[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124016-en":3,"doc-seo-124016-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124016,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems - An Overview","This review surveys current strategies for solving imaging inverse problems using variational methods and machine learning. Emphasis is placed on point estimators and their robustness to adversarial perturbations, supported by numerical experiments on a one-dimensional toy problem that empirically confirm theoretical guarantees. A further theme is exploring the subspace of data-consistent solutions via explicit guidance to enforce semantic or textural properties, bridging stability, robustness, and controllable solution spaces.","arXiv :2402 . 12072v2 [ ee ss .IV] 9 Jul 2024  \nDOI: xxx/xxxx  \nREVIEW ARTICLE  \nRobustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview  \nAlexander Auras*†1 | Kanchana Vaishnavi Gandikota†1 | Hannah Droege2 | Michael Moeller 1 † These authors contributed equally  \n1Institute for Vision and Graphics, University of Siegen, NRW, Germany  \n2Institute of Computer Science, Rheinische Friedrich-Wilhelms-Universität Bonn, NRW, Germany  \nCorrespondence  \n*Alexander Auras,  \nInstitute for Vision and Graphics, University of Siegen,  \nAdolf-Reichwein-Straße 2a, 57076 Siegen,  \nGermany.  \nEmail: alexander.auras@uni-siegen.de  \nAbstract  \nThis paper provides an overview of current approaches for solving inverse problems in imaging using variational methods and machine learning. A special focus lies on point estimators and their robustness against adversarial perturbations. In this context results of numerical experiments for a one-dimensional toy problem are provided, showing the robustness of different approaches and empirically verifying theoretical guarantees. Another focus of this review is the exploration of the subspace of dataconsistent solutions through explicit guidance to satisfy specific semantic or textural properties.  \nKEYWORDS:  \nInverse Problems; Machine Learning; Robustness; Explorability;  \n1  INTRODUCTION  \nThe goal of image reconstruction is to recover an unknown image from indirect or distorted measurements, i.e., to recover aground truth image 􀁵 from measurements  \n􀁦 = 􀁁 (􀁵) + 􀁮 . (1)  \nfor a forward operator 􀁦 and (additive) noise 􀁮. When the forward measurement process is linear, recovering 􀁵 becomes a linear inverse problem, which is what we focus on in this paper. Simple approaches compute reconstructions 􀁵 for (1) linearly via least-squares estimates, possibly including an additional smoothing or regularization. Examples of this approach include filtered back projection [58] for tomographic image reconstruction, and Wiener filtering for image deconvolution, which incorporates regularization through linear filtering in Fourier space. Variational approaches (c.f. [21]) to such problems find the minimizer of a suitable cost function, typically consisting of a data fidelity term 􀁅 (􀁁, 􀁵, 􀁦 ) that measures the discrepancy from the observation model (1) and a regularizer 􀁒(􀁵) that incorporates prior knowledge about the image to be recovered,  \n􀁵̂ in 􀁅(􀁁, 􀁵, 􀁦 ) + 􀁒(􀁵) . (2)  \nAt least in finite dimensions, the above perspective relates to the Bayesian approach to inverse problems (see e.g. [50]), via the concept of maximum a-posteriori probability (MAP) estimates, if the regularizer (with regularization strength 􀀋) in the form exp(−􀀋􀁒(􀁵)) isintegrable w.r.t. 􀁵. By modeling the unknown image 􀁵 and the measurements 􀁦 as realizations of random variables with respective distributions 􀁰(􀁵) and 􀁰(􀁦 ), and computing the MAP estimate as the argument that maximizes the posterior distribution 􀁰(􀁵|􀁦 ), the application of Bayes law yields (2) with 􀁅 (􀁁, 􀁵, 􀁦 ) = − log 􀁰(􀁦 |􀁵) and 􀁒(􀁵) = − log(􀁰(􀁵)) . While (2)  \nis a point estimate, the Bayesian approach to inverse problems inverse models or learns the entire posterior probability 􀁰(􀁵|􀁦 ) and/or sampling schemes for it. Though their notion of solutions is different, both approaches consider the inverse problem tobe well-posed if a unique solution exists and depends on the data continuously.  \nClassical approaches have thoroughly analyzed ill-posed problems and a large body of work provides stability and convergence guarantees, [e.g. by](e.g. by) selecting (noise-level-dependent) regularizers with suitable properties in (2). Yet, these regularizers are typically not expressive enough to model the distribution of natural/realistic images faithfully. In the past decade, deep learning has achieved remarkable success in image reconstruction through the ability to capture data-dependent structures, showing great improvements in reconstr","cbCaiiOPUJHkgFTR","https://ap.wps.com/l/cbCaiiOPUJHkgFTR","pdf",2519207,1,28,"English","en",105,"# Introduction\n## Overview of deep learning for inverse problems in imaging\n## Deep learning for point estimates to inverse problems\n## Stochastic sampling from the posterior","[{\"question\":\"What inverse-problem setting does the paper focus on?\",\"answer\":\"It focuses on image reconstruction from indirect or distorted measurements, formulating the recovery as a (linear) inverse problem under a forward operator and additive noise.\"},{\"question\":\"How does the review address robustness in inverse imaging?\",\"answer\":\"It centers on point estimators and studies their robustness against adversarial perturbations, including experiments on a one-dimensional toy problem that verify theoretical guarantees.\"},{\"question\":\"What does the paper mean by explorability of solutions?\",\"answer\":\"It explores the subspace of data-consistent solutions by providing explicit guidance to satisfy specific semantic or textural properties.\"}]","Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems - An Overview | PDF",1785819858,71,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"robustness-and-exploration-of-variational-and-machine-learning-approaches-to-inverse-problems-an-overview","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/robustness-and-exploration-of-variational-and-machine-learning-approaches-to-inverse-problems-an-overview/124016/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What inverse-problem setting does the paper focus on?","Question",{"text":75,"@type":76},"It focuses on image reconstruction from indirect or distorted measurements, formulating the recovery as a (linear) inverse problem under a forward operator and additive noise.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the review address robustness in inverse imaging?",{"text":80,"@type":76},"It centers on point estimators and studies their robustness against adversarial perturbations, including experiments on a one-dimensional toy problem that verify theoretical guarantees.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the paper mean by explorability of solutions?",{"text":84,"@type":76},"It explores the subspace of data-consistent solutions by providing explicit guidance to satisfy specific semantic or textural properties.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]