{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T11:55:50Z","timestamp":1790164550295,"version":"4.1.0"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem. The finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images. They have been successfully applied in image processing, e.g., for solving the Perona\u2013Malik equation or curvature-driven level set equations, where the nonlinearities are represented by a scalar function dependent on a solution gradient. Design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence-enhancing image smoothing. We provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor, and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh. This method is well known as the diamond-cell method owing to the choice of covolume as a diamond-shaped polygon. Further, we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem. The proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction. Finally computational results illustrated in figures are discussed.<\/jats:p>","DOI":"10.1137\/070685038","type":"journal-article","created":{"date-parts":[[2007,12,12]],"date-time":"2007-12-12T18:02:46Z","timestamp":1197482566000},"page":"37-60","source":"Crossref","is-referenced-by-count":26,"title":["Convergence Analysis of Finite Volume Scheme for Nonlinear Tensor Anisotropic Diffusion in Image Processing"],"prefix":"10.1137","volume":"46","author":[{"given":"Olga","family":"Drbl\u00edkov\u00e1","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karol","family":"Mikula","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,12,12]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0729012"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/060651203"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:1999149"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"R. Eymard, T. Gallou\u00ebt, and R. Herbin,\n                      Finite Volume Methods\n                      , in Handbook for Numerical Analysis 7 Ph. Ciarlet and J. L. Lions, eds., Elsevier, New York, 2000.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dri036"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2002.800883"},{"key":"R7","first-page":"79","volume":"74","author":"Handlovi\u010dov\u00e1 A.","year":"2005","journal-title":"Acta Math. Univ. Comenian. (N.S.)"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100374"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"B. J\u00e4hne,\n                      Spatio-temporal Image Processing\n                      , Lecture Notes in Comput. Sci.  751, Springer, Berlin, 1993.","DOI":"10.1007\/3-540-57418-2"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.2478\/cmam-2005-0009"},{"key":"R11","unstructured":"Z. Kriv\u00e1 and K. Mikula,\n                      An adaptive finite volume method in processing of color images\n                      , in Proceedings of ALGORITMY 2000, Conference on Scientific Computing, Podbansk\u00e9, 2000, pp. 174\u2013187."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1006\/jvci.2001.0502"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cagd.2005.06.010"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"K. Mikula,\n                      Image processing with partial differential equations\n                      , Modern Methods in Scientific Computing and Applications, A. Bourlioux and M. J. Gander, eds., Springer, Berlin, 2002, pp. 283\u2013321.","DOI":"10.1007\/978-94-010-0510-4_8"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005479"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"K. Mikula, A. Sarti, and F. Sgallari,\n                      Semi-implicit co-volume level set method in medical image segmentation\n                      , in Handbook of Biomedical Image Analysis: Segmentation and Registration Models, J. Suri et al., eds., Springer, New York, 2005, pp. 583\u2013626.","DOI":"10.1007\/0-306-48551-6_11"},{"key":"R17","unstructured":"P. Perona and J. Malik,\n                      Scale space and edge detection using anisotropic diffusion\n                      , in Proceedings of the IEEE Computer Society Workshop on Computer Vision, 1987."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1006\/jvci.1999.0444"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139901384662"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.110135797"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1109\/42.781012"},{"key":"R22","unstructured":"J. Weickert,\n                      Anisotropic Diffusion in Computer Vision\n                      , Teubner, Stuttgart, 1998."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0262-8856(98)00102-4"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008009714131"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1006\/jvci.2001.0495"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902417042"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070685038","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:09:00Z","timestamp":1787324940000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070685038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,12]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070685038"],"URL":"https:\/\/doi.org\/10.1137\/070685038","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,12,12]]}}}