{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T03:40:02Z","timestamp":1778816402265,"version":"3.51.4"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2000,2,1]],"date-time":"2000-02-01T00:00:00Z","timestamp":949363200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2000,2,1]],"date-time":"2000-02-01T00:00:00Z","timestamp":949363200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Computer Vision"],"published-print":{"date-parts":[[2000,2]]},"DOI":"10.1023\/a:1008152115986","type":"journal-article","created":{"date-parts":[[2002,12,22]],"date-time":"2002-12-22T08:17:47Z","timestamp":1040545067000},"page":"149-161","source":"Crossref","is-referenced-by-count":152,"title":["Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case"],"prefix":"10.1007","volume":"36","author":[{"given":"Bei","family":"Tang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guillermo","family":"Sapiro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vicent","family":"Caselles","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"257242_CR1","series-title":"Nato ASI Series","first-page":"1","volume-title":"Nematics","author":"F. Alouges","year":"1991","unstructured":"Alouges, F. 1991. An energy decreasing algorithm for harmonic maps. In Nematics, J.M. Coron et al. (Eds.), Nato ASI Series, Kluwer Academic Publishers: Netherlands, pp. 1\u201313."},{"key":"257242_CR2","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1137\/0729052","volume":"29","author":"L. Alvarez","year":"1992","unstructured":"Alvarez, L., Lions, P.L., and Morel, J.M. 1992. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29:845\u2013866.","journal-title":"SIAM J. Numer. Anal."},{"key":"257242_CR3","doi-asserted-by":"crossref","unstructured":"Angenent, S., Haker, S., Tannenbaum, A., and Kikinis, R. 1998. Laplace-Beltrami operator and brain flattening. University of Minnesota ECE Report.","DOI":"10.1007\/10704282_30"},{"key":"257242_CR4","doi-asserted-by":"crossref","unstructured":"Anzellotti, G. 1985. The Euler equation for functionals with linear growth. Transactions of the American Mathematical Society 483\u2013501.","DOI":"10.1090\/S0002-9947-1985-0792808-4"},{"key":"257242_CR5","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/0022-1236(88)90065-1","volume":"80","author":"F. Bethuel","year":"1988","unstructured":"Bethuel, F. and Zheng, X. 1988. Density of smooth functions between two manifolds in Sobolev spaces. Journal of Functional Analysis 80:60\u201375.","journal-title":"Journal of Functional Analysis"},{"key":"257242_CR6","doi-asserted-by":"crossref","unstructured":"Black, M. and Anandan, P. 1993. A framework for the robust estimation of optical flow. In Fourth International Conf. on Computer Vision, Berlin, Germany, pp. 231\u2013236.","DOI":"10.1109\/ICCV.1993.378214"},{"issue":"3","key":"257242_CR7","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1109\/83.661192","volume":"7","author":"M. Black","year":"1998","unstructured":"Black, M., Sapiro, G., Marimont, D., and Heeger, D. 1998. Robust anisotropic diffusion. IEEE Trans. Image Processing 7(3):421\u2013432.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR8","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1109\/83.661180","volume":"7","author":"P. Blomgren","year":"1998","unstructured":"Blomgren, P. and Chan, T. 1998. Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Processing 7:304\u2013309.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR9","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/BF01205490","volume":"107","author":"H. Brezis","year":"1986","unstructured":"Brezis, H., Coron, J.M., and Lieb, E.H. 1986. Harmonic maps with defects. Communications in Mathematical Physics 107:649\u2013705.","journal-title":"Communications in Mathematical Physics"},{"issue":"4","key":"257242_CR10","first-page":"263","volume":"27","author":"B. Cabral","year":"1993","unstructured":"Cabral, B. and Leedom, C. 1993. Imaging vector fields using line integral convolution. ACM Computer Graphics (SIGGRAPH' 93) 27(4):263\u2013272.","journal-title":"ACM Computer Graphics (SIGGRAPH' 93)"},{"key":"257242_CR11","doi-asserted-by":"crossref","unstructured":"Caselles, V. and Sapiro, G. 1998. Vector median filters, morphology, and PDE': Theoretical connections, ECE-University of Minnesota Technical Report. Also in Caselles, V., Sapiro, G., and Chung, D.H. Vector median filters, inf-sup operations, and coupled PDE': Theoretical connections. Journal of Mathematical Imaging and Vision, pp. 109- 120, April 2000.","DOI":"10.1023\/A:1008310305351"},{"key":"257242_CR12","unstructured":"Chan, T. and Shen, J. 1999. Variational restoration of non-flat image features: Models and algorithms. UCLA CAM-TR 99-20."},{"key":"257242_CR13","doi-asserted-by":"crossref","first-page":"507","DOI":"10.4310\/jdg\/1214448751","volume":"36","author":"K.C. Chang","year":"1992","unstructured":"Chang, K.C., Ding, W.Y., and Ye, R. 1992. Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differential Geometry 36:507\u2013515.","journal-title":"J. Differential Geometry"},{"key":"257242_CR14","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF01161995","volume":"201","author":"Y. Chen","year":"1989","unstructured":"Chen, Y. 1989. The weak solutions of the evolution problems of harmonic maps. Math. Z. 201:69\u201374.","journal-title":"Math. Z."},{"key":"257242_CR15","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF02571698","volume":"205","author":"Y. Chen","year":"1994","unstructured":"Chen, Y., Hong, M.C., and Hungerbuhler, N. 1994. Heat flow of p-harmonic maps with values into spheres. Math. Z. 205:25\u201335.","journal-title":"Math. Z."},{"key":"257242_CR16","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1002\/cpa.3160480403","volume":"XLVIII","author":"Y. Chen","year":"1995","unstructured":"Chen, Y., Li, J., and Lin, F.H. 1995. Partial regularity for weak heat flows into spheres. Communications on Pure and Applied Mathematics XLVIII:429\u2013448.","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"257242_CR17","first-page":"99","volume-title":"Minimum energy configurations for liquid crystals: Computational results","author":"R. Cohen","year":"1987","unstructured":"Cohen, R., Hardt, R.M., Kinderlehrer, D., Lin, S.Y., and Luskin, M. 1987. In Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer (Eds.), Minimum energy configurations for liquid crystals: Computational results, IMA Volumes in Mathematics and its Applications. Springer-Verlag: New York, pp. 99\u2013121."},{"issue":"4","key":"257242_CR18","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/s0294-1449(16)30295-5","volume":"7","author":"J.M. Coron","year":"1990","unstructured":"Coron, J.M. 1990. Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincar\u00e9, Analyse Non Lin\u00e9aire 7(4):335\u2013344.","journal-title":"Ann. Inst. H. Poincar\u00e9, Analyse Non Lin\u00e9aire"},{"key":"257242_CR19","first-page":"82","volume":"401","author":"J.M. Coron","year":"1989","unstructured":"Coron, J.M. and Gulliver, R. 1989. Minimizing p-harmonic maps into spheres. J. Reine Angew. Mathem. 401:82\u2013100.","journal-title":"J. Reine Angew. Mathem."},{"key":"257242_CR20","doi-asserted-by":"crossref","unstructured":"Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., and Stuetzle. W. 1995. Multiresolution analysis of arbitrary meshes. Computer Graphics (SIGGRAPH' 95 Proceedings), pp. 173\u2013182.","DOI":"10.1145\/218380.218440"},{"issue":"1","key":"257242_CR21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/blms\/10.1.1","volume":"10","author":"J. Eells","year":"1978","unstructured":"Eells, J. and Lemarie, L. 1978. A report on harmonic maps. Bull. London Math. Soc. 10(1):1\u201368.","journal-title":"Bull. London Math. Soc."},{"issue":"5","key":"257242_CR22","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1112\/blms\/20.5.385","volume":"20","author":"J. Eells","year":"1988","unstructured":"Eells, J. and Lemarie, L. 1988. Another report on harmonic maps. Bull. London Math. Soc. 20(5):385\u2013524.","journal-title":"Bull. London Math. Soc."},{"key":"257242_CR23","doi-asserted-by":"crossref","first-page":"109","DOI":"10.2307\/2373037","volume":"86","author":"J. Eells","year":"1964","unstructured":"Eells, J. and Sampson, J.H. 1964. Harmonic mappings of Riemannian manifolds. Am. J. Math. 86:109\u2013160.","journal-title":"Am. J. Math."},{"key":"257242_CR24","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1080\/03605309408821034","volume":"19","author":"M. Feldman","year":"1994","unstructured":"Feldman, M. 1994. Partial regularity for harmonic maps of evolutions into spheres. Comm. in Partial Differential Equations 19:761\u2013790.","journal-title":"Comm. in Partial Differential Equations"},{"key":"257242_CR25","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01190893","volume":"3","author":"A. Freire","year":"1995","unstructured":"Freire, A. 1995. Uniqueness for the harmonic map flow in two dimensions. Calc. Var. 3:95\u2013105.","journal-title":"Calc. Var."},{"key":"257242_CR26","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF02163266","volume":"1","author":"M. Giaquinta","year":"1993","unstructured":"Giaquinta, M., Modica, G., and Soucek, J. 1993. Variational problems for maps of bounded variation with values in S\n1. Cal. Var. 1:87\u2013121.","journal-title":"Cal. Var."},{"key":"257242_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2377-9","volume-title":"Signal Processing for Computer Vision","author":"G.H. Granlund","year":"1995","unstructured":"Granlund, G.H. and Knutsson, H. 1995. Signal Processing for Computer Vision. Kluwer: Boston, MA."},{"key":"257242_CR28","unstructured":"Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., and Halle, M. 1999. Conformal surface parametrization for texture mapping. University of Minnesota IMA Preprint Series 1611."},{"issue":"1","key":"257242_CR29","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1090\/S0273-0979-97-00692-7","volume":"34","author":"R.M. Hardt","year":"1997","unstructured":"Hardt, R.M. 1997. Singularities of harmonic maps. Bulletin of the American Mathematical Society 34(1):15\u201334.","journal-title":"Bulletin of the American Mathematical Society"},{"key":"257242_CR30","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1002\/cpa.3160400503","volume":"XL","author":"R.M. Hardt","year":"1987","unstructured":"Hardt, R.M. and Lin, F.H. 1987. Mappings minimizing the L\np norm of the gradient. Communications on Pure and Applied Mathematics XL:555\u2013588.","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"257242_CR31","volume-title":"The Finite Element Method","author":"T. Hughes","year":"1987","unstructured":"Hughes, T. 1987. The Finite Element Method. Prentice-Hall: New Jersey."},{"key":"257242_CR32","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF00336961","volume":"50","author":"J.J. Koenderink","year":"1984","unstructured":"Koenderink, J.J. 1984. The structure of images. Biological Cybernetics 50:363\u2013370.","journal-title":"Biological Cybernetics"},{"key":"257242_CR33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-6465-9","volume-title":"Scale-Space Theory in Computer Vision","author":"T. Lindeberg","year":"1994","unstructured":"Lindeberg, T. 1994. Scale-Space Theory in Computer Vision. Kluwer: The Netherlands."},{"key":"257242_CR34","unstructured":"Osher, S. and Vese, L. 1999. Personal communication."},{"key":"257242_CR35","unstructured":"Pardo, A. and Sapiro, G. 1999. Vector probability diffusion. University of Minnesota, IMA preprint 1649, October 1999."},{"key":"257242_CR36","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1109\/83.661195","volume":"7","author":"P. Perona","year":"1998","unstructured":"Perona, P. 1998. Orientation diffusion. IEEE Trans. Image Processing 7:457\u2013467.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR37","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1109\/34.56205","volume":"12","author":"P. Perona","year":"1990","unstructured":"Perona, P. and Malik, J. 1990. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern. Anal. Machine Intell. 12:629\u2013639.","journal-title":"IEEE Trans. Pattern. Anal. Machine Intell."},{"key":"257242_CR38","series-title":"Nato ASI Series","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/978-94-011-3428-6_23","volume-title":"Nematics","author":"L.M. Pismen","year":"1991","unstructured":"Pismen, L.M. and Rubinstein, J. 1991. Dynamics of defects. In Nematics, J.M. Coron et al. (Eds.), Nato ASI Series: Kluwer Academic Publishers: Netherlands, pp. 303\u2013326."},{"issue":"2","key":"257242_CR39","doi-asserted-by":"crossref","first-page":"297","DOI":"10.4310\/CAG.1995.v3.n2.a4","volume":"3","author":"J. Qing","year":"1995","unstructured":"Qing, J. 1995. On singularities of the heat flow for harmonic maps from surfaces into spheres. Communications in Analysis and Geometry 3(2):297\u2013315.","journal-title":"Communications in Analysis and Geometry"},{"key":"257242_CR40","first-page":"31","volume":"I","author":"L.I. Rudin","year":"1994","unstructured":"Rudin, L.I. and Osher, S. 1994. Total variation based image restoration with free local constraints. Proc. IEEE-ICIP\nI, pp. 31\u201335, Austin, Texas.","journal-title":"Proc. IEEE-ICIP"},{"key":"257242_CR41","first-page":"259","volume":"60","author":"L.I. Rudin","year":"1992","unstructured":"Rudin, L.I., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. PhysicaD 60:259\u2013268.","journal-title":"PhysicaD"},{"key":"257242_CR42","doi-asserted-by":"crossref","first-page":"1582","DOI":"10.1109\/83.541429","volume":"5","author":"G. Sapiro","year":"1996","unstructured":"Sapiro, G. and Ringach, D. 1996. Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Processing 5:1582\u20131586.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR43","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1109\/83.661181","volume":"7","author":"N. Sochen","year":"1998","unstructured":"Sochen, N., Kimmel, R., and Malladi, R. 1998. A general framework for low level vision. IEEE Trans. Image Processing 7:310\u2013318.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR44","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1007\/BF02567432","volume":"60","author":"M. Struwe","year":"1985","unstructured":"Struwe, M. 1985. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici 60:558\u2013581.","journal-title":"Comment. Math. Helvetici"},{"key":"257242_CR45","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-02624-3","volume-title":"Variational Methods","author":"M. Struwe","year":"1990","unstructured":"Struwe, M. 1990. Variational Methods. Springer Verlag: New York."},{"key":"257242_CR46","unstructured":"Tang, B., Sapiro, G., and Caselles, V. 1999. Color image enhancement via chromaticity diffusion, pre-print."},{"key":"257242_CR47","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1109\/83.503905","volume":"5","author":"P.E. Trahanias","year":"1996","unstructured":"Trahanias, P.E., Karakos, D., and Venetsanopoulos, A.N. 1996. Directional processing of color images: Theory and experimental results. IEEE Trans. Image Processing 5:868\u2013880.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR48","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1109\/83.242362","volume":"2","author":"P.E. Trahanias","year":"1993","unstructured":"Trahanias, P.E. and Venetsanopoulos, A.N. 1993. Vector directional filters\u2014A new class of multichannel image processing filters. IEEE Trans. Image Processing 2:528\u2013534.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR49","first-page":"283","volume":"76","author":"J. Weickert","year":"1996","unstructured":"Weickert, J. 1996. Foundations and applications of nonlinear anisotropic diffusion filtering. Zeitscgr. Angewandte Math. Mechan. 76:283\u2013286.","journal-title":"Zeitscgr. Angewandte Math. Mechan."},{"key":"257242_CR50","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0262-8856(98)00102-4","volume":"17","author":"J. Weickert","year":"1999","unstructured":"Weickert, J. 1999. Coherence-enhancing diffusion of color images. Image and Vision Computing 17:201\u2013212.","journal-title":"Image and Vision Computing"},{"key":"257242_CR51","volume-title":"Geometry Driven Diffusion in ComputerVision","author":"R.T. Whitaker","year":"1983","unstructured":"Whitaker, R.T. and Gerig, G. 1983. Vector-valued diffusion. In Geometry Driven Diffusion in ComputerVision, Ber B. Haar Romeny, (Ed.). Kluwer: Boston, MA."},{"key":"257242_CR52","first-page":"1019","volume":"2","author":"A.P. Witkin","year":"1983","unstructured":"Witkin, A.P. 1983. Scale-space filtering. Int. Joint. Conf. Artificial Intelligence 2:1019\u20131021.","journal-title":"Int. Joint. Conf. Artificial Intelligence"},{"key":"257242_CR53","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1109\/83.661184","volume":"7","author":"A. Yezzi","year":"1998","unstructured":"Yezzi, A. 1998. Modified curvature motion for image smoothing and enhancement. IEEE Trans. Image Processing 7:345\u2013352.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR54","doi-asserted-by":"crossref","first-page":"1539","DOI":"10.1109\/83.541424","volume":"5","author":"Y.L. You","year":"1996","unstructured":"You, Y.L., Xu, W., Tannenbaum, A., and Kaveh, M. 1996. Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Processing 5:1539\u20131553.","journal-title":"IEEE Trans. Image Processing"},{"key":"257242_CR55","unstructured":"Zhang, D. and Hebert, M. 1999. Harmonic maps and their applications in surface matching. Proc. CVPR' 99. Colorado."}],"container-title":["International Journal of Computer Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1008152115986.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1023\/A:1008152115986\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1008152115986.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T09:32:56Z","timestamp":1754904776000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1023\/A:1008152115986"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,2]]},"references-count":55,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2000,2]]}},"alternative-id":["257242"],"URL":"https:\/\/doi.org\/10.1023\/a:1008152115986","relation":{},"ISSN":["0920-5691","1573-1405"],"issn-type":[{"value":"0920-5691","type":"print"},{"value":"1573-1405","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,2]]}}}