{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T15:32:51Z","timestamp":1786116771640,"version":"build-2736575974"},"reference-count":34,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110443","type":"journal-article","created":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T23:24:58Z","timestamp":1782343498000},"page":"110443","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P1","title":["An innovative deep learning approach to shadow removal with anisotropic PDEs"],"prefix":"10.1016","volume":"163","author":[{"given":"Fakhr-eddine","family":"LIMAMI","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4851-3617","authenticated-orcid":false,"given":"Amine","family":"LAGHRIB","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aissam","family":"HADRI","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lekbir","family":"AFRAITES","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"10","key":"10.1016\/j.cnsns.2026.110443_bib0001","doi-asserted-by":"crossref","first-page":"13143","DOI":"10.1109\/TNNLS.2023.3278866","article-title":"Hyperspectral image denoising: from model-driven, data-driven, to model-data-driven","volume":"35","author":"Zhang","year":"2023","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"10.1016\/j.cnsns.2026.110443_bib0002","doi-asserted-by":"crossref","first-page":"6356","DOI":"10.1109\/TIP.2022.3211471","article-title":"Cooperated spectral low-rankness prior and deep spatial prior for HSI unsupervised denoising","volume":"31","author":"Zhang","year":"2022","journal-title":"IEEE Trans Image Process"},{"key":"10.1016\/j.cnsns.2026.110443_bib0003","series-title":"Advances in neural information processing systems","article-title":"Imagenet classification with deep convolutional neural networks","volume":"vol. 25","author":"Krizhevsky","year":"2012"},{"key":"10.1016\/j.cnsns.2026.110443_bib0004","series-title":"International conference on medical image computing and computer-assisted intervention","first-page":"234","article-title":"U-net: convolutional networks for biomedical image segmentation","author":"Ronneberger","year":"2015"},{"key":"10.1016\/j.cnsns.2026.110443_bib0005","doi-asserted-by":"crossref","first-page":"4788","DOI":"10.1109\/TIP.2021.3074804","article-title":"Deraincyclegan: rain attentive cyclegan for single image deraining and rainmaking","volume":"30","author":"Wei","year":"2021","journal-title":"IEEE Trans Image Process"},{"issue":"11","key":"10.1016\/j.cnsns.2026.110443_bib0006","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1145\/3422622","article-title":"Generative adversarial networks","volume":"63","author":"Goodfellow","year":"2020","journal-title":"Commun ACM"},{"key":"10.1016\/j.cnsns.2026.110443_bib0007","series-title":"Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition","first-page":"14049","article-title":"Shadowdiffusion: when degradation prior meets diffusion model for shadow removal","author":"Guo","year":"2023"},{"key":"10.1016\/j.cnsns.2026.110443_bib0008","series-title":"Proceedings of the IEEE conference on computer vision and pattern recognition","first-page":"4067","article-title":"Deshadownet: a multi-context embedding deep network for shadow removal","author":"Qu","year":"2017"},{"key":"10.1016\/j.cnsns.2026.110443_bib0009","series-title":"Proceedings of the IEEE\/CVF international conference on computer vision","first-page":"10213","article-title":"Argan: attentive recurrent generative adversarial network for shadow detection and removal","author":"Ding","year":"2019"},{"key":"10.1016\/j.cnsns.2026.110443_bib0010","unstructured":"Dosovitskiy A., Beyer L., Kolesnikov A., Weissenborn D., Zhai X., Unterthiner T., et al. An image is worth 16x16 words: transformers for image recognition at scale. 2020. https:\/\/arxiv.org\/abs\/2010.11929."},{"issue":"3","key":"10.1016\/j.cnsns.2026.110443_bib0011","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s00138-024-01532-4","article-title":"Tensor-guided learning for image denoising using anisotropic PDEs","volume":"35","author":"Limami","year":"2024","journal-title":"Mach Vis Appl"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110443_bib0012","first-page":"424","article-title":"Fractional optimal control for deep convolutional neural networks exploring ODE-based solutions for image denoising","volume":"19","author":"Limami","year":"2025","journal-title":"Inverse Probl Imaging"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110443_bib0013","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2025.108315","article-title":"Enhanced shadow removal using space-variant anisotropic PDEs and tensor-based osmosis models","volume":"363","author":"Limami","year":"2026","journal-title":"J Frankl Inst"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110443_bib0014","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1007\/s11760-020-01831-z","article-title":"A novel image denoising approach based on a non-convex constrained PDE: application to ultrasound images","volume":"15","author":"Hadri","year":"2021","journal-title":"Signal Image Video Process"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110443_bib0015","doi-asserted-by":"crossref","DOI":"10.1088\/1361-6420\/ad0b26","article-title":"Bilevel optimal parameter learning for a high-order nonlocal multiframe super-resolution problem","volume":"40","author":"Laghrib","year":"2023","journal-title":"Inverse Probl"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110443_bib0016","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1002\/cpa.20075","article-title":"Variational image inpainting","volume":"58","author":"Chan","year":"2005","journal-title":"Commun Pure Appl Math: J Issued Courant Inst Math Sci"},{"issue":"11","key":"10.1016\/j.cnsns.2026.110443_bib0017","doi-asserted-by":"crossref","first-page":"2127","DOI":"10.1109\/TMI.2013.2274734","article-title":"Interactive medical image segmentation using PDE control of active contours","volume":"32","author":"Karasev","year":"2013","journal-title":"IEEE Trans Med Imaging"},{"key":"10.1016\/j.cnsns.2026.110443_bib0018","series-title":"International conference on scale space and variational methods in computer vision","first-page":"532","article-title":"Novel schemes for hyperbolic pdes using osmosis filters from visual computing","author":"Hagenburg","year":"2011"},{"key":"10.1016\/j.cnsns.2026.110443_bib0019","first-page":"1","article-title":"Three-dimension spatial-spectral attention transformer for hyperspectral image denoising","volume":"62","author":"Zhang","year":"2024","journal-title":"IEEE Trans Geosci Remote Sens"},{"key":"10.1016\/j.cnsns.2026.110443_bib0020","series-title":"International workshop on energy minimization methods in computer vision and pattern recognition","first-page":"26","article-title":"Linear osmosis models for visual computing","author":"Weickert","year":"2013"},{"key":"10.1016\/j.cnsns.2026.110443_bib0021","series-title":"International conference on scale space and variational methods in computer vision","first-page":"368","article-title":"A fully discrete theory for linear osmosis filtering","author":"Vogel","year":"2013"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110443_bib0022","doi-asserted-by":"crossref","DOI":"10.1088\/1361-6420\/ab08d2","article-title":"Anisotropic osmosis filtering for shadow removal in images","volume":"35","author":"Parisotto","year":"2019","journal-title":"Inverse Probl"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110443_bib0023","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s10851-013-0446-3","article-title":"Sparse non-negative stencils for anisotropic diffusion","volume":"49","author":"Fehrenbach","year":"2014","journal-title":"J Math Imaging Vis"},{"key":"10.1016\/j.cnsns.2026.110443_bib0024","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.camwa.2022.12.015","article-title":"Fast and stable schemes for non-linear osmosis filtering","volume":"133","author":"Calatroni","year":"2023","journal-title":"Comput Math Appl"},{"key":"10.1016\/j.cnsns.2026.110443_bib0025","unstructured":"Kingma D.P., Ba J., Adam: a method for stochastic optimization. 2014. https:\/\/arxiv.org\/abs\/1412.6980."},{"issue":"2","key":"10.1016\/j.cnsns.2026.110443_bib0026","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.acha.2007.05.004","article-title":"Locally analytic schemes: a link between diffusion filtering and wavelet shrinkage","volume":"24","author":"Welk","year":"2008","journal-title":"Appl Comput Harmon Anal"},{"key":"10.1016\/j.cnsns.2026.110443_bib0027","unstructured":"Schrader K., Weickert J., Krause M.. Anisotropic diffusion stencils: from simple derivations over stability estimates to resnet implementations. 2023. arXiv: 2309.05575."},{"key":"10.1016\/j.cnsns.2026.110443_bib0028","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s10851-022-01106-x","article-title":"Connections between numerical algorithms for PDEs and neural networks","volume":"65","author":"Alt","year":"2023","journal-title":"J Math Imaging Vis"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110443_bib0029","first-page":"1","article-title":"Exploiting residual and illumination with GANS\u2019s for shadow detection and shadow removal","volume":"19","author":"Zhang","year":"2023","journal-title":"ACM Trans Multimed Comput Commun Appl"},{"key":"10.1016\/j.cnsns.2026.110443_bib0030","doi-asserted-by":"crossref","DOI":"10.1016\/j.patcog.2024.111126","article-title":"Image shadow removal via multi-scale deep retinex decomposition","volume":"159","author":"Huang","year":"2025","journal-title":"Pattern Recognit"},{"key":"10.1016\/j.cnsns.2026.110443_bib0031","series-title":"TSPLNet: a three-stage progressive lightweight network for shadow removal","volume":"vol. 31","author":"Hu","year":"2025"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110443_bib0032","first-page":"424","article-title":"Fractional optimal control for deep convolutional neural networks exploring ODE-based solutions for image denoising","volume":"19","author":"Limami","year":"2025","journal-title":"Inverse Probl Imaging"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110443_bib0033","first-page":"2956","article-title":"Paired regions for shadow detection and removal","volume":"35","author":"Ruiqi","year":"2012","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"10.1016\/j.cnsns.2026.110443_bib0034","series-title":"Quality & quantity","first-page":"1","article-title":"Image discrimination robustness of classical image quality metrics: an analysis on MAE, MSE, ERGAS, LMSE, SSIM, SAM, UIQI, PSNR, NIQE, PIQE, SNR, VIF, FSIM, GMSD, and NCC","author":"Civicioglu","year":"2025"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426007975?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426007975?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T12:21:40Z","timestamp":1786018900000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426007975"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":34,"alternative-id":["S1007570426007975"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110443","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"An innovative deep learning approach to shadow removal with anisotropic PDEs","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110443","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110443"}}