE and PNSR versus m for different values of sampling ratio are shown in Fig. 9 . The same tendency as in the previous experiments is visible that the decrease in the MSE values and the increase in the PNSR values occur with the increase of the value of sampling ratio for different values of m ; the optimal value of m varies in the range between 2.0 and 3.0 and is data dependent. The corresponding values of performance indexes MSE and PSNR versus c when m is set to 2.0 are shown in Fig. 10. It is evident from these plots that best reconstruction abilities are achieved when the values of c are around 4 or 5, beyond which the values of MSE and PNSR do not change significantly. The fuzzy encoding-decoding mechanism produces much lower value of the reconstruction error compared with Boolean encoding-decoding (the case when c = 1). The improvements expressed in terms of MSE brought by fuzzy encoding-decoding are 23.4%, 25.8%, and 29.1% for p = 0.15, 0.25 and 0.35, respectively. Images of leaf and airplane Here, we run experiments on images leaf (of size 360*480) and airplane (of size 512*512) following the same setup as in previous experiments. These images are also from the internet. When the sampling ratio p is set to 0.25, the value of c is set to 3 (which is determined experimentally) and the value of fuzzification coefficient m is set to 2.0, the values of MSE for images leaf and airplane are 18.05 and 13.15 while the values of PNSR of the reconstructed images are 35.57dB and 36.94dB, respectively. The fuzzy encoding-decoding shows an obvious advantage over the encoding-decoding completed in Boolean style and the ratios of improvement of MSE are 30.7% and 27.7% for the images of leaf and plane in this case. The original images and the corresponding reconstructed images are shown in Fig. 11. The optimal value of c for the image leaf is equal to 3, and the optimal value of c for the image airplane varies between 2 and 4. The range of values of m resulting in better reconstruction quality given specified values of c and sampling ratio p is located between 2.0 and 3.3. Vi. Conclusions We have proposed a novel compression algorithm based on sampling and fuzzy encoding. The compression mechanism proposed in this study is quite distinct from the existing ones as the membership degrees of the gray level of each pixel to the gray levels of the selected prototypes are implied by the Manhattan distance between pixels during the encoding and decoding processes. Since each pixel and prototype have fixed coordinates in the two-dimensional plane, the membership degrees between them could be calculated on-the-fly, which avoids unnecessary space cost and could help achieve a much high compression ratio. Experimental results show that the proposed method could achieve very high compression ratio and at the same time ensure the quality of the decompressed image. We also have discussed how its parameters affect the reconstruction quality of the decompressed images. While this study has focused on the compression of grayscale images, one could extend this approach to compressing color images in future study. The developed approach could offer an inspiration to other types of data compressing fields, such as dynamic image compression, audio compression. Declarations Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant Nos. 62076189, 62006184, the Recruitment Program of Global Experts, Canada Research Chair (CRC), Natural Sciences and Engineering Research council of Canada (NSERC) and the Science and Technology Development Fund, MSAR, under Grant No. 0012/2019/A1, and the National Key R&D Program of China under Grant 2018YFB1700104. Ethical approval Ethics approval was not required for this research. Conflict of interest There is no conflict of interest with any of the suggested reviewers. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed. Informed Consent We the undersigned declare that this manuscript entitled “A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images” is original, has not been published before and is not currently being considered for publication elsewhere. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. Signed by all authors as follows: Dan Wang, Xiubin Zhu, Witold Pedrycz, Zhenhua Yu, and Zhiwu Li Authorship contributions Dan Wang: Investigation, Methodology, Coding and Writing Xiubin Zhu: Experimental studies, Writing and Reviewing Witold Pedrycz: Methodology and Reviewing Zhenhua Yu: Reviewing Zhiwu Li: Investigation, Reviewing References L. F. R. Lucas, N. M. M. Rodrigues, L. A. da Silva Cruz and S. M. M. de Faria, “Lossless compression of medical images using 3-D predictors,” IEEE Transactions on Medical Imaging , vol. 36, no. 11, pp. 2250-2260, Nov. 2017. A. C. Johnsy and G. Schirinzi, “A lossless coding scheme for maps using binary wavelet transform,” European Journal of Remote Sensing , vol. 50, no. 1, pp. 77-86, 2017. N. R. M. Noor and T. Vladimirova, “Investigation into lossless hyperspectral image compression for satellite remote sensing,” International Journal for Remote Sensing , vol. 34, no. 14, pp. 5072-5104, 2013. H. Bekkouche, M. Barret, and J. 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Cite Share Download PDF Status: Published Journal Publication published 19 Aug, 2023 Read the published version in Soft Computing → Version 1 posted Reviewers agreed at journal 31 Jan, 2023 Reviewers invited by journal 05 Jan, 2023 Editor assigned by journal 06 Jan, 2022 First submitted to journal 03 Jan, 2022 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. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1219938","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":165255393,"identity":"e80156f4-676d-441e-a7a1-f0fe5c926c96","order_by":0,"name":"Dan Wang","email":"","orcid":"","institution":"Xi'an University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Dan","middleName":"","lastName":"Wang","suffix":""},{"id":165255394,"identity":"435b9085-d11a-4d06-b960-afd6c0a27381","order_by":1,"name":"Xiu-bin Zhu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVRIiWNgGAWjYDACCQY2BoYKBgY2diAngU0CKkhQyxmgFmaStDC2ARkgLSDtBLUY3G5/9ph33jZ5PmYGNokHZRby5gzMB2/zMNjl4dIiOedAujHvttuGbcwMzAYJ5yQMdzawJVvzMCQX49LCL5FwTBqohRGohfFBYpsE44YDPGbSPAwHEhtwaGGTSGyT5p1z2x6oBaisTcJ+wwH+b3i18Esks0nzNtxOhNmSCLSFDa8WyTnH2ID4dnIbM2MzyC/JGw6zGVvOMUjGqQUUYhJvam7bzm9vPib5o6zOdsPx5oc33lTY4dSCBBihasARZEBY/SgYBaNgFIwC3AAAFjNMeLysrw8AAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-7947-8749","institution":"Xidian University","correspondingAuthor":true,"prefix":"","firstName":"Xiu-bin","middleName":"","lastName":"Zhu","suffix":""},{"id":165255395,"identity":"4f0913d8-a2c3-480a-87b5-4c6de8a363e1","order_by":2,"name":"Witold Pedrycz","email":"","orcid":"","institution":"University of Alberta","correspondingAuthor":false,"prefix":"","firstName":"Witold","middleName":"","lastName":"Pedrycz","suffix":""},{"id":165255396,"identity":"62057d17-7f2b-428d-9f5b-afa634ffad98","order_by":3,"name":"Zhenhua Yu","email":"","orcid":"","institution":"Xi'an University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Zhenhua","middleName":"","lastName":"Yu","suffix":""},{"id":165255397,"identity":"df8c2ff1-cae2-46e4-90eb-9f70f958d364","order_by":4,"name":"Zhiwu Li","email":"","orcid":"","institution":"Xidian University","correspondingAuthor":false,"prefix":"","firstName":"Zhiwu","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2021-12-31 15:34:36","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1219938/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1219938/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00500-023-09106-8","type":"published","date":"2023-08-19T22:00:09+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":31315332,"identity":"e75a59bb-236c-4878-98fb-b0380c43b6d5","added_by":"auto","created_at":"2023-01-09 16:05:17","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":105038,"visible":true,"origin":"","legend":"\u003cp\u003eAn overview of the proposed compression method.\u003c/p\u003e","description":"","filename":"Fig1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/7128319cfc051d0b635d158f.jpg"},{"id":31314466,"identity":"233a6fb9-6887-4d9d-a69f-c25e07e13035","added_by":"auto","created_at":"2023-01-09 15:57:17","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":237974,"visible":true,"origin":"","legend":"\u003cp\u003eSegmentation of an input image of size \u003cem\u003eM\u003c/em\u003e ×\u003cem\u003eN\u003c/em\u003e into smaller blocks of size \u003cem\u003em\u003c/em\u003e×\u003cem\u003en\u003c/em\u003e.\u003c/p\u003e","description":"","filename":"Fig2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/b1e6901ce0ad74a50715777e.jpg"},{"id":31315333,"identity":"9452c58c-e2f3-4fd7-b900-c7bea7f3d150","added_by":"auto","created_at":"2023-01-09 16:05:17","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":214545,"visible":true,"origin":"","legend":"\u003cp\u003eSampling of prototypes (black spots) in block and the determination of distances of a pixel (black circle) to the \u003cem\u003ec\u003c/em\u003e nearest prototypes.\u003c/p\u003e","description":"","filename":"Fig3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/b24e99a29a147267899fd0a0.jpg"},{"id":31315941,"identity":"9d4ce46b-892d-4cd7-8a3d-2e055215d9fe","added_by":"auto","created_at":"2023-01-09 16:13:17","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":264369,"visible":true,"origin":"","legend":"\u003cp\u003eOriginal image and reconstructed image with different values of sampling ratios (\u003cem\u003ec\u003c/em\u003e=3 and \u003cem\u003em\u003c/em\u003e=2.0)\u003c/p\u003e\n\u003cp\u003e(a) Original image (b) \u003cem\u003ep\u003c/em\u003e = 0.15, (c) \u003cem\u003ep\u003c/em\u003e = 0.25, and (d) \u003cem\u003ep\u003c/em\u003e = 0.35.\u003c/p\u003e","description":"","filename":"Fig4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/a9984f53b12b700b4f65fac8.jpg"},{"id":31314461,"identity":"c2c0ee21-3a54-44fe-8405-6da711abca06","added_by":"auto","created_at":"2023-01-09 15:57:17","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":59800,"visible":true,"origin":"","legend":"\u003cp\u003eValues of MSE obtained in successive generations of sampling.\u003c/p\u003e","description":"","filename":"Fig5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/ba85458bbb3482f5d4df750c.jpg"},{"id":31317568,"identity":"ca2b7f19-a8e1-4e26-aaff-95f260db090e","added_by":"auto","created_at":"2023-01-09 16:29:17","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":205565,"visible":true,"origin":"","legend":"\u003cp\u003eMSE and PNSR curves versus the values of the fuzzification coefficient.\u003c/p\u003e\n\u003cp\u003e(a) MSE and (b) PNSR.\u003c/p\u003e","description":"","filename":"Fig6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/ffb99ec7db76020f657e1d13.jpg"},{"id":31315338,"identity":"3f26fe6f-9b6b-4e7e-8565-341af61cb3ff","added_by":"auto","created_at":"2023-01-09 16:05:17","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":158968,"visible":true,"origin":"","legend":"\u003cp\u003ePlots of values of MSE(a) and PNSR(b) of the reconstructed images with different sampling ratios versus \u003cem\u003ec\u003c/em\u003e(\u003cem\u003em\u003c/em\u003e = 2.0).\u003c/p\u003e","description":"","filename":"Fig7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/f8901f42542dc6758d1ddfdd.jpg"},{"id":31317221,"identity":"c2606c61-f0e3-49a0-b6ca-0f87bc6c4cbe","added_by":"auto","created_at":"2023-01-09 16:21:17","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":357842,"visible":true,"origin":"","legend":"\u003cp\u003eOriginal image and reconstructed images obtained with different values of sampling ratios when \u003cem\u003ec\u003c/em\u003e = 3 and \u003cem\u003em\u003c/em\u003e = 2.0 (a) Original image (b) \u003cem\u003ep\u003c/em\u003e = 0.15, (c) \u003cem\u003ep\u003c/em\u003e = 0.25, and (d) \u003cem\u003ep\u003c/em\u003e = 0.35.\u003c/p\u003e","description":"","filename":"Fig8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/7fb9d196be4d2ec4d29eee71.jpg"},{"id":31314463,"identity":"824a9f44-ea3a-439e-9e33-9e8beb28aa4b","added_by":"auto","created_at":"2023-01-09 15:57:17","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":199123,"visible":true,"origin":"","legend":"\u003cp\u003ePlots of MSE (a) and PNSR (b) versus \u003cem\u003em\u003c/em\u003e and selected values of \u003cem\u003ep\u003c/em\u003e(the value of \u003cem\u003ec\u003c/em\u003e is set to 4).\u003c/p\u003e","description":"","filename":"Fig9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/9329af6959f879b19b042deb.jpg"},{"id":31314468,"identity":"0f23a7ac-6649-4cf7-b46d-012d0aa20f21","added_by":"auto","created_at":"2023-01-09 15:57:17","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":154397,"visible":true,"origin":"","legend":"\u003cp\u003ePlots of MSE curves (a) and PNSR curves (b) as a function of \u003cem\u003ec \u003c/em\u003efor selected values of \u003cem\u003ep\u003c/em\u003e when \u003cem\u003em\u003c/em\u003e = 2.0.\u003c/p\u003e","description":"","filename":"Fig10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/16dec06858ec139218a996ac.jpg"},{"id":31315335,"identity":"25cc281f-9a87-437c-8fb7-fcfc3b7eff03","added_by":"auto","created_at":"2023-01-09 16:05:17","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":262059,"visible":true,"origin":"","legend":"\u003cp\u003eOriginal images and the reconstructed images when \u003cem\u003ep\u003c/em\u003e=0.25, \u003cem\u003ec\u003c/em\u003e=3 and \u003cem\u003em\u003c/em\u003e=2.0. (a) leaf and (b) airplane.\u003c/p\u003e","description":"","filename":"Fig11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/de4045f2f43e09fad5192412.jpg"},{"id":44736275,"identity":"feced1ac-3ae8-4175-97f1-7ac7013c16aa","added_by":"auto","created_at":"2023-10-16 22:29:47","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1255432,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1219938/v1/ee80cd68-903e-40a8-8fed-933d9f9e0a3f.pdf"}],"financialInterests":"","formattedTitle":"A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images","fulltext":[{"header":"I. Introduction","content":"\u003cp\u003eDigital images play an import role in multimedia and information intercommunication by carrying a wealth of information. Images may represent objects in the form of picture-like bitmaps, vector-based drawings or 3D rendering. With the rapid development of information technology, a large amount of digital image data is generated at high pace. Although in recent years we have witnessed the increase of bandwidth of the communication networks and the decrease of the price of storage devices, digital images produced by various technologies and applications have established higher demands with regard to the bandwidth and storage capacity. In order to overcome the existing limitations on the bandwidth of communication networks and storage capacity, the problem of digital image compression has been extensively researched. We have been witnessing a highly diversified plethora of approaches to image compressing. Image compression is a data compression technology which aims at reducing the cost of transmission and storage. Generally, there are two categories of compression algorithms: lossy compression and lossless compression [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Compared with lossless compression, lossy compression may lead to loss of fidelity, but it is especially useful when being used at low bit rates while loss of fidelity is acceptable in order to fulfill the bandwidth of transmission. Lossy compression algorithms usually tend to achieve a trade-off between the bit rate of the transmission system and the quality of the images being processed. Some lossy compression techniques may lead to an extremely high compression ratio.\u003c/p\u003e \u003cp\u003eExisting image compression techniques usually make use of visual perception or exploit statistical properties of image data to reach the aim of compressing data. The motivation behind the proposed algorithm presented in this study is to use a certain number of neighboring prototypes to represent one pixel using fuzzy set-based encoding, given the fact that significant correlations or redundancy usually exist among consecutive or neighboring pixels. We present a novel image compression algorithm which makes use of the sampled pixels (prototypes) and membership degrees of each pixel to a group of prototypes with respect to the gray levels to reconstruct image. The membership values used for image reconstruction are implied by the distance of the pixels to be reconstructed to the selected prototypes. Given a digital image of fixed size, it could be represented by a two-dimensional plane where \u003cem\u003ex\u003c/em\u003e and \u003cem\u003ey\u003c/em\u003e are spatial coordinates, and any pair of coordinates (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) represents a specific pixel. When fuzzy encoding / decoding is involved, the membership degrees of the gray level of this pixel to the brightness of the prototypes become necessary. But since each pixel has its unique coordinate, the distances between different pixels are determined and could be calculated on the fly. Once a group of prototypes has been determined and membership degrees computed with respect to coordinates could approximate the membership grades between values of gray levels, the requirement for storage space could be reduced significantly, as we only need to store the index (coordinates) and values of gray-levels of the prototypes after compression is completed. Given the coordinates of the prototypes and their gray levels, it is enough to reconstruct the whole image, which means that we are able to achieve a very high compression ratio.\u003c/p\u003e \u003cp\u003eIt is worth noting that in current image compression techniques (no matter which compression algorithm has been adopted), the inner relationship between the gray levels of pixels and their coordinates has not been fully exploited. The method proposed in this study has not been explored in the existing literatures. The developed approach exhibits a tangible level of originality in the sense that a novel method which fully utilize the information implicit in the coordinates of each pixel to reconstruct the image. An overview of the overall framework highlighting the essence of the method is visualized in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The completion of the compression process gives rise to a collection of prototypes whose coordinates and gray levels are stored for further decompression. In the decompression process, the reconstruction of the original image is based on these prototypes and the membership degrees of each pixel to these prototypes implied in their coordinates. In this study, we are interested in the compression of grayscale images. Grayscale images are usually stored in the form of two-dimensional arrays of gray-level values. The only colors of a grayscale image are the shades of gray and each pixel in the image is a scalar proportional to the intensity of gray level.\u003c/p\u003e \u003cp\u003eThis study is organized as follows. In Section II, we offer a brief introduction to the existing image compression algorithms with some focus on the techniques involving fuzzy sets. In Section III, we elaborate on the detailed image compression algorithm. Performance indexes used to evaluate the quality of reconstructed images are discussed in Section IV. A series of experiments and ensuing parametric analysis are reported in Section V while conclusions are offered in Section VI.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"Ii.\tCompression Of Images","content":"\u003cp\u003eImage compression technologies have been utilized in many diverse applications, such as remote sensing, medical imaging, digital television, and video surveillance systems. The key to image compression technology is to remove redundant data from the original image, and later the compressed image is decompressed to exactly rebuild the original image or to reconstruct an approximation one. There are several different types of data redundancies, for example, spatial redundancy, temporal redundancy, coding redundancy, psychovisual redundancy and knowledge redundancy. General compression systems try to remove those redundancies to meet the objective of compressing images.\u003c/p\u003e \u003cp\u003eGenerally, image compression methods could be divided into two categories: lossless methods and lossy methods, depending on whether the compressed data supports error-free data reconstruction. A general compression system is usually constructed in the encoder-decoder architecture. The encoder removes redundancy information from the original image and enhances the immunity to noise while the decoder recovers/reconstructs the original image as faithfully as possible. Lossless compression is a necessary choice when degradation of original image is not acceptable. For example, in medical fields, the details of medical images are very important to diagnose illness and choose a right therapeutic scheme [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Other areas in which lossless compression are extensively adopted includes compression of thermal images taken by satellites [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], remote sensing images [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], and fingerprints images [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] etc. Many powerful lossless image compression algorithms have been developed. It is needless to say that there is no ideal lossless compression tool, which could efficiently compress all possible images [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e][\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]. For this reason, many different compression algorithms have been designed with specific assumptions about what kinds of redundancy the original images are likely to contain. Some of the most commonly used lossless compression algorithms include Huffman coding [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], arithmetic coding [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and Run Length Coding [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. In recent years, we also have witnessed a lot of such studies reported in the literature [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e][\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. The main drawback of lossless image compression algorithms is that they cannot guarantee the compressed image being smaller in size than the original one.\u003c/p\u003e \u003cp\u003eCompared with lossless compression algorithms, lossy image compression algorithms can reach higher compression rate at the expense of losing some information of the original images. While the degradation of the uncompressed image could be quantified through certain performance indexes, such as Mean Square Error (MSE) or Peak-Signal-To-Noise ratio (PSNR), the efficiency of the compression algorithm could also be assessed in terms of data compression ability and implementation complexity of the algorithm. Lossy compression is widely used, albeit imperfect, especially when the main goal is to reach a given compression rate to meet the bandwidth requirement or to overcome the restrictions on storage or to reduce encoding-decoding time. There are a number of approaches to lossy image compression, such as vector quantisation, predictive coding, and transform coding [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. These algorithms are carried in the following aspects to achieve the goal of compressing image: i) quantification of the color space to reduce dimensionality of the problem, ii) implementing lower resolution for chroma information than for luma information when coding images, iii) fractal compression and iv) transform coding, which is also the most commonly used method. Fuzzy sets and Fuzzy relational equations also have been used in image compression/decompression [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], which exploit a new field for the research of image processing. In these studies, images are represented as fuzzy relations since fuzzy sets are able to capture the essence of the problem from information granule point of view and come with a sound algorithmic framework. In the area of lossy image compression, we can refer to some recent studies presented in the literature [\u003cspan additionalcitationids=\"CR15\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e][\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e] [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]. In recent year, there also has been an interest in compressing images using neural networks, refer to some recent studies [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. The performance of compression algorithms can be quantified in terms of the following criteria: image quality, compression ratio, and compression speed, which is affected by computational complexity and memory requirements.\u003c/p\u003e"},{"header":"Iii. Fuzzy Compression-decompression Algorithm","content":"\u003cp\u003eIn what follows, we elaborate on the underlying compression-decompression mechanism based on sampling and fuzzy encoding completed on a basis of coordinated-based membership degrees. This will help us stress a systematic and coherent development process being proposed in this study as well as highlight the novelty along with the proposed mechanism. The algorithm is run for some specific values of setting-namely, the value of sampling ratio, the number of prototypes for reconstructing one pixel and the fuzzification coefficient. The effect of these values on the quality of the reconstruction will be studied later.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eA. Segmentation of image\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eBefore an image could be processed, it needs to be digitalized, encoded and stored. Grayscale images could be defined as a two-dimensional function, \u003cem\u003ef\u003c/em\u003e(\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e), where \u003cem\u003ex\u003c/em\u003e and \u003cem\u003ey\u003c/em\u003e are spatial coordinate in horizontal and vertical directions, respectively. The gray level (brightness) of the image at any point (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) is denoted by its amplitude \u003cem\u003ef\u003c/em\u003e. A sample of the image at a certain point is usually referred to as a pixel. An image consists of a finite number of pixels, each of which has a particular location and a gray-level value. There are several different ways of coding the values of gray-level at each point depending on the number of bits to represent one pixel. More bits are used to represent one pixel, more colors could be represented, and more memory and wider bandwidth are demanded when being processed. 8-bit color image is the most commonly one, which allows up to 256 colors (gray levels). A continuous digital representation of image is transformed into digital format and the values of gray-level of each square could be arranged in a two dimensional matrix.\u003c/p\u003e\n\u003cp\u003eSuppose that the original image is represented by a matrix of pixels of size \u003cem\u003eM\u003c/em\u003e\u0026times;\u003cem\u003eN\u003c/em\u003e, and the gray level of each pixel is stored as an 8-bit integer value which therefore allows for 256 different shades of brightness ranging from 0 to 255. We set value 0 to black, 255 to white and distribute the gray scale linearly in between. The first step is to partition an image into a number of non-overlapping blocks which are adjacent to each other. Each block is a rectangle with a fixed-size \u003cem\u003em\u003c/em\u003e\u0026times;\u003cem\u003en\u003c/em\u003e where \u003cem\u003em\u003c/em\u003e\u0026lt;\u0026lt;\u003cem\u003eM\u003c/em\u003e and \u003cem\u003en\u003c/em\u003e\u0026lt;\u0026lt;\u003cem\u003eN\u003c/em\u003e, as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Note that the sizes of the blocks on the right or the bottom of the image may be smaller than \u003cem\u003em\u003c/em\u003e\u0026times;\u003cem\u003en\u003c/em\u003e. These blocks are treated in the same way as others for uniformity. The choice of suitable values of m and n requires some attention. If the size of a block is too large, we may face the problem of explosion of search space. On the other hand, undersized block may hinder the visual quality of reconstructed image, especially in the areas near the boundaries of the blocks.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eB.\u0026nbsp;Selection of prototypes\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eIn each block, we randomly pick up a sample of pixels, say \u003cem\u003eg\u003c/em\u003e pixels; we will refer to them as prototypes and call them \u003cspan class=\"BoldItalic\"\u003ev\u003c/span\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cspan class=\"BoldItalic\"\u003ev\u003c/span\u003e\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cspan class=\"BoldItalic\"\u003ev\u003c/span\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e. The coordinates of these points are marked as (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e), (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e), \u0026hellip;, (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e) while the values of their gray levels are denoted as \u003cem\u003ef\u003c/em\u003e(\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e), \u003cem\u003ef\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e), \u0026hellip;, \u003cem\u003ef\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e), respectively. These sampling points are treated as \u0026ldquo;anchor\u0026rdquo; points based on which the block is reconstructed. The sampling ratio is described as the number of sampling points to the total number of pixels in this block and then forms the following expression \u003cem\u003ep\u003c/em\u003e = \u003cem\u003eg\u003c/em\u003e/(\u003cem\u003em\u003c/em\u003e\u0026times;\u003cem\u003en\u003c/em\u003e). The sampling percentage \u003cem\u003ep\u003c/em\u003e influences the quality of the reconstruction and compression ratio. As there is an inherent factor of randomness, the whole process is repeated a certain number of iterations and the prototypes producing the lowest reconstruction error are treated as representatives of current block. As to the sampling ratio, one option is to adopt the same sampling ratio in each sub-block; another option worth considering is to choose different values of sampling ratio for different blocks depending upon the level of details presented in the given block and this ratio could be determined based on the reconstruction criterion.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eC. Determination of membership degrees\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eLet us assume that for each block of pixels, we are provided a collection of prototypes \u003cem\u003ev\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e, as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. For any pixel \u003cem\u003es\u003c/em\u003e located at coordinates (\u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e) in this block that is not designated as prototype, we calculate its membership degrees to the \u003cem\u003ec\u003c/em\u003e nearest prototypes among \u003cem\u003ev\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e, which are marked as \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e. The corresponding coordinates of these \u003cem\u003ec\u003c/em\u003e prototypes are marked as (\u003cem\u003ex\u0026rsquo;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ey\u0026rsquo;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e), (\u003cem\u003ex\u0026rsquo;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ey\u0026rsquo;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e), \u0026hellip;, (\u003cem\u003ex\u0026rsquo;\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003ey\u0026rsquo;\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e). The choice of parameter \u003cem\u003ec\u003c/em\u003e will influence the visual quality of the reconstructed image. The values of membership degrees are determined in the same manner as for Fuzzy C-Means (FCM) clustering method [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e][\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e][\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e]\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$${u_i}(s)=\\frac{1}{{\\sum\\limits_{{j=1}}^{c} {{{\\left( {\\frac{{||s - v{'_i}||}}{{||s - v{'_j}||}}} \\right)}^{2/(m - 1)}}} }}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e) stands for the value of membership degree of pixel \u003cem\u003es\u003c/em\u003e to prototype \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e. Here ||.|| denotes a certain distance function (we use a Manhattan distance in this study, any other form of the distance could also be considered), namely, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(||s - v{'_i}||={\\text{|}}a - x{'_i}{\\text{|}}+{\\text{|}}b - y{'_i}{\\text{|}}\\)\u003c/span\u003e\u003c/span\u003e. In any image, each pixel of an image can be identified by its \u003cem\u003ex\u003c/em\u003e and \u003cem\u003ey\u003c/em\u003e coordinates. Given a number of prototypes \u003cem\u003ev\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e and a specific pixel \u003cem\u003es\u003c/em\u003e (\u003cem\u003ea, b\u003c/em\u003e), the vector of membership grades [\u003cem\u003eu\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e), \u003cem\u003eu\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e),\u0026hellip;, \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e)] of this pixel to the \u003cem\u003ec\u003c/em\u003e prototypes that are closest to it do not need to be stored since their values could be calculated on the fly. The values of membership degree in this vector depend upon a number of parameters, such as \u003cem\u003em\u003c/em\u003e and \u003cem\u003ec\u003c/em\u003e. The values of \u003cem\u003em\u003c/em\u003e (fuzzification coefficient in the FCM method) influence membership values. The most commonly used value of \u003cem\u003em\u003c/em\u003e is equal to 2 while higher values that 2 make the resulting membership functions become \u0026ldquo;spiky\u0026rdquo; and lower values of fuzzification coefficient produce Boolean-like (binary) membership regions [\u003cspan class=\"CitationRef\"\u003e10\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eD. Reconstruction of image through fuzzy decoding\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eHaving a collection of pixels which are considered as prototypes, we can formulate a reconstruction problem: determine the value of gray level of each pixel in the block given a group of prototypes \u003cem\u003ev\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eg\u003c/em\u003e\u003c/sub\u003e. We identify two categories of pixels: (a) Pixels that have been identified as prototypes. Since the values of their gray levels have been stored in advance, the reconstruction doesn\u0026rsquo;t require any specialized handing. (b) Pixels that are not selected as prototypes, denoted as \u003cem\u003es\u003c/em\u003e, whose coordinate is represented as (\u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e) and the gray level of the reconstructed pixel is denoted as F(\u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e). The reconstruction problem is solved in the same way as in the FCM thus the reconstruction result (value of gray level) is determined as:\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$$F(a,b)=\\frac{{\\sum\\limits_{{i=1}}^{c} {{u_i}{{(s)}^m}f(x{'_i},y{'_i})} }}{{\\sum\\limits_{{i=1}}^{c} {{u_{^{i}}}{{(s)}^m}} }}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere (\u003cem\u003ex\u003c/em\u003e\u0026rsquo;\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u0026rsquo;\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e), \u003cem\u003ei\u003c/em\u003e=1, 2, \u0026hellip;, \u003cem\u003ec\u003c/em\u003e, represents the coordinate of the prototype used for reconstructing \u003cem\u003es\u003c/em\u003e(\u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e). The quality of the compression-decompression depends on a number of parameters and they can be optimized. In general, we could regard compression-decompression problem as an optimization of selection of prototypes guided by a minimization of some certain performance index which quantifies a departure of the values of gray level of the reconstructed pixels from their original ones, namely \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((F({x_s},{y_s}) - f({x_s},{y_s}))\\)\u003c/span\u003e\u003c/span\u003e where \u003cem\u003es\u003c/em\u003e denotes the pixel in current block. The performance index of the mean squared error (MSE) considered here comes in the following form.\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$${\\text{MS}}{{\\text{E}}_{{\\text{block}}}}{\\text{=}}\\frac{1}{{m \\times n}}{\\sum\\limits_{{s \\in current{\\text{ }}block}} {(F({x_s},{y_s}) - f({x_s},{y_s}))} ^2}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003cp class=\"EquationNumber\"\u003e\u003cem\u003eE. Reconstruction of the overall image\u003c/em\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eOne the prototypes for each block have been determined, they form a backbone of the basis for the reconstruction of the overall image. The reconstruction the overall image is processed block by block. After the prototypes for each sub-block have been determined, for the pixels in each block that have not been identified as prototypes, the decompression (reconstruction) process consists of three steps: (a) determination of the \u003cem\u003ec\u003c/em\u003e nearest prototypes \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e2\u003c/sub\u003e, \u0026hellip;, \u003cem\u003ev\u003c/em\u003e\u0026rsquo;\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e from the prototypes that represent current block according to the distances of this pixel to the prototypes, (b) calculation of membership degrees (matching levels) of this pixel to the selected \u003cem\u003ec\u003c/em\u003e nearest prototypes and (c) aggregation of the gray levels of the selected prototypes weighted by the membership degrees.\u003c/p\u003e"},{"header":"Iv. Performance Indexes","content":"\u003cp\u003eThe performance index of the Mean Square Error MSE\u003csub\u003eall\u003c/sub\u003e, which is used to evaluate the quality of the overall compression-decompression is expressed in the following form:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$MS{E_{all}}=\\frac{1}{{M \\times N}}{\\sum\\limits_{{s \\in {\\text{who}}le\\_image}} {(F({x_s},{y_s}) - f({x_s},{y_s}))} ^2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003es\u003c/em\u003e represents the pixel that belongs to the whole image, \u003cem\u003ef\u003c/em\u003e(\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) denotes original gray level of pixel local at (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) while \u003cem\u003eF\u003c/em\u003e(\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) represented the value of the reconstructed gray level.\u003c/p\u003e \u003cp\u003eBesides performance index MSE that we use to quantify the amount of resulting distortion in the reconstruction image, performance index Peak-Signal-to Noise- Ratio (PSNR) is defined as\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$PSNR=10{\\log _{10}}\\frac{{x_{{peak}}^{2}}}{{\\sigma _{q}^{2}}}=10{\\log _{10}}\\frac{{{{({2^{{\\text{bits }}}} - {\\text{1)}}}^{\\text{2}}}}}{{\\frac{1}{{MN}}\\sum\\limits_{{j=1}}^{N} {\\sum\\limits_{{i=1}}^{M} {{{(F(i,j) - f(i,j))}^2}} } }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the value of bits is equal to 8. These two performance indexes are categorized as objective criteria. Compared with subjective criteria which rely on human\u0026rsquo;s visual perception to judge the quality of an image [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], objective criteria are much easier to generate and seemingly unbiased. The lower the value of MSE is, the closer the reconstructed image is to the original one. On the other hand, higher values of PSNR indicate better image quality.\u003c/p\u003e \u003cp\u003eWith regard to the compression of image, we are also interested in the compression ratio (CR) which is expressed as ratio of the original uncompressed input length to the sequence length of the compressed output.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$CR=\\frac{{{\\text{original image\u0026#039;s file size}}}}{{{\\text{compressed file size}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAnother performance index worth considering is bits per pixel (BPP) which is determined as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$BPP=\\frac{{{\\text{Number of bits}}}}{{{\\text{Number of pixels}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHigher values of CR indicate better compression ability. Since each pixel has a range of values from 0 to 256, its values could be stored in a one byte array. Suppose the sampling ratio is equal to \u003cem\u003ep\u003c/em\u003e (0\u0026lt;p\u0026lt;1), we need to allocate about \u003cem\u003ep\u003c/em\u003e\u0026times;\u003cem\u003eM\u003c/em\u003e\u0026times;\u003cem\u003eN\u003c/em\u003e\u0026times;8 bits to store the values of gray-level of the sampled pixels in total. Beyond that, we also need to mark which points have been selected as prototypes. The most direct and simplest way is to use chromosome representation in which one bit is corresponding to one pixel in the image (0 means unselected while 1 means selected). Therefore, a storage space of \u003cem\u003eM\u003c/em\u003e\u0026times;\u003cem\u003eN\u003c/em\u003e bits needs to be allocated. In addition, a few \u0026ldquo;tokens\u0026rdquo; to store the relevant information about current image, such as its size, sampling ratio and the values of relative parameters. Since the very limited number of tokens, they are almost negligible when calculating CR and BPP. On the basis of these considerations, the values of CR and BPP can be approximated as:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${\\text{CR}} \\approx \\frac{{M \\times N \\times 8}}{{p \\times M \\times N \\times 8+M \\times N}}=\\frac{8}{{8p+1}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${\\text{BPP}} \\approx \\frac{{p \\times M \\times N \\times 8+M \\times N}}{{M \\times N}}=8p+1$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"V.\tExperimental Studies","content":"\u003cp\u003eIn this section, we report on a series of experiments concerning the compression of grayscale images. Our goal is to quantify the performance of the compression mechanism, as well as evaluate the impact being brought by the sampling ratio \u003cem\u003ep\u003c/em\u003e, the number of prototypes \u003cem\u003ec\u003c/em\u003e used for representing / reconstructing each pixel, and the value of fuzzification coefficient \u003cem\u003em\u003c/em\u003e. The influence of sampling ratios on the reconstruction quality in terms of performance index MSE is obvious; by intuition, higher sampling ratio will improve the quality of the reconstructed image. Given a certain sampling ratio \u003cem\u003ep\u003c/em\u003e, the effects of parameters \u003cem\u003em\u003c/em\u003e and \u003cem\u003ec\u003c/em\u003e on the reconstruction quality require some careful attention. The optimal values of \u003cem\u003em\u003c/em\u003e and \u003cem\u003ec\u003c/em\u003e are likely linked to the nature of the image. When the value of \u003cem\u003ec\u003c/em\u003e equals to 1, it becomes to a Boolean encoding-decoding style compression. In this case, instead of using several nearest \u003cem\u003ec\u003c/em\u003e (\u003cem\u003ec\u003c/em\u003e\u0026gt;1) prototypes to represent and reconstruct one pixel (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e), we choose the prototype for which this pixel is the closet and use this prototype to represent/reconstruct pixel (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e). At the same time, we determine the membership value with respect to the closest prototype as 1 while the membership grades of pixel (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e) to other prototypes are determined as 0. The results of experiments show that suitable edge length of the sub-block ranges from 10 to 20. In the process of the experiments, when performing image segmentation, the size of each block is set as 15\u0026times;15. The prototypes in each block are determined as the best family of randomly selected prototypes resulting in the best reconstruction quality of current block. For each block, the random selection is repeated 100 times. We sweep through the values of \u003cem\u003ep\u003c/em\u003e starting from 0.05 and moving to the upper bound, which is 0.50 in this case. At the same time, the corresponding values of MSE and PSNR are recorded.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eGrayscale image-girl\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThis image is from the network (URL: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://imgcache.cjmx.com/star/201605/20160518145710516.jpg\u003c/span\u003e\u003c/span\u003e) and has been converted to grayscale. We perform the experiments on a grayscale image whose size is 400\u0026times;300. Fig.\u0026nbsp;4 shows the original image and the reconstructed images when the value of \u003cem\u003ec\u003c/em\u003e is set to 4, the value of fuzzification coefficient \u003cem\u003em\u003c/em\u003e is equal to 2.0, and sampling ratios are set to 0.15, 0.25 and 0.35, respectively. It is evident that the increase of the value of sampling ratio results in a visible improvement of the reconstructed image. The progression of the optimization of prototypes in each sub-block is quantified in terms of the MSE obtained in successive generations. The values of MSE obtained when \u003cem\u003ep\u003c/em\u003e=0.35 for 312-th sub-block versus the successive generations are shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThe plots of MSE and PSNR (when \u003cem\u003ec\u003c/em\u003e equals to 3) versus the fuzzification coefficient \u003cem\u003em\u003c/em\u003e for different values of sampling ratio are shown in Fig.\u0026nbsp;6. It is evident that the range of optimal values of \u003cem\u003em\u003c/em\u003e is positioned slightly beyond the typical value of 2.0. But as the value of \u003cem\u003em\u003c/em\u003e keeps increasing, the improvement in the quality of the reconstructed image is not overly critical when the value of \u003cem\u003em\u003c/em\u003e is larger than 3.0. The tendency is also evident that with the increase of the value of sampling ratio, the values of MSE decrease and the values of PSNR keep increasing, which indicates an improvement in the quality of the reconstructed images.\u003c/p\u003e\n\u003cp\u003eThe plots of the performance indexes MSE and PNSR versus \u003cem\u003ec\u003c/em\u003e for different values of sampling ratios are shown in Fig.\u0026nbsp;7 (the value of fuzzification coefficient \u003cem\u003em\u003c/em\u003e is set to 2.0). The performance of using fuzzy encoding-decoding is superior over the Boolean style compression (the case when \u003cem\u003ec\u003c/em\u003e = 1) in all cases. The improvements of performance expressed in terms of MSE are 28.6%, 30.4% and 34% when applying fuzzy encoding-decoding compared with Boolean encoding-decoding mechanism for \u003cem\u003ep\u003c/em\u003e = 0.15, 0.25 and 0.35, respectively. The optimal value of \u003cem\u003ec\u003c/em\u003e is positioned around 4 or 5 at which point the least reconstruction error is obtained. Beyond this region, we may see a slightly decrease in the quality of the reconstructed image.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eImage of Lena\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eIn the experiments on gray image of Lena with a size of 256\u0026times;256, we follow the same setup as in the previous image and report the results. The original image and the reconstructed ones for different values of sampling ratio are shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e. With the increase of the value of sampling ratio, the visual quality of the reconstructed keeps increasing. The values of MSE and PNSR versus \u003cem\u003em\u003c/em\u003e for different values of sampling ratio are shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e. The same tendency as in the previous experiments is visible that the decrease in the MSE values and the increase in the PNSR values occur with the increase of the value of sampling ratio for different values of \u003cem\u003em\u003c/em\u003e; the optimal value of \u003cem\u003em\u003c/em\u003e varies in the range between 2.0 and 3.0 and is data dependent. The corresponding values of performance indexes MSE and PSNR versus \u003cem\u003ec\u003c/em\u003e when \u003cem\u003em\u003c/em\u003e is set to 2.0 are shown in Fig.\u0026nbsp;10. It is evident from these plots that best reconstruction abilities are achieved when the values of \u003cem\u003ec\u003c/em\u003e are around 4 or 5, beyond which the values of MSE and PNSR do not change significantly. The fuzzy encoding-decoding mechanism produces much lower value of the reconstruction error compared with Boolean encoding-decoding (the case when \u003cem\u003ec\u003c/em\u003e = 1). The improvements expressed in terms of MSE brought by fuzzy encoding-decoding are 23.4%, 25.8%, and 29.1% for \u003cem\u003ep\u003c/em\u003e = 0.15, 0.25 and 0.35, respectively.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eImages of leaf and airplane\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eHere, we run experiments on images leaf (of size 360*480) and airplane (of size 512*512) following the same setup as in previous experiments. These images are also from the internet. When the sampling ratio \u003cem\u003ep\u003c/em\u003e is set to 0.25, the value of \u003cem\u003ec\u003c/em\u003e is set to 3 (which is determined experimentally) and the value of fuzzification coefficient \u003cem\u003em\u003c/em\u003e is set to 2.0, the values of MSE for images leaf and airplane are 18.05 and 13.15 while the values of PNSR of the reconstructed images are 35.57dB and 36.94dB, respectively. The fuzzy encoding-decoding shows an obvious advantage over the encoding-decoding completed in Boolean style and the ratios of improvement of MSE are 30.7% and 27.7% for the images of leaf and plane in this case. The original images and the corresponding reconstructed images are shown in Fig.\u0026nbsp;11. The optimal value of \u003cem\u003ec\u003c/em\u003e for the image leaf is equal to 3, and the optimal value of \u003cem\u003ec\u003c/em\u003e for the image airplane varies between 2 and 4. The range of values of \u003cem\u003em\u003c/em\u003e resulting in better reconstruction quality given specified values of \u003cem\u003ec\u003c/em\u003e and sampling ratio \u003cem\u003ep\u003c/em\u003e is located between 2.0 and 3.3.\u003c/p\u003e"},{"header":"Vi. Conclusions","content":"\u003cp\u003eWe have proposed a novel compression algorithm based on sampling and fuzzy encoding. The compression mechanism proposed in this study is quite distinct from the existing ones as the membership degrees of the gray level of each pixel to the gray levels of the selected prototypes are implied by the Manhattan distance between pixels during the encoding and decoding processes. Since each pixel and prototype have fixed coordinates in the two-dimensional plane, the membership degrees between them could be calculated on-the-fly, which avoids unnecessary space cost and could help achieve a much high compression ratio. Experimental results show that the proposed method could achieve very high compression ratio and at the same time ensure the quality of the decompressed image. We also have discussed how its parameters affect the reconstruction quality of the decompressed images. While this study has focused on the compression of grayscale images, one could extend this approach to compressing color images in future study. The developed approach could offer an inspiration to other types of data compressing fields, such as dynamic image compression, audio compression.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAcknowledgment\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China under Grant Nos. 62076189, 62006184, the Recruitment Program of Global Experts, Canada Research Chair (CRC), Natural Sciences and Engineering Research council of Canada (NSERC) and the Science and Technology Development Fund, MSAR, under Grant No.\u0026nbsp;0012/2019/A1, and the National Key R\u0026amp;D Program of China under Grant 2018YFB1700104.\u003c/p\u003e\n\u003cp\u003eEthical approval\u003c/p\u003e\n\u003cp\u003eEthics approval was not required for this research.\u003c/p\u003e\n\u003cp\u003eConflict of interest\u003c/p\u003e\n\u003cp\u003eThere is no conflict of interest with any of the suggested reviewers. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.\u003c/p\u003e\n\u003cp\u003eInformed Consent\u003c/p\u003e\n\u003cp\u003eWe the undersigned declare that this manuscript entitled \u0026ldquo;A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images\u0026rdquo; is original, has not been published before and is not currently being considered for publication elsewhere.\u003c/p\u003e\n\u003cp\u003eWe confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.\u003c/p\u003e\n\u003cp\u003eSigned by all authors as follows: Dan Wang, Xiubin Zhu, Witold Pedrycz, Zhenhua Yu, and Zhiwu Li\u003c/p\u003e\n\u003cp\u003eAuthorship contributions\u003c/p\u003e\n\u003cp\u003eDan Wang: Investigation, Methodology, Coding and Writing\u003c/p\u003e\n\u003cp\u003eXiubin Zhu: Experimental studies, Writing and Reviewing\u003c/p\u003e\n\u003cp\u003eWitold Pedrycz: Methodology and Reviewing\u003c/p\u003e\n\u003cp\u003eZhenhua Yu:\u0026nbsp;Reviewing\u003c/p\u003e\n\u003cp\u003eZhiwu Li: Investigation, Reviewing\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eL. F. R. Lucas, N. M. M. Rodrigues, L. A. da Silva Cruz and S. M. 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In this study, we present a novel method of lossy compression based on sampling and fuzzy encoding for grayscale images and discuss the problem of their reconstruction. First, an image is divided into a number of non-overlapping blocks of pixels. Next, we perform multiple rounds of random sampling. In each round, a number of pixels are selected as prototypes for the representing the corresponding block. Each pixel in the block is reconstructed based on the gray-levels of the prototypes and membership degrees computed with respect to the distances of each pixel to the prototypes. The reconstruction abilities delivered by the prototypes are quantified by a certain objective fidelity criterion and the prototypes leading to lowest reconstruction error are determined as representatives of current block. Finally, once the representatives in each block have been determined, we reconstruct the whole image based on these prototypes. Experimental studies as well as visual evaluations show that the proposed algorithm is able to achieve high compression ratios while preserving the overall fidelity in the decompressed images.\u003c/p\u003e","manuscriptTitle":"A Coordinate-based Fuzzy Encoding Strategy for Compressing Grayscale Images","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-01-09 15:57:12","doi":"10.21203/rs.3.rs-1219938/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2023-01-31T06:03:17+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-01-05T11:11:28+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2022-01-06T08:56:46+00:00","index":"","fulltext":""},{"type":"submitted","content":"Soft Computing","date":"2022-01-03T22:53:57+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"soft-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"soco","sideBox":"Learn more about [Soft Computing](https://www.springer.com/journal/500)","snPcode":"500","submissionUrl":"https://submission.nature.com/new-submission/500/3","title":"Soft Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"f662260c-e0f9-493a-9243-e92d10d71239","owner":[],"postedDate":"January 9th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-10-16T22:20:23+00:00","versionOfRecord":{"articleIdentity":"rs-1219938","link":"https://doi.org/10.1007/s00500-023-09106-8","journal":{"identity":"soft-computing","isVorOnly":false,"title":"Soft Computing"},"publishedOn":"2023-08-19 22:00:09","publishedOnDateReadable":"August 19th, 2023"},"versionCreatedAt":"2023-01-09 15:57:12","video":"","vorDoi":"10.1007/s00500-023-09106-8","vorDoiUrl":"https://doi.org/10.1007/s00500-023-09106-8","workflowStages":[]},"version":"v1","identity":"rs-1219938","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1219938","identity":"rs-1219938","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}