{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T13:47:34Z","timestamp":1764251254960},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2010,11,19]],"date-time":"2010-11-19T00:00:00Z","timestamp":1290124800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2011,5]]},"DOI":"10.1007\/s10851-010-0239-x","type":"journal-article","created":{"date-parts":[[2010,11,18]],"date-time":"2010-11-18T11:27:40Z","timestamp":1290079660000},"page":"1-19","source":"Crossref","is-referenced-by-count":12,"title":["Control of the Effects of Regularization on Variational Optic Flow\u00a0Computations"],"prefix":"10.1007","volume":"40","author":[{"given":"Z.","family":"Belhachmi","sequence":"first","affiliation":[]},{"given":"F.","family":"Hecht","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2010,11,19]]},"reference":[{"key":"239_CR1","first-page":"1349","volume-title":"Proc. XVI Congreso de Ecuaciones Diferenciales y Aplicationes","author":"L. Alvarez","year":"1999","unstructured":"Alvarez, L., Esclarin, J., Lefebure, M., Sanchez, J.: A PDE model for computing the optic flow. In: Proc. XVI Congreso de Ecuaciones Diferenciales y Aplicationes, Las Palmas de Gran Canaria, Spain, pp. 1349\u20131346 (1999)"},{"issue":"1","key":"239_CR2","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1023\/A:1008170101536","volume":"39","author":"P. Anandan","year":"2000","unstructured":"Anandan, P.: A computational framework and an algorithm for the measurement of visual motion. Int. J. Comput. Vis. 39(1), 41\u201356 (2000)","journal-title":"Int. J. Comput. Vis."},{"key":"239_CR3","first-page":"290","volume-title":"Proc. Second Asian Conference on Computer Vision","author":"G. Aubert","year":"1995","unstructured":"Aubert, G., Derriche, R., Kornprobst, P.: Optic flow estimation while preserving its discontinuities: a variational approach. In: Proc. Second Asian Conference on Computer Vision, vol. 2, Singapore, 2000. pp. 290\u2013295. Springer, Berlin (1995)"},{"issue":"1","key":"239_CR4","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1137\/S0036139998340170","volume":"60","author":"G. Aubert","year":"1999","unstructured":"Aubert, G., Deriche, R., Kornprobst, P.: Computing optical flow via variational techniques. SIAM J. Appl. Math. 60(1), 156\u2013182 (1999)","journal-title":"SIAM J. Appl. Math."},{"issue":"1\u20133","key":"239_CR5","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s10915-005-9035-y","volume":"27","author":"Z. Belhachmi","year":"2006","unstructured":"Belhachmi, Z., Bernardi, C., Deparis, S., Hecht, F.: An efficient discretization of the Navier-Stokes equations in an axisymmetric domain. J. Sci. Comput. 27(1\u20133), 97\u2013110 (2006)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"239_CR6","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1051\/m2an:2007035","volume":"41","author":"Z. Belhachmi","year":"2007","unstructured":"Belhachmi, Z., Bernardi, C., Karageorghis, A.: Mortar spectral element discretization of nonhomogeneous and anisotropic Laplace and Darcy equations. M2AN 41(4), 801\u2013824 (2007)","journal-title":"M2AN"},{"issue":"1","key":"239_CR7","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1137\/S0363012904441179","volume":"46","author":"Z. Belhachmi","year":"2007","unstructured":"Belhachmi, Z., Bucur, D.: Stability and uniqueness for the crack identification problem. SIAM J. Control. Optim. 46(1), 253\u2013273 (2007)","journal-title":"SIAM J. Control. Optim."},{"issue":"8","key":"239_CR8","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1109\/34.85668","volume":"13","author":"J. Big\u00fcn","year":"1991","unstructured":"Big\u00fcn, J., Granlund, H., Wiklund, J.: Multidimensional orientation estimation with applications to textures analysis and optic flow. IEEE Trans. Pattern Anal. Mach. Intel. 13(8), 775\u2013790 (1991)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intel."},{"key":"239_CR9","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/PL00005393","volume":"85","author":"C. Bernardi","year":"2000","unstructured":"Bernardi, C., Verf\u00fcrth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579\u2013608 (2000)","journal-title":"Numer. Math."},{"issue":"3","key":"239_CR10","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1137\/S1064827501386481","volume":"24","author":"A. Borzi","year":"2002","unstructured":"Borzi, A., Ito, K., Kunisch, K.: Optimal control formulation for determining the optical flow. SIAM J. Sci. Comput. 24(3), 818\u2013847 (2002)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"239_CR11","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1137\/S1540345902410482","volume":"1","author":"M. Braack","year":"2003","unstructured":"Braack, M., Ern, A.: A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221\u2013238 (2003)","journal-title":"Multiscale Model. Simul."},{"key":"239_CR12","unstructured":"Bruhn, A.: Variational optic flow computation: accurate modelling and efficient numerics. Ph.D. thesis in Computer Science (2006). Saarland University, Saarbr\u00fccken, Germany"},{"key":"239_CR13","volume-title":"Geometric Properties from Incomplete Data, Computational Imaging and Vision","author":"A. Bruhn","year":"2005","unstructured":"Bruhn, A., Weickert, J.: Confidence measure for variational optic flow methods. In: Klette, R., Kozera, R., Noakes, L., Weickert, J. (eds.) Geometric Properties from Incomplete Data, Computational Imaging and Vision. Springer, Dordrecht (2005)"},{"key":"239_CR14","series-title":"Lecture Notes in Computer Sciences","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1007\/3-540-45783-6_55","volume-title":"Pattern Recognition","author":"A. Bruhn","year":"2002","unstructured":"Bruhn, A., Weickert, J., Schn\u00f6rr, C.: Combining the advantages of local and global optic flow methods. In: Van Gool, L. (ed.) Pattern Recognition. Lecture Notes in Computer Sciences, vol. 2449, pp.\u00a0454\u2013462. Springer, Berlin (2002)"},{"issue":"3","key":"239_CR15","first-page":"211","volume":"61","author":"A. Bruhn","year":"2004","unstructured":"Bruhn, A., Weickert, J., Schn\u00f6rr, C.: Lucas\/Kanade meets Horn\/Schunck: combining local and global optic flow methods. Int. J. Comput. Vis. 61(3), 211\u2013231 (2004)","journal-title":"Int. J. Comput. Vis."},{"issue":"5","key":"239_CR16","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1109\/TIP.2005.846018","volume":"15","author":"A. Bruhn","year":"2005","unstructured":"Bruhn, A., Weickert, J., Feddern, C., Kohlberger, T., Schn\u00f6rr, C.: Real-time optic flow computation with variational methods. IEEE Trans. Image Process. 15(5), 608\u2013615 (2005)","journal-title":"IEEE Trans. Image Process."},{"key":"239_CR17","first-page":"17","volume-title":"Handbook of Numerical Analysis","author":"P.G. Ciarlet","year":"1991","unstructured":"Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol.\u00a0II, pp. 17\u2013351. North-Holland, Amsterdam (1991)"},{"key":"239_CR18","first-page":"523","volume-title":"Proc. Eighth Scandinavian Conference on Image Analysis","author":"I. Cohen","year":"1993","unstructured":"Cohen, I.: Nonlinear variational method for optical flow computation. In: Proc. Eighth Scandinavian Conference on Image Analysis, vol.\u00a01, pp.\u00a0523\u2013530. Tromso, Norway, (1993)"},{"key":"239_CR19","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1090\/S0025-5718-1980-0559195-7","volume":"34","author":"T. Dupont","year":"1980","unstructured":"Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441\u2013463 (1980)","journal-title":"Math. Comput."},{"key":"239_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-1740-8","volume-title":"Regularization of Inverse Problems","author":"H. Engl","year":"1996","unstructured":"Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996)"},{"key":"239_CR21","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1088\/0266-5611\/5\/4\/007","volume":"5","author":"H. Engl","year":"1989","unstructured":"Engl, H., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl. 5, 523\u2013540 (1989)","journal-title":"Inverse Probl."},{"key":"239_CR22","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/019","volume-title":"Partial Differential Equations","author":"L.C. Evans","year":"2010","unstructured":"Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn., vol.\u00a019. American Mathematical Society, Providence (2010)","edition":"2"},{"key":"239_CR23","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms","author":"V. Girault","year":"1986","unstructured":"Girault, V., Raviart, P.-A.: Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)"},{"key":"239_CR24","unstructured":"Hecht, F., Pironneau, O.: FreeFem++. See www.freefem.org"},{"issue":"1\/2","key":"239_CR25","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1081\/NFA-120004011","volume":"23","author":"W. Hinterberger","year":"2002","unstructured":"Hinterberger, W., Scherzer, O., Schn\u00f6rr, C., Weickert, J.: Analysis of optical ow models in the framework of calculus of variations. Numer. Funct. Anal. Optim. 23(1\/2), 69\u201389 (2002)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"239_CR26","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0004-3702(81)90024-2","volume":"17","author":"B. Horn","year":"1981","unstructured":"Horn, B., Schunck, B.: Determining optic flow. Artif. Intell. 17, 185\u2013203 (1981)","journal-title":"Artif. Intell."},{"key":"239_CR27","series-title":"Lectures Notes in Computer Science","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1007\/978-3-540-88690-7_22","volume-title":"Proc. European Conference on Computer Vision","author":"C. Kondermann","year":"2008","unstructured":"Kondermann, C., Mester, R., Grabe, C.: Statistical confidence measure for optical flows. In: Proc. European Conference on Computer Vision. Lectures Notes in Computer Science, vol. 5304, pp.\u00a0290\u2013301. Springer, Berlin (2008)"},{"key":"239_CR28","series-title":"Lectures Notes in Computer Science","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/978-3-540-74936-3_14","volume-title":"Proc. DAGM Symposium on Pattern Recognition","author":"C. Kondermann","year":"2007","unstructured":"Kondermann, C., Kondermann, D., Jahne, B., Grabe, C.: An adaptive confidence measure for optical flows based on linear subspace projections. In: Proc. DAGM Symposium on Pattern Recognition. Lectures Notes in Computer Science, vol. 4713, pp. 132\u2013141. Springer, Berlin (2007)"},{"issue":"4","key":"239_CR29","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1109\/83.491336","volume":"5","author":"A. Kumar","year":"1996","unstructured":"Kumar, A., Tannenbaum, A.R., Balas, G.J.: Optic flow: a curve evolution approach. IEEE Trans. Image Process. 5(4), 598\u2013610 (1996)","journal-title":"IEEE Trans. Image Process."},{"key":"239_CR30","unstructured":"Lucas, B.: Generalized image matching by the method of diffrences. PhD thesis, Carnegie Mellon University (1984)"},{"key":"239_CR31","first-page":"674","volume-title":"Proc. Seventh International Joint Conference on Artificial Intelligence","author":"B. Lucas","year":"1981","unstructured":"Lucas, B., Kanade, T.: An iterative image registration technique with an application to image stereo. In: Proc. Seventh International Joint Conference on Artificial Intelligence, Vancouver, pp. 674\u2013679 (1981)"},{"key":"239_CR32","first-page":"933","volume-title":"Proc. Sixth International Conference on Computer Vision","author":"E. M\u00e9min","year":"1998","unstructured":"M\u00e9min, E., P\u00e9rez, P.: A multigrid approach for hierarchical motion estimation. In: Proc. Sixth International Conference on Computer Vision, Bombay, India, pp. 933\u2013938. Narosa Publishing House, Delhi (1998)"},{"key":"239_CR33","series-title":"Lectures Notes in Computer Sciences","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/BFb0014860","volume-title":"Computer Vision","author":"H.H. Nagel","year":"1990","unstructured":"Nagel, H.H.: Extending the oriented smoothness constraint into the temporal domain and the estimation of the derivatives of optical flow. In: Faugeras, O. (ed.) Computer Vision, ECCV\u201990. Lectures Notes in Computer Sciences, vol. 427, pp. 139\u2013148. Springer, Berlin (1990)"},{"key":"239_CR34","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1109\/TPAMI.1986.4767833","volume":"8","author":"H.H. Nagel","year":"1986","unstructured":"Nagel, H.H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. Pattern Anal. Mach. Intell. 8, 565\u2013593 (1986)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"239_CR35","series-title":"Springer Series in Computational Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03206-0","volume-title":"Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems","author":"H.-G. Roos","year":"1996","unstructured":"Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer Series in Computational Mathematics, vol.\u00a024. Springer, Berlin (1996)"},{"key":"239_CR36","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","volume":"60","author":"L.I. Rudin","year":"1992","unstructured":"Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259\u2013268 (1992)","journal-title":"Physica D"},{"key":"239_CR37","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1109\/ICPR.1994.576391","volume-title":"Proc. Twelfth International Conference on Pattern and Recognition","author":"C. Schn\u00f6rr","year":"1994","unstructured":"Schn\u00f6rr, C.: Segmentation of visual motion by minimizing convex non quadratic-functionals. In: Proc. Twelfth International Conference on Pattern and Recognition, vol.\u00a0A, pp. 661\u2013663. IEEE Comput. Soc. Press, Los Alamitos (1994)"},{"key":"239_CR38","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF01249896","volume":"4","author":"C. Schn\u00f6rr","year":"1994","unstructured":"Schn\u00f6rr, C.: Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals. J. Math. Imaging Vis. 4, 189\u2013198 (1994)","journal-title":"J. Math. Imaging Vis."},{"issue":"4","key":"239_CR39","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1016\/j.media.2006.04.002","volume":"10","author":"M. Sermesant","year":"2006","unstructured":"Sermesant, M., Moireau, P., Camara, O., Sainte-Marie, J., Andriantsimiavona, R., Cimrman, R., Hill, D.L., Chapelle, D., Razavi, R.: Cardiac function estimation from MRI using a heart model and data assimilation: advances and difficulties. Med. Image Anal. 10(4), 642\u2013656 (2006)","journal-title":"Med. Image Anal."},{"key":"239_CR40","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1109\/CVPR.1991.139707","volume-title":"Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition","author":"E.P. Simoncelli","year":"1991","unstructured":"Simoncelli, E.P., Adelson, E.H., Heeger, D.J.: Probability distributions of optical flow. In: Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 310\u2013315. (1991)"},{"key":"239_CR41","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1109\/WVM.1989.47097","volume-title":"Proc. Workshop on Visual Motion","author":"D. Shulman","year":"1989","unstructured":"Shulman, D., Herv\u00e9, J.-Y.: Regularization of discontinuous fields. In: Proc. Workshop on Visual Motion, pp. 81\u201386 (1989)"},{"key":"239_CR42","series-title":"Lectures Notes in Computer Science","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/978-3-540-88690-7_7","volume-title":"Proc. European Conference on Computer Vision","author":"D. Sun","year":"2008","unstructured":"Sun, D., Roth, S., Black, J.M.: Learning optical flow. In: Proc. European Conference on Computer Vision. Lectures Notes in Computer Science, vol. 5304, pp. 83\u201397. Springer, Berlin (2008)"},{"issue":"2","key":"239_CR43","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1109\/TPAMI.1986.4767767","volume":"8","author":"D. Terzopoulos","year":"1986","unstructured":"Terzopoulos, D.: Image analysis using multigrid relaxation. IEEE Trans. Pattern Anal. Mach. Intell. 8(2), 129\u2013139 (1986)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"239_CR44","first-page":"1035","volume":"4","author":"A.N. Tikhonov","year":"1963","unstructured":"Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035\u20131038 (1963)","journal-title":"Sov. Math. Dokl."},{"key":"239_CR45","volume-title":"A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques","author":"R. Verf\u00fcrth","year":"1996","unstructured":"Verf\u00fcrth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner, Amsterdam (1996)"},{"key":"239_CR46","series-title":"Mathematics in Industry","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/978-3-540-34767-5_5","volume-title":"Mathematical Models for Registration and Applications to Medical Imaging","author":"J. Weickert","year":"2006","unstructured":"Weickert, J., Bruhn, A., Brox, T., Papenberg, N.: A survey on variational methods for small displacements. In: Scherzer, O. (ed.) Mathematical Models for Registration and Applications to Medical Imaging. Mathematics in Industry, vol.\u00a010, pp. 103\u2013136. Springer, Berlin (2006)"},{"issue":"3","key":"239_CR47","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1023\/A:1013614317973","volume":"45","author":"J. Weickert","year":"2001","unstructured":"Weickert, J., Schn\u00f6rr, C.: A theoretical framework for convex regularizers in PDE-based computation of image motion. Int. J. Comput. Vis. 45(3), 245\u2013264 (2001)","journal-title":"Int. J. Comput. Vis."},{"key":"239_CR48","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1023\/A:1011286029287","volume":"14","author":"J. Weickert","year":"2001","unstructured":"Weickert, J., Schn\u00f6rr, C.: Variational optic flow computations with a spatio-temporal smoothness contraint. J. Math. Imaging Vis. 14, 245\u2013255 (2001)","journal-title":"J. Math. Imaging Vis."},{"key":"239_CR49","volume-title":"Proc. British Machine Vision Conference, London","author":"M. Werlberger","year":"2002","unstructured":"Werlberger, M., Trobin, W., Pock, T., Wedel, A., Cremers, D., Bischof, H.: Anisotropic Huber-L1 optical flow. In: Proc. British Machine Vision Conference, London (2002)"},{"key":"239_CR50","series-title":"Lectures Notes in Computer Science","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/978-3-642-03641-5_16","volume-title":"Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition","author":"H. Zimmer","year":"2009","unstructured":"Zimmer, H., Bruhn, A., Weickert, J., Valgaerts, L., Salgado, A., Rosenhahn, B., Seidel, H.P.: Complementary optical flow. In: Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition. Lectures Notes in Computer Science, vol.\u00a05681, pp. 207\u2013220. Springer, Berlin (2009)"}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-010-0239-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10851-010-0239-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-010-0239-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T22:31:13Z","timestamp":1559255473000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10851-010-0239-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,11,19]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,5]]}},"alternative-id":["239"],"URL":"https:\/\/doi.org\/10.1007\/s10851-010-0239-x","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"value":"0924-9907","type":"print"},{"value":"1573-7683","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,11,19]]}}}