[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82711-en":3,"doc-seo-82711-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82711,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Directionally Weighted Total Variation for Inverse Problems","Weighted total variation (TV) regularization is studied for inverse problems where the forward operator has a large null space, a regime where standard unweighted TV yields systematic reconstruction artifacts and spatial bias. Spatially varying weights are proposed from sensitivity analysis of the forward operator via an associated Green’s function, rebalancing the TV penalty to match inhomogeneous sensitivity. Two models are analyzed: an isotropic worst-case directional weighted TV and a refinement using the solution’s actual jump direction. Optimality conditions and exact recovery results are proved under structural assumptions, and numerical experiments show improved artifact reduction and better localization and size recovery, highlighting the importance of spatial weighting.","arXiv :2607 .03054v1 [math .NA] 3 Jul 2026  \nDirectionally Weighted Total Variation for Inverse Problems  \nOle Løseth Elvetun∗and Bjørn Fredrik Nielsen†  \nJuly 7, 2026  \nAbstract  \nWe study weighted total variation (TV) regularization for inverse problems in which the forward operator has a large null space, a setting in which standard (unweighted) TV is known to produce systematic reconstruction artifacts such as spatial bias. To address this issue, we consider spatially varying weights derived from a sensitivity analysis of the forward operator expressed via an associated Green’s function. These weights rebalance the TV penalty and thereby compensate for the inhomogeneous sensitivity of the forward operator.  \nWe introduce and analyze two related models: an isotropic weighted TV functional, where the weight reflects worst-case directional sensitivity, and a directionally weighted refinement in which the weighting depends on the actual jump direction of the solution. In addition to the standard Tikhonov formulation, we also study corresponding basis-pursuit problems. We derive optimality conditions and prove exact recovery results under suitable structural assumptions. Furthermore, we investigate how recoverability depends on the geometry of the solution and its distance to the observation boundary.  \nNumerical experiments for inverse source problems and related applications demonstrate that the proposed weighted formulations significantly reduce artifacts present in unweighted TV reconstructions, yielding improved localization and size recovery. These results highlight the crucial role of spatial weighting in TV regularization for inverse problems with large null spaces.  \n1 Introduction  \nIn this paper we are concerned with inverse problems for which the forward operator K has a significant null space. More precisely, we assume that K :  \n∗ Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 ˚As, Norway. Email: [ole.elvetun@nmbu.no](ole.elvetun@nmbu.no).  \n†Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, NO-1432 ˚As, Norway. Email: [bjorn.f.nielsen@nmbu.no](bjorn.f.nielsen@nmbu.no).  \nL2 (Ω) → L2 (∂Ω) is a linear operator and consider the weighted problem  \n􀀸 􀀹  \n􀀾 􀀾  \n􀀾 􀀾  \nf∈n(Ω)  12 ∥Kf − d∥2L2 (∂Ω) + α Ω w (y) | Df | (y}) +β Z∂Ω w∂ (y)|f(y)|ds(y)  .  \n􀀾 {z 􀀾  \n􀀺 =TVw (f) 􀀻  \n(1)  \nHere, d ∈ L2 (∂Ω) is the measured data, Ω ⊂ R2 is a bounded Lipschitz domain,α > 0 is a regularization parameter, Df is the vector-valued Radon measure representation of the ”gradient” of f, and w is the weight function  \nw (y) = sup ∥K(∇G(·;y) · d)∥Lp (∂Ω) , (2)  \nd , |d| 2 =1  \nwhere G is a suitable Green’s function which we define below. The presence of the boundary term, i.e., the term involving the parameter β, is motivated by the following observation: Without this term, it becomes ”cheap” to obtain a good data fit with a source which is (approximately) constant close to ∂Ω since the total variation of a constant function is zero. Hence, a boundary-bias occurs. We will return to this issue below and explain how it is linked to the choice of boundary conditions to use in the definition of the Green’s function G.  \nOperators that map interior distributions to boundary observations, i.e. , K : L2 (Ω) → L2 (∂Ω), arise naturally in inverse source problems for elliptic partial differential equations, see [22, 16] and the references therein. Concrete examples include source identification in groundwater modelling, the inverse problem of electrocardiography [24], and certain regimes of electrical impedance tomography [5] . A defining feature of such operators is that the dimension of the null space is typically infinite: many distinct interior sources produce identical boundary data. Stable reconstruction therefore requires regularization, and the regularization functional plays a particularly active role since it must determine the component of the solution th","cbCaijERrlj2C2p4","https://ap.wps.com/l/cbCaijERrlj2C2p4","pdf",3441622,2,1,25,"English","en",105,"# Abstract\n# Introduction\n## Weighted TV formulation\n## Operators with large null spaces and artifacts\n## Related work and paper contributions","[{\"question\":\"Why does unweighted total variation produce artifacts when the forward operator has a large null space?\",\"answer\":\"Because the data-fidelity term decreases more strongly in highly sensitive regions than in nearly blind regions. This makes the TV penalty effectively cheaper in sensitive areas, shifting recovered features and causing spatially dependent intensity changes or missing reconstructions.\"},{\"question\":\"How are the spatially varying TV weights constructed?\",\"answer\":\"Weights are derived from a sensitivity analysis of the forward operator expressed through an associated Green’s function. The weight at a point is defined using a supremum involving the action of K on directional perturbations of the Green’s function.\"},{\"question\":\"What are the two weighted TV models proposed in the paper?\",\"answer\":\"One uses isotropic weighting based on worst-case directional sensitivity. The other refines this by weighting according to the actual jump direction of the solution, rather than the worst-case over directions.\"}]",1784182436,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"directionally-weighted-total-variation-for-inverse-problems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/directionally-weighted-total-variation-for-inverse-problems/82711/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why does unweighted total variation produce artifacts when the forward operator has a large null space?","Question",{"text":75,"@type":76},"Because the data-fidelity term decreases more strongly in highly sensitive regions than in nearly blind regions. This makes the TV penalty effectively cheaper in sensitive areas, shifting recovered features and causing spatially dependent intensity changes or missing reconstructions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the spatially varying TV weights constructed?",{"text":80,"@type":76},"Weights are derived from a sensitivity analysis of the forward operator expressed through an associated Green’s function. The weight at a point is defined using a supremum involving the action of K on directional perturbations of the Green’s function.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two weighted TV models proposed in the paper?",{"text":84,"@type":76},"One uses isotropic weighting based on worst-case directional sensitivity. The other refines this by weighting according to the actual jump direction of the solution, rather than the worst-case over directions.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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"]