{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:37:56Z","timestamp":1759333076382,"version":"3.37.3"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,3,3]],"date-time":"2021-03-03T00:00:00Z","timestamp":1614729600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,3,3]],"date-time":"2021-03-03T00:00:00Z","timestamp":1614729600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s40314-021-01462-1","type":"journal-article","created":{"date-parts":[[2021,3,3]],"date-time":"2021-03-03T19:15:08Z","timestamp":1614798908000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A nonlinear fourth-order PDE for image denoising in Sobolev spaces with variable exponents and its numerical algorithm"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3301-1385","authenticated-orcid":false,"given":"Hamdi","family":"Houichet","sequence":"first","affiliation":[]},{"given":"Anis","family":"Theljani","sequence":"additional","affiliation":[]},{"given":"Maher","family":"Moakher","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,3]]},"reference":[{"issue":"4","key":"1462_CR1","doi-asserted-by":"publisher","first-page":"1120","DOI":"10.1051\/cocv\/2014061","volume":"21","author":"G Aubert","year":"2015","unstructured":"Aubert G, Drogoul A (2015) Topological gradient for a fourth order operator used in image analysis. ESAIM Control Optim Calc Variat 21(4):1120\u20131149","journal-title":"ESAIM Control Optim Calc Variat"},{"issue":"11","key":"1462_CR2","doi-asserted-by":"publisher","first-page":"2191","DOI":"10.1016\/j.mcm.2008.07.002","volume":"49","author":"D Auroux","year":"2009","unstructured":"Auroux D (2009) From restoration by topological gradient to medical image segmentation via an asymptotic expansion. Math Comput Model 49(11):2191\u20132205","journal-title":"Math Comput Model"},{"issue":"2\u20133","key":"1462_CR3","first-page":"251","volume":"25","author":"D Auroux","year":"2006","unstructured":"Auroux D, Masmoudi M (2006) A one-shot inpainting algorithm based on the topological asymptotic analysis. Computat Appl Math 25(2\u20133):251\u2013267","journal-title":"Computat Appl Math"},{"issue":"2","key":"1462_CR4","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1007\/s10851-008-0121-2","volume":"33","author":"D Auroux","year":"2008","unstructured":"Auroux D, Masmoudi M (2008) Image processing by topological asymptotic expansion. J Math Imaging Vis 33(2):122","journal-title":"J Math Imaging Vis"},{"issue":"2","key":"1462_CR5","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1016\/j.aeue.2014.10.022","volume":"69","author":"A Baghaie","year":"2015","unstructured":"Baghaie A, Zeyun Y (2015) Structure tensor based image interpolation method. AEU Int J Elect Commun 69(2):515\u2013522","journal-title":"AEU Int J Elect Commun"},{"issue":"1","key":"1462_CR6","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1002\/mrm.20334","volume":"53","author":"PG Batchelor","year":"2005","unstructured":"Batchelor PG, Moakher M, Atkinson D, Calamante C, Connelly A (2005) A rigorous framework for diffusion tensor calculus. Magn Reson Med 53(1):221\u2013225","journal-title":"Magn Reson Med"},{"issue":"1","key":"1462_CR7","doi-asserted-by":"publisher","first-page":"253","DOI":"10.5269\/bspm.v34i1.25626","volume":"34","author":"K Ben\u00a0Haddouch","year":"2016","unstructured":"Ben\u00a0Haddouch K, El\u00a0Allali Z, Tsouli N, El\u00a0Habib S, Kissi F (2016) Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions. Bol Soc Paran Math 34(1):253\u2013272","journal-title":"Bol Soc Paran Math"},{"issue":"2","key":"1462_CR8","doi-asserted-by":"publisher","first-page":"413","DOI":"10.4310\/CMS.2011.v9.n2.a4","volume":"9","author":"A Bertozzi","year":"2011","unstructured":"Bertozzi A, Sch\u00f6nlieb C-B (2011) Unconditionally stable schemes for higher order inpainting. Commun Math Sci 9(2):413\u2013457","journal-title":"Commun Math Sci"},{"issue":"3","key":"1462_CR9","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/090769521","volume":"3","author":"K Bredies","year":"2010","unstructured":"Bredies K, Kunisch K, Pock T (2010) Total generalized variation. SIAM J Imaging Sci 3(3):492\u2013526","journal-title":"SIAM J Imaging Sci"},{"issue":"2","key":"1462_CR10","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1137\/S1064827598344169","volume":"22","author":"T Chan","year":"2000","unstructured":"Chan T, Marquina A, Mulet P (2000) High-order total variation-based image restoration. SIAM J Sci Comput 22(2):503\u2013516","journal-title":"SIAM J Sci Comput"},{"issue":"04","key":"1462_CR11","doi-asserted-by":"publisher","first-page":"1350012","DOI":"10.1142\/S0219530513500127","volume":"11","author":"PG Ciarlet","year":"2013","unstructured":"Ciarlet PG, Dinca G, Matei P (2013) Fr\u00e9chet differentiability of the norm in a sobolev space with a variable exponent. Anal Appl 11(04):1350012","journal-title":"Anal Appl"},{"key":"1462_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18363-8","volume-title":"Lebesgue and Sobolev spaces with variable exponents","author":"L Diening","year":"2011","unstructured":"Diening L, Harjulehto P, H\u00e4st\u00f6 P, Ruzicka M (2011) Lebesgue and Sobolev spaces with variable exponents. Springer, Berlin"},{"issue":"4","key":"1462_CR13","doi-asserted-by":"publisher","first-page":"2700","DOI":"10.1137\/140967374","volume":"7","author":"A Drogoul","year":"2016","unstructured":"Drogoul A (2016) Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in imaging. SIAM J Imaging Sci 7(4):2700\u20132731","journal-title":"SIAM J Imaging Sci"},{"issue":"3","key":"1462_CR14","doi-asserted-by":"publisher","first-page":"267","DOI":"10.4064\/sm-143-3-267-293","volume":"143","author":"DE Edmunds","year":"2000","unstructured":"Edmunds DE, R\u00e1kosn\u00edk J (2000) Sobolev embeddings with variable exponent. Stud Math 143(3):267\u2013293","journal-title":"Stud Math"},{"issue":"1","key":"1462_CR15","doi-asserted-by":"publisher","first-page":"179","DOI":"10.5269\/bspm.v31i1.15148","volume":"31","author":"AR El Amrouss","year":"2012","unstructured":"El Amrouss AR, Ourraoui A (2012) Existence of solutions for a boundary problem involving p (x)-biharmonic operator. Boletim da sociedade paranaense de matem\u00e1tica 31(1):179\u2013192","journal-title":"Boletim da sociedade paranaense de matem\u00e1tica"},{"issue":"153","key":"1462_CR16","first-page":"1","volume":"2009","author":"A El Amrouss","year":"2009","unstructured":"El Amrouss A, Moradi F, Moussaoui M (2009) Existence of solutions for fourth-order PDEs with variable exponents. Electron J Differ Equ 2009(153):1\u201313","journal-title":"Electron J Differ Equ"},{"issue":"5","key":"1462_CR17","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1051\/m2an\/2013074","volume":"47","author":"M Elsey","year":"2013","unstructured":"Elsey M, Wirth B (2013) A simple and efficient scheme for phase field crystal simulation. ESAIM Math Model Numer Anal 47(5):1413\u20131432","journal-title":"ESAIM Math Model Numer Anal"},{"unstructured":"Eyre DJ (1998) An unconditionally stable one-step scheme for gradient systems. Unpublished article","key":"1462_CR18"},{"issue":"2","key":"1462_CR19","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1006\/jmaa.2000.7617","volume":"263","author":"X Fan","year":"2001","unstructured":"Fan X, Zhao D (2001) On the spaces $${L}^{ p (x)}$$ and $${W}^{ m, p (x)}$$. J Math Anal Appl 263(2):424\u2013446","journal-title":"J Math Anal Appl"},{"doi-asserted-by":"crossref","unstructured":"Fillard P, Arsigny V, Ayache N, Pennec X (2005) A riemannian framework for the processing of tensor-valued images. In: International Workshop on Deep Structure, Singularities, and Computer Vision, pages 112\u2013123. Springer","key":"1462_CR20","DOI":"10.1007\/11577812_10"},{"issue":"3","key":"1462_CR21","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s00033-016-0668-5","volume":"67","author":"S Heidarkhani","year":"2016","unstructured":"Heidarkhani S, Afrouzi GA, Moradi S, Caristi G, Ge B (2016) Existence of one weak solution for $$p(x)$$-biharmonic equations with navier boundary conditions. Zeitschrift f\u00fcr angewandte Mathematik und Physik 67(3):73","journal-title":"Zeitschrift f\u00fcr angewandte Mathematik und Physik"},{"issue":"4","key":"1462_CR22","doi-asserted-by":"publisher","first-page":"421","DOI":"10.20852\/ntmsci.2019.383","volume":"7","author":"H Houichet","year":"2019","unstructured":"Houichet H, Moakher M, Rjaibi B (2019) Noise removal and edge detection in ultrasound images by the topological gradient method. New Trends Math Sci 7(4):421\u2013440","journal-title":"New Trends Math Sci"},{"issue":"4","key":"1462_CR23","doi-asserted-by":"publisher","first-page":"394","DOI":"10.4208\/jms.v52n4.19.03","volume":"52","author":"H Houichet","year":"2019","unstructured":"Houichet H, Theljani A, Rjaibi B, Moakher M (2019) A nonstandard higher-order variational model for speckle noise removal and thin-structure detection. J Math Study 52(4):394\u2013424","journal-title":"J Math Study"},{"unstructured":"Houichet H, Theljani A, Mohamed A (2019) Well-posedness of high-order non-standard parabolic partial differential equation for image denoising. Submitted, page\u00a025","key":"1462_CR24"},{"issue":"11","key":"1462_CR25","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1016\/j.enganabound.2008.01.004","volume":"32","author":"L Jaafar\u00a0Belaid","year":"2008","unstructured":"Jaafar\u00a0Belaid L, Jaoua M, Masmoudi M, Siala ML (2008) Application of the topological gradient to image restoration and edge detection. Eng Anal Bound Elements J 32(11):891\u2013899","journal-title":"Eng Anal Bound Elements J"},{"issue":"1","key":"1462_CR26","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1007\/s10851-010-0225-3","volume":"39","author":"Z Jin","year":"2011","unstructured":"Jin Z, Yang X (2011) A variational model to remove the multiplicative noise in ultrasound images. J Math Imaging Vis 39(1):62\u201374","journal-title":"J Math Imaging Vis"},{"issue":"1","key":"1462_CR27","doi-asserted-by":"publisher","first-page":"145","DOI":"10.4134\/BKMS.b150980","volume":"54","author":"J Kim","year":"2017","unstructured":"Kim J, Shin J (2017) An unconditionally gradient stable numerical method for the Ohta\u2013Kawasaki model. Bull Korean Math Soc 54(1):145\u2013158","journal-title":"Bull Korean Math Soc"},{"doi-asserted-by":"crossref","unstructured":"Krissian K, Kikinis R, Westin CF, Vosburgh K (2005) Speckle-constrained filtering of ultrasound images. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR\u201905), 2:547\u2013552","key":"1462_CR28","DOI":"10.1109\/CVPR.2005.331"},{"issue":"4\u20135","key":"1462_CR29","first-page":"777","volume":"26","author":"H Michael","year":"2011","unstructured":"Michael H, Michael H, Hicham TM (2011) An adaptive finite-element Moreau\u2013Yosida-based solver for a non-smooth Cahn\u2013Hilliard problem. Optim Methods Softw 26(4\u20135):777\u2013811","journal-title":"Optim Methods Softw"},{"issue":"12","key":"1462_CR30","doi-asserted-by":"publisher","first-page":"5220","DOI":"10.1109\/TIP.2015.2479471","volume":"24","author":"VBS Prasath","year":"2015","unstructured":"Prasath VBS, Vorotnikov D, Pelapur R, Jose S, Seetharaman G, Palaniappan Kannappan (2015) Multiscale tikhonov-total variation image restoration using spatially varying edge coherence exponent. IEEE Trans Image Process 24(12):5220\u20135235","journal-title":"IEEE Trans Image Process"},{"issue":"1","key":"1462_CR31","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","volume":"60","author":"LI Rudin","year":"1992","unstructured":"Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenom 60(1):259\u2013268","journal-title":"Physica D Nonlinear Phenom"},{"issue":"2","key":"1462_CR32","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1137\/1036054","volume":"36","author":"AM Stuart","year":"1994","unstructured":"Stuart AM, Humphries AR (1994) Model problems in numerical stability theory for initial value problems. SIAM Rev 36(2):226\u2013257","journal-title":"SIAM Rev"},{"issue":"3","key":"1462_CR33","first-page":"337","volume":"40","author":"VB Surya Prasath","year":"2016","unstructured":"Surya Prasath VB (2016) Adaptive coherence-enhancing diffusion flow for color images. Informatica 40(3):337","journal-title":"Informatica"},{"unstructured":"Theljani A (2020) Multi-scale non-standard fourth-order pde in image denoising and its fixed point algorithm. Int J Numer Anal Model","key":"1462_CR34"},{"key":"1462_CR35","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/j.nonrwa.2018.10.013","volume":"47","author":"A Theljani","year":"2019","unstructured":"Theljani A, Belhachmi Z, Moakher M (2019) High-order anisotropic diffusion operators in spaces of variable exponents and application to image inpainting and restoration problems. Nonlinear Anal Real World Appl 47:251\u2013271","journal-title":"Nonlinear Anal Real World Appl"},{"issue":"1\u20132","key":"1462_CR36","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/jvci.2001.0495","volume":"13","author":"J Weickert","year":"2002","unstructured":"Weickert J, Scharr H (2002) A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J Vis Commun Image Represent 13(1\u20132):103\u2013118","journal-title":"J Vis Commun Image Represent"},{"issue":"4","key":"1462_CR37","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1162\/08997660360581958","volume":"15","author":"AL Yuille","year":"2003","unstructured":"Yuille AL, Rangarajan A (2003) The concave-convex procedure. Neural Computat 15(4):915\u2013936","journal-title":"Neural Computat"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01462-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s40314-021-01462-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01462-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T22:38:23Z","timestamp":1617921503000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s40314-021-01462-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,3]]},"references-count":37,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["1462"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01462-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2021,3,3]]},"assertion":[{"value":"21 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 February 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"70"}}