{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T22:46:10Z","timestamp":1778798770726,"version":"3.51.4"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,7,17]],"date-time":"2020-07-17T00:00:00Z","timestamp":1594944000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,17]],"date-time":"2020-07-17T00:00:00Z","timestamp":1594944000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by \u201clocalization technique\u201d using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds<jats:inline-formula><jats:alternatives><jats:tex-math>$$S^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>S<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and<jats:italic>SO<\/jats:italic>(3).<\/jats:p>","DOI":"10.1007\/s00211-020-01134-y","type":"journal-article","created":{"date-parts":[[2020,7,17]],"date-time":"2020-07-17T14:31:29Z","timestamp":1594996289000},"page":"181-217","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A new numerical scheme for discrete constrained total variation flows and its convergence"],"prefix":"10.1007","volume":"146","author":[{"given":"Yoshikazu","family":"Giga","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Koya","family":"Sakakibara","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kazutoshi","family":"Taguchi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Masaaki","family":"Uesaka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,17]]},"reference":[{"key":"1134_CR1","doi-asserted-by":"crossref","unstructured":"Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds, pp. xvi+224, Princeton University Press, Princeton (2008)","DOI":"10.1515\/9781400830244"},{"key":"1134_CR2","unstructured":"Ambrosio, L., Gigli, N., Savare, G.: Gradient flows in metric spaces and in the space of probability measures. In: Lectures in Mathematics ETH Z\u00fcrich, pp. viii+333, Birkh\u00e4user, Basel (2005)"},{"key":"1134_CR3","doi-asserted-by":"publisher","first-page":"1471","DOI":"10.1137\/070680825","volume":"40","author":"JW Barrett","year":"2008","unstructured":"Barrett, J.W., Feng, X., Prohl, A.: On $$p$$-harmonic map heat flows for $$1 \\le p< \\infty $$ and their finite element approximations. SIAM J. Math. Anal. 40, 1471\u20131498 (2008)","journal-title":"SIAM J. Math. Anal."},{"key":"1134_CR4","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/S0006-3495(94)80775-1","volume":"66","author":"PJ Basser","year":"1994","unstructured":"Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259\u2013267 (1994)","journal-title":"Biophys. J."},{"key":"1134_CR5","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s00526-007-0153-2","volume":"32","author":"R Dal Passo","year":"2008","unstructured":"Dal Passo, R., Giacomelli, L., Moll, S.: Rotationally symmetric 1-harmonic maps from $$D^2$$ to $$S^2$$. Calc. Var. Partial Differ. Equ. 32, 533\u2013554 (2008)","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"1134_CR6","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1016\/j.na.2016.05.007","volume":"143","author":"A Di Castroa","year":"2016","unstructured":"Di Castroa, A., Giacomelli, L.: The 1-harmonic flow with values into a smooth planar curve. Nonlinear Anal. 143, 174\u2013192 (2016)","journal-title":"Nonlinear Anal."},{"key":"1134_CR7","first-page":"251","volume":"2002","author":"C Chefd\u2019hotel","year":"2002","unstructured":"Chefd\u2019hotel, C., Tschumperl\u00e9, D., Deriche, R., Faugeras, O.: Constrained flows of matrix-valued functions: application to diffusion tensor regularization. Comput. Vis. ECCV 2002, 251\u2013265 (2002)","journal-title":"Comput. Vis. ECCV"},{"key":"1134_CR8","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1023\/B:JMIV.0000011324.14508.fb","volume":"20","author":"C Chefd\u2019hotel","year":"2004","unstructured":"Chefd\u2019hotel, C., Tschumperl\u00e9, D., Deriche, R., Faugeras, O.: Regularizing flows for constrained matrix-valued images. J. Math. Imaging Vis. 20, 147\u2013162 (2004)","journal-title":"J. Math. Imaging Vis."},{"key":"1134_CR9","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1155\/2007\/27432","volume":"2007","author":"O Christiansen","year":"2007","unstructured":"Christiansen, O., Lee, T.-M., Lie, J., Sinha, U., Chan, T.F.: Total variation regularization of matrix-valued images. Int. J. Biomed. Imaging 2007, 11 (2007)","journal-title":"Int. J. Biomed. Imaging"},{"key":"1134_CR10","doi-asserted-by":"publisher","first-page":"265","DOI":"10.2307\/2373376","volume":"93","author":"MG Crandall","year":"1971","unstructured":"Crandall, M.G., Liggett, T.M.: Generation of semi-groups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265\u2013298 (1971)","journal-title":"Am. J. Math."},{"key":"1134_CR11","first-page":"389","volume":"6","author":"X Feng","year":"2009","unstructured":"Feng, X., Yoon, M.: Finite element approximation of the gradient flow for a class of linear growth energies with applications to color image denoising. Int. J. Numer. Anal. Model. 6, 389\u2013401 (2009)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"1134_CR12","first-page":"153","volume":"92","author":"RL Foote","year":"1984","unstructured":"Foote, R.L.: Regularity of the distance function. Proc. Am. Math. Soc. 92, 153\u2013155 (1984)","journal-title":"Proc. Am. Math. Soc."},{"key":"1134_CR13","doi-asserted-by":"publisher","first-page":"1723","DOI":"10.1137\/12088402X","volume":"45","author":"L Giacomelli","year":"2013","unstructured":"Giacomelli, L., Maz\u00f3n, J.M., Moll, S.: The 1-harmonic flow with values into $$\\mathbb{S}^{1}$$. SIAM J. Math. Anal. 45, 1723\u20131740 (2013)","journal-title":"SIAM J. Math. Anal."},{"key":"1134_CR14","doi-asserted-by":"publisher","first-page":"627","DOI":"10.2140\/apde.2014.7.627","volume":"7","author":"L Giacomelli","year":"2014","unstructured":"Giacomelli, L., Maz\u00f3n, J.M., Moll, S.: The 1-harmonic flow with values in a hyperoctant of the $$N$$-sphere. Anal. PDE 7, 627\u2013671 (2014)","journal-title":"Anal. PDE"},{"key":"1134_CR15","doi-asserted-by":"crossref","unstructured":"Giacomelli, L., \u0141asica, M., Moll, S.: Regular 1-harmonic flow. Calc. Var. Partial Diff. Equ. 58, Paper No. 82, 24 pp. (2019)","DOI":"10.1007\/s00526-019-1526-z"},{"key":"1134_CR16","doi-asserted-by":"crossref","unstructured":"Giga, M.-H., Giga, Y., Kobayashi, R.: Very singular diffusion equations. In: Maruyama, M., Sunada, T. (eds.) Proceedings of the last Taniguchi Symposium. Adv. Stud. Pure Math. 31, 93\u2013125 (2001)","DOI":"10.2969\/aspm\/03110093"},{"key":"1134_CR17","doi-asserted-by":"crossref","first-page":"253","DOI":"10.4310\/MAA.2003.v10.n2.a6","volume":"10","author":"Y Giga","year":"2003","unstructured":"Giga, Y., Kobayashi, R.: On constrained equations with singular diffusivity. Methods Appl. Anal. 10, 253\u2013278 (2003)","journal-title":"Methods Appl. Anal."},{"key":"1134_CR18","first-page":"9","volume":"22","author":"Y Giga","year":"2004","unstructured":"Giga, Y., Kuroda, H.: On breakdown of solutions of a constrained gradient system of total variation. Bol. Soc. Parana. Mat. 22, 9\u201320 (2004)","journal-title":"Bol. Soc. Parana. Mat."},{"key":"1134_CR19","doi-asserted-by":"publisher","first-page":"121","DOI":"10.3934\/cpaa.2015.14.121","volume":"14","author":"Y Giga","year":"2015","unstructured":"Giga, Y., Kuroda, H.: A counterexample to finite time stopping property for one-harmonic map flow. Commun. Pure Appl. Anal. 14, 121\u2013125 (2015)","journal-title":"Commun. Pure Appl. Anal."},{"key":"1134_CR20","first-page":"199","volume":"11","author":"Y Giga","year":"2005","unstructured":"Giga, Y., Kuroda, H., Yamazaki, N.: An existence result for a discretized constrained gradient system of total variation flow in color image processing. Interdiscip. Inform. Sci. 11, 199\u2013204 (2005)","journal-title":"Interdiscip. Inform. Sci."},{"key":"1134_CR21","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/978-3-7643-7719-9_21","volume":"154","author":"Y Giga","year":"2006","unstructured":"Giga, Y., Kuroda, H., Yamazaki, N.: Global solvability of constrained singular diffusion equation associated with essential variation. Int. Ser. Numer. Mathods 154, 209\u2013218 (2006)","journal-title":"Int. Ser. Numer. Mathods"},{"key":"1134_CR22","doi-asserted-by":"publisher","first-page":"109114","DOI":"10.1016\/j.jcp.2019.109114","volume":"405","author":"Y Giga","year":"2020","unstructured":"Giga, Y., Ueda, Y.: Numerical computations of split Bregman method for fourth order total variation flow. J. Comput. Phys. 405, 109114 (2020)","journal-title":"J. Comput. Phys."},{"key":"1134_CR23","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1137\/080725891","volume":"2","author":"T Goldstein","year":"2009","unstructured":"Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323\u2013343 (2009)","journal-title":"SIAM J. Imaging Sci."},{"key":"1134_CR24","unstructured":"Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Mathematics, pp. xv+628, Academic Press\/Harcourt Brace Jovanovich, Publishers, New York\/London (1978)"},{"key":"1134_CR25","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.physa.2005.05.024","volume":"356","author":"R Kobayashi","year":"2005","unstructured":"Kobayashi, R., Warren, J.A.: Modeling the formation and dynamics of polycrystals in 3D. Physica A 356, 127\u2013132 (2005)","journal-title":"Physica A"},{"key":"1134_CR26","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/S0167-2789(00)00023-3","volume":"140","author":"R Kobayashi","year":"2000","unstructured":"Kobayashi, R., Warren, J.A., Carter, W.C.: A continuum model of grain boundaries. Physica D 140, 141\u2013150 (2000)","journal-title":"Physica D"},{"key":"1134_CR27","first-page":"680","volume":"2016","author":"A Kovnatsky","year":"2016","unstructured":"Kovnatsky, A., Glashoff, K., Bronstein, M.M.: MADMM: a generic algorithm for non-smooth optimization on manifolds. Comput. Vis. ECCV 2016, 680\u2013696 (2016)","journal-title":"Comput. Vis. ECCV"},{"key":"1134_CR28","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/s10915-013-9740-x","volume":"58","author":"R Lai","year":"2014","unstructured":"Lai, R., Osher, S.J.: A splitting method for orthogonality constrained problems. J. Sci. Comput. 58, 431\u2013449 (2014)","journal-title":"J. Sci. Comput."},{"key":"1134_CR29","doi-asserted-by":"publisher","first-page":"1691","DOI":"10.1137\/16M1103610","volume":"10","author":"M \u0141asica","year":"2017","unstructured":"\u0141asica, M., Moll, S., Mucha, P.B.: Total variation denoising in $$\\ell ^1$$ anisotropy. SIAM J. Imaging Sci. 10, 1691\u20131723 (2017)","journal-title":"SIAM J. Imaging Sci."},{"key":"1134_CR30","doi-asserted-by":"publisher","first-page":"2434","DOI":"10.1137\/140998135","volume":"25","author":"G Li","year":"2015","unstructured":"Li, G., Pong, T.K.: Global convergence of splitting methods for nonconvex composite optimization. SIAM J. Optim. 25, 2434\u20132460 (2015)","journal-title":"SIAM J. Optim."},{"key":"1134_CR31","doi-asserted-by":"publisher","first-page":"637","DOI":"10.4310\/CMS.2011.v9.n3.a1","volume":"9","author":"A Oberman","year":"2011","unstructured":"Oberman, A., Osher, S., Takei, R., Tsai, R.: Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation. Commun. Math. Sci. 9, 637\u2013662 (2011)","journal-title":"Commun. Math. Sci."},{"key":"1134_CR32","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s11263-005-3222-z","volume":"66","author":"X Pennec","year":"2006","unstructured":"Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66, 41\u201366 (2006)","journal-title":"Int. J. Comput. Vis."},{"key":"1134_CR33","unstructured":"Po\u017e\u00e1r, N.: On the self-similar solutions of the crystalline mean curvature flow in three dimensions, p. 28 (2018). arXiv:1806.02482"},{"key":"1134_CR34","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1137\/0733005","volume":"33","author":"J Rulla","year":"1996","unstructured":"Rulla, J.: Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33, 68\u201387 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"1134_CR35","unstructured":"Setzer, S.: Splitting methods in image processing. Ph.D. Thesis, Universitat Mannheim, p. 174 (2009)"},{"key":"1134_CR36","first-page":"81","volume":"27","author":"K Taguchi","year":"2018","unstructured":"Taguchi, K.: On discrete one-harmonic map flows with values into an embedded manifold on a multi-dimensional domain. Adv. Math. Sci. Appl. 27, 81\u2013113 (2018)","journal-title":"Adv. Math. Sci. Appl."},{"key":"1134_CR37","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1109\/83.918563","volume":"10","author":"B Tang","year":"2001","unstructured":"Tang, B., Sapiro, G., Caselles, V.: Color image enhancement via chromaticity diffusion. IEEE Trans. Image Process. 10, 701\u2013707 (2001)","journal-title":"IEEE Trans. Image Process."},{"key":"1134_CR38","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1137\/S0036142901396715","volume":"40","author":"LA Vese","year":"2002","unstructured":"Vese, L.A., Osher, S.J.: Numerical methods for $$p$$-harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40, 2085\u20132104 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"1134_CR39","doi-asserted-by":"publisher","first-page":"2226","DOI":"10.1137\/130951075","volume":"7","author":"A Weinmann","year":"2014","unstructured":"Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7, 2226\u20132257 (2014)","journal-title":"SIAM J. Imaging Sci."},{"key":"1134_CR40","first-page":"35","volume":"78","author":"Y Wang","year":"2018","unstructured":"Wang, Y., Yin, W., Zeng, J.: Global convergence of ADMM in nonconvex nonsmooth optimization. J. Sci. Comput. 78, 35 (2018)","journal-title":"J. Sci. Comput."},{"key":"1134_CR41","doi-asserted-by":"crossref","unstructured":"Zhang, J., Ma, S., Zhang, S.: Primal-dual optimization algorithms over Riemannian manifolds: an iteration complexity analysis. Math. Program. pp. 46 (2019)","DOI":"10.1007\/s10107-019-01418-8"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01134-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-020-01134-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01134-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T04:11:36Z","timestamp":1667448696000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-020-01134-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,17]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1134"],"URL":"https:\/\/doi.org\/10.1007\/s00211-020-01134-y","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,17]]},"assertion":[{"value":"6 December 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 March 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}