Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography

Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography | 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 Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography Yuzhe Ling, Xiongwen Ke, Huihui Wang, Qingping Zhou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7586896/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Diffusion models have become particularly prominent for generating high-quality samples. Consequently, they have become widely adopted for modeling priors in image reconstruction. However, integrating diffusion priors into nonlinear inverse problems presents significant computational challenges, primarily due to the inherent complexity of nonlinear forward operators. This work presents a new approach for incorporating a diffusion prior into the split Gibbs sampler, denoted as DP-SGS, to sample from the posterior distribution in nonlinear settings. DP-SGS simplifies the complex posterior sampling problem by introducing an auxiliary variable that decomposes it into two simple conditional distributions: a forward model that handles the likelihood term and a prior model implicitly determined by the pre-trained diffusion model.Both conditional distributions are efficiently approximated as Gaussian distributions using first-order Taylor expansion for the prior model and Laplace approximation for the forward model, thereby ensuring computational tractability despite the nonlinear forward operator.Experiments on the challenging electrical impedance tomography (EIT) imaging problem demonstrate that our method enhances solution accuracy while achieving rapid convergence rates, confirming its effectiveness for solving nonlinear inverse problems. inverse problems diffusion models split Gibbs sampler electrical impedance tomography Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-7586896","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":515537210,"identity":"fc0becd2-b1af-4373-9124-dfb6ed2cb99c","order_by":0,"name":"Yuzhe Ling","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Yuzhe","middleName":"","lastName":"Ling","suffix":""},{"id":515537211,"identity":"87fdf5b2-1358-4a1e-a57c-70998e5a0909","order_by":1,"name":"Xiongwen Ke","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Xiongwen","middleName":"","lastName":"Ke","suffix":""},{"id":515537213,"identity":"0f501f13-8c52-4cda-8aa7-a724c9a4dab0","order_by":2,"name":"Huihui Wang","email":"","orcid":"","institution":"King's College London","correspondingAuthor":false,"prefix":"","firstName":"Huihui","middleName":"","lastName":"Wang","suffix":""},{"id":515537215,"identity":"16539ca2-cb28-494f-8e4d-d2f5c611880f","order_by":3,"name":"Qingping Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVRIiWNgGAWjYHACNiC2gTB5SNCSRrqWwyRo4bvd/uzBx7bz8vIzEhgfvG1jkDcnpEXyzoF0w5lttw033EhgNpzbxmC4s4GAFoMbCcekedtuJxhIJLABGQwJBgcIaklsA6o8lwB0GPtvIrUkgww/kMBwI4GNmSgtkneOsUnOOJdsuOHMw2bJOeckDDcQ0gIKMYkPZXby8u3JBz+8KbORJ2gLww04i7EBSEgQUo+iZRSMglEwCkYBDgAAqCQ+12kUPEIAAAAASUVORK5CYII=","orcid":"","institution":"Central South University","correspondingAuthor":true,"prefix":"","firstName":"Qingping","middleName":"","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2025-09-11 02:23:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7586896/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7586896/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91986144,"identity":"aff70df1-a041-4d62-a219-6273bb44b781","added_by":"auto","created_at":"2025-09-23 11:56:37","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5589,"visible":true,"origin":"","legend":"","description":"","filename":"0288625c1097472faea12835697c4438.json","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/8754c64ffeb7061ddcdadc3b.json"},{"id":91985037,"identity":"4fd0b4d4-9bd2-4bac-8833-03fb0b63f625","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"xml","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":113717,"visible":true,"origin":"","legend":"","description":"","filename":"0288625c1097472faea12835697c44381enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/c5abb2480b01a0b95ad2bf25.xml"},{"id":91984986,"identity":"e787d5c3-26be-49ac-9e21-0084f1534aba","added_by":"auto","created_at":"2025-09-23 11:48:36","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":37284,"visible":true,"origin":"","legend":"","description":"","filename":"CoverLetter2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/4ac97c59fb1804627ef4c0e5.pdf"},{"id":91985056,"identity":"571fa71d-97e7-4f47-8cdd-a4fa48394998","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"png","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":63947,"visible":true,"origin":"","legend":"","description":"","filename":"SSIM.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/d80dcb32707966358144d62d.png"},{"id":91985067,"identity":"f8d330d1-6788-4f47-8fbd-d7b912b79b0a","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"pdf","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3581769,"visible":true,"origin":"","legend":"","description":"","filename":"SplitGibbssamplerwithimplicitdiffusionpriorwithapplicationtoelectricalimpedancetomography2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/eaff182cedc11c2952600ed8.pdf"},{"id":91985009,"identity":"48aa6ada-92e6-4236-914e-1087713f2fec","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"png","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":351800,"visible":true,"origin":"","legend":"","description":"","filename":"interval.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/103408b9a1f77bc0add96111.png"},{"id":91985052,"identity":"5a9ebdf2-964e-4fb3-89a1-a00066a1eb0d","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"png","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":904409,"visible":true,"origin":"","legend":"","description":"","filename":"mean.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/405886a2e0e2481f038e0674.png"},{"id":91985064,"identity":"b82a60ab-d850-4130-ac4f-fb3aa5438051","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"png","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":486855,"visible":true,"origin":"","legend":"","description":"","filename":"mean1.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/1e8c43b37e5a8b42a6919038.png"},{"id":91985044,"identity":"4e58d4c9-68df-4cae-aa26-ddc64b051139","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"png","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":669475,"visible":true,"origin":"","legend":"","description":"","filename":"mean2.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/96dbf5ee283f5d77eeb0ac90.png"},{"id":91985062,"identity":"39912857-db0e-4700-87fe-5abd8d24b9fd","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"png","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":106462,"visible":true,"origin":"","legend":"","description":"","filename":"msessim.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/d358ca596d480743b58e73dc.png"},{"id":91985041,"identity":"346ae828-7fdb-4a98-a306-fe66ed3f65b4","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"png","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":302429,"visible":true,"origin":"","legend":"","description":"","filename":"obs.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/bff09b40af355b995befffa6.png"},{"id":91985006,"identity":"7f9bb892-71a4-475e-9442-8440082c6904","added_by":"auto","created_at":"2025-09-23 11:48:36","extension":"bst","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":146013,"visible":true,"origin":"","legend":"","description":"","filename":"snapacite.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/f5acc12a7a95f406c1460474.bst"},{"id":91986150,"identity":"307fb6c6-eefb-4d97-8e97-ca19987384fd","added_by":"auto","created_at":"2025-09-23 11:56:39","extension":"bst","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":29828,"visible":true,"origin":"","legend":"","description":"","filename":"snaps.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/e47a0d22067ae78e0171b2c5.bst"},{"id":91985035,"identity":"b979f33a-aa6a-4c74-a508-28482c961991","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"bst","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":35515,"visible":true,"origin":"","legend":"","description":"","filename":"snbasic.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/f218a8201433059276a0db7b.bst"},{"id":91985012,"identity":"b1a3e922-5f8e-48f7-9864-5218a560f01f","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"bst","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":33968,"visible":true,"origin":"","legend":"","description":"","filename":"snchicago.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/d2b69ca9e249e889dee177d3.bst"},{"id":91985007,"identity":"03c1b269-38fd-4e70-9871-7631a8e7dd62","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"cls","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":55857,"visible":true,"origin":"","legend":"","description":"","filename":"snjnl.cls","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/b8a9cce7c23c311cb9e760c5.cls"},{"id":91985063,"identity":"3704dce7-e59d-4dae-8c72-184a9394dace","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"bst","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64023,"visible":true,"origin":"","legend":"","description":"","filename":"snmathphysay.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/38283df4d1ca308d1d129c0e.bst"},{"id":91985013,"identity":"458a8dd3-fb8f-4ac8-af02-79f1c8fe4636","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"bst","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64166,"visible":true,"origin":"","legend":"","description":"","filename":"snmathphysnum.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/3cd83c2e85e565512ae951ea.bst"},{"id":91985011,"identity":"7d78fc99-8bf3-48a6-a7f6-a23c35be02c7","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"bst","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":37333,"visible":true,"origin":"","legend":"","description":"","filename":"snnature.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/39fb1c82afa73f8e277a5cd7.bst"},{"id":91985055,"identity":"60f472d0-d9cf-4d0a-8947-442fecdd4ad5","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"bst","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39951,"visible":true,"origin":"","legend":"","description":"","filename":"snvancouveray.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/f9ae86af324905bf3b214441.bst"},{"id":91985014,"identity":"9c06bd23-b5d1-4884-971d-bd5755d9e930","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"bst","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":40758,"visible":true,"origin":"","legend":"","description":"","filename":"snvancouvernum.bst","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/83dd09d46ce090bbc8153527.bst"},{"id":91985040,"identity":"806090df-eb22-4301-8efb-2f3783e52bfd","added_by":"auto","created_at":"2025-09-23 11:48:38","extension":"png","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":622570,"visible":true,"origin":"","legend":"","description":"","filename":"std1.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/38b53812f28ce985e27aefb5.png"},{"id":91985032,"identity":"597864c6-fba7-4c15-b79f-227a3a72c77b","added_by":"auto","created_at":"2025-09-23 11:48:37","extension":"png","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":910918,"visible":true,"origin":"","legend":"","description":"","filename":"std2.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/3b81326433a6dac98c161855.png"},{"id":91986151,"identity":"13a4ac7f-2564-4a4a-af1c-b83f63ec4f84","added_by":"auto","created_at":"2025-09-23 11:56:39","extension":"png","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":786949,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinemean.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/6b42e9eb08b321c922429fbf.png"},{"id":91985058,"identity":"e248bef1-9933-46dd-969d-b0435d04e5ae","added_by":"auto","created_at":"2025-09-23 11:48:39","extension":"png","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":110893,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinemsessim.png","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/e31df5ec269632bec67e43f0.png"},{"id":91984984,"identity":"f272ba57-d0f6-43ea-8e6c-76fd4d08f5d8","added_by":"auto","created_at":"2025-09-23 11:48:36","extension":"xml","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":123574,"visible":true,"origin":"","legend":"","description":"","filename":"0288625c1097472faea12835697c44381structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/f4996c91d231490270ba3c58.xml"},{"id":91986145,"identity":"da8e99c6-d918-471d-be58-710bcf3f7f5d","added_by":"auto","created_at":"2025-09-23 11:56:37","extension":"html","order_by":33,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":141048,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1/389ad3859f0e3dfee51da4f6.html"},{"id":100548856,"identity":"078c6f38-b0a8-47c1-9066-b8b0f61a19da","added_by":"auto","created_at":"2026-01-19 08:21:14","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1687951,"visible":true,"origin":"","legend":"","description":"","filename":"SplitGibbssamplerwithimplicitdiffusionpriorwithapplicationtoelectricalimpedancetomography2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7586896/v1_covered_e5360a4f-6357-4887-8e70-38f9f9e95307.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"inverse problems, diffusion models, split Gibbs sampler, electrical impedance tomography","lastPublishedDoi":"10.21203/rs.3.rs-7586896/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7586896/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDiffusion models have become particularly prominent for generating high-quality samples. Consequently, they have become widely adopted for modeling priors in image reconstruction. However, integrating diffusion priors into nonlinear inverse problems presents significant computational challenges, primarily due to the inherent complexity of nonlinear forward operators. This work presents a new approach for incorporating a diffusion prior into the split Gibbs sampler, denoted as DP-SGS, to sample from the posterior distribution in nonlinear settings. DP-SGS simplifies the complex posterior sampling problem by introducing an auxiliary variable that decomposes it into two simple conditional distributions: a forward model that handles the likelihood term and a prior model implicitly determined by the pre-trained diffusion model.Both conditional distributions are efficiently approximated as Gaussian distributions using first-order Taylor expansion for the prior model and Laplace approximation for the forward model, thereby ensuring computational tractability despite the nonlinear forward operator.Experiments on the challenging electrical impedance tomography (EIT) imaging problem demonstrate that our method enhances solution accuracy while achieving rapid convergence rates, confirming its effectiveness for solving nonlinear inverse problems.\u003c/p\u003e","manuscriptTitle":"Split Gibbs diffusion posterior sampling for nonlinear inverse problems with application to electrical impedance tomography","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-23 11:48:05","doi":"10.21203/rs.3.rs-7586896/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0972ded8-d719-4b07-b4fe-97bf93048e77","owner":[],"postedDate":"September 23rd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-18T00:08:27+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-23 11:48:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7586896","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7586896","identity":"rs-7586896","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}