Skip to main content
SpringerLink
Log in

Orthonormal Vector Sets Regularization with PDE's and Applications

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

We are interested in regularizing fields of orthonormal vector sets, using constraint-preserving anisotropic diffusion PDE's. Each point of such a field is defined by multiple orthogonal and unitary vectors and can indeed represent a lot of interesting orientation features such as direction vectors or orthogonal matrices (among other examples). We first develop a general variational framework that solves this regularization problem, thanks to a constrained minimization of φ-functionals. This leads to a set of coupled vector-valued PDE's preserving the orthonormal constraints. Then, we focus on particular applications of this general framework, including the restoration of noisy direction fields, noisy chromaticity color images, estimated camera motions and DT-MRI (Diffusion Tensor MRI) datasets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acton, S.T. 1998. Multigrid anisotropic diffusion. IEEE Transactions on Image Processing, 7:280–291.

    Google Scholar 

  • Alvarez, L., Deriche, R., Weickert, J., and Sànchez, J. 2002. Dense disparity map estimation respecting image discontinuities: A PDE and scale-space based approach. International Journal of Visual Communication and Image Representation, Special Issue on PDE in Image Processing, Computer Vision and Computer Graphics, 13(1/2):3–21.

    Google Scholar 

  • Alvarez, L., Lions, P.L., and Morel, J.M. 1992. Image se-lective smoothing and edge detection by nonlinear diffusion (II). SIAM Journal of Numerical Analysis, 29:845–866.

    Google Scholar 

  • Bertalmio, M., Sapiro, G., Caselles, V., and Ballester, C. 2000. Image inpainting. In Proceedings of the SIGGRAPH, Kurt Akeley (Ed.), ACM Press, pp. 417–424.

  • Bertalmio, M., Sapiro, G., Cheng, L.T., and Osher, S. 2001. Variational problems and PDE's on implicit surfaces. In IEEE Work-shop on Variational and Level Set Methods, Vancouver, Canada, pp. 186–193.

  • Blomgren, P. and Chan, T.F. 1998. Color TV: Total variation methods for restoration of vector-valued images. IEEE Trans.Imag.Proc., 7(3):304–309. Special issue on PDE and Geometry-Driven Diffusion in Image Processing and Analysis.

    Google Scholar 

  • Caselles, V., Morel, J.M., Sapiro, G., and Tannenbaum, A. 1998. Introduction to the special issue on PDE and geometry-driven diffusion in image processing and analysis. IEEE Transactions on Image Processing, 7(3):269–273.

    Google Scholar 

  • Chambolle, A. and Lions, P.L. 1997. Image recovery via total variation minimization and related problems. Nümerische Mathematik, 76(2):167–188.

    Google Scholar 

  • Chan, T., Kang, S.H., and Shen, J. 2000. Total variation denoising and enhancement color images based on the CB and HSV color models. Journal of Visual Communication and Image Representation, 12(4).

  • Chan, T., Sandberg, B.Y., and Vese, L. 2000. Active contours without edges for vector-valued images. Journal of Visual Communication and Image Representation, 11:130–141.

    Google Scholar 

  • Chan, T. and Shen, J. 2000. Non-texture inpaintings by curvature-driven diffusions. Journal of Visual Communication and Image Representation, 12(4):436–449.

    Google Scholar 

  • Chan, T. and Shen, J. 2001a. Variational restoration of nonflat image features: Models and algorithms. SIAM Journal on Applied Mathematics, 61(4):1338–1361.

    Google Scholar 

  • Chan, T. and Shen, J. 2001b. Mathematical models for local nontexture inpaintings SIAM Journal on Applied Mathematics, 62(3):1019–1043.

    Google Scholar 

  • Charbonnier, P., Aubert, G., Blanc-Féraud, M., and Barlaud, M. 1994. Two deterministic half-quadratic regularization algorithms for computed imaging. In Proceedings of the International Conference on Image Processing, 2:168–172.

    Google Scholar 

  • Chefd'hotel, C., Tschumperlé, D., Deriche, R., and Faugeras, O. 2002. Constrained flows on matrix-valued functions: Application to diffusion tensor regularization. In Proceedings of ECCV'02.

  • Cohen, L. 1995. Auxiliary variables and two-step iterative algorithms in computer vision problems. International Conference on Computer Vision.

  • Cottet, G.H. and Germain, L. 1993. Image processing through reaction combined with nonlinear diffusion. Mathematics of Computation, 61(204):659–673.

    Google Scholar 

  • Coulon, O., Alexander, D.C., and Arridge, S.R. 2001a. Ageometrical approach to 3D diffusion tensor magnetic resonance image regularisation. Department of Computer Science, University College London. Technical Report.

  • Coulon, O., Alexander, D.C., and Arridge, S.R. 2001b. A regular-ization scheme for diffusion tensor magnetic resonance images. In XVIIth International Conferenceon Information Processing in Medical Imaging.

  • Dibos, F. and Koepfler, G. 1998. Global total variation minimization. CEREMADE (URACNRS 749), Technical Report 9801.

  • Faugeras, O. 1993. Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press: Cambridge, MA.

    Google Scholar 

  • Geman, S. and McClure, D.E. 1985. Bayesian image analysis: An application to single photon emission tomography. Amer.Statist.Assoc., pp. 12–18.

  • Granlund, G.H. and Knutsson, H. 1995. Signal Processing for Computer Vision. Kluwer Academic Publishers.

  • Kimmel, R., Malladi, R., and Sochen, N. 2000. Images as embedded maps and minimal surfaces: Movies, color, texture, and volumetric medical images. International Journal of Computer Vision, 39(2):111–129.

    Google Scholar 

  • Kimmel, R. and Sochen, N. 2002. Orientation diffusion or how to comb a porcupine. Journal of Visual Communication and Image Representation, Special Issue on PDE in Image Processing, Computer Vision and Computer Graphics, 13(1/2):238–248.

    Google Scholar 

  • Kornprobst, P., Deriche, R., and Aubert, G. 1997a. Image coupling, restoration and enhancement via PDE's. In Proceedings of the International Conference on Image Processing, Santa Barbara, California, vol. 4, pp. 458–461.

    Google Scholar 

  • Kornprobst, P., Deriche, R., and Aubert, G. 1997b. Nonlinear operators in image restoration. In Proceedings of the International Conference on Computer Vision and Pattern Recognition, Puerto Rico, IEEE Computer Society, pp. 325–331.

    Google Scholar 

  • Kornprobst, P., Deriche, R., and Aubert, G. 1998. EDP, débruitage et réhaussement en traitement d'image: Analyse et contributions. In 11éme Congrés RFIA, AFCET, vol. 1, pp. 277–286.

    Google Scholar 

  • Koschan, A. 1995. A comparative study on color edge detection. In Proceedings of the 2nd Asian Conference on Computer Vision, Singapore, vol. 3, pp. 574–578.

    Google Scholar 

  • Le Bihan, D. 2000. Methods and applications of diffusion mri. In Magnetic Resonance Imaging and Spectroscopy in Medicine and Biology, I.R. Young (Ed.). John Wiley and Sons.

  • Malladi, R. and Sethian, J.A. 1996. Image processing: Flows under min/max curvature and mean curvature. Graphical Models and Image Processing, 58(2):127–141.

    Google Scholar 

  • Morel, J.M. and Solimini, S. 1988. Segmentation of images by variational methods: A constructive approach. Rev.Math.Univ.Complut.Madrid, 1:169–182.

    Google Scholar 

  • Mumford, D. and Shah, J. 1989. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42:577–684.

    Google Scholar 

  • Nagel, H.H. and Enkelmann, W. 1986. An investigation of smoothness constraint for the estimation of displacement vector fiels from images sequences. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8:565–593.

    Google Scholar 

  • Nikolova, M. and Ng, M. 2001. Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning. In Proceedings of the International Conference on Image Processing. IEEE Signal Processing Society.

  • Nordström, N. 1990. Biased anisotropic diffusion—A unified regularization and diffusion approach to edge detection. Image and Vision Computing, 8(11):318–327.

    Google Scholar 

  • Osher, S. and Rudin, L.I. 1990. Feature-oriented image enhancement using shock filters. SIAM Journal of Numerical Analysis, 27(4):919–940.

    Google Scholar 

  • Paragios, N. and Deriche, R. 2002. Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision. International Journal of Visual Communication and Image Representation, Special Issue on PDE in Image Processing, Computer Vision and Computer Graphics, 13(1/2):249–268.

    Google Scholar 

  • Perona, P. 1998. Orientation diffusions. IEEE Transactions on Image Processing, 7(3):457–467.

    Google Scholar 

  • Perona, P. and Malik, J. 1990. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7):629–639.

    Google Scholar 

  • Poupon, C. 1999. Détection des faisceaux de fibres de la substance blanche pour l'étude de la connectivité anatomique cérébrale. Ph.D. thesis, Ecole Nationale Supérieure des Télécommunications.

  • Poupon, C., Mangin, J.F., Frouin, V., Regis, J., Poupon, F., Pachot-Clouard, M., Le Bihan, D., and Bloch, I. 1998. Regularization of mr diffusion tensor maps for tracking brain white matter bundles. In W.M. Wells, A. Colchester, and S. Delp (Ed.), Medical Image Computing and Computer-Assisted Intervention-MICCAI'98, Cambridge, MA, USA, vol. 1496 in Lecture Notes in Computer Science. Springer: Berlin, pp. 489–498.

    Google Scholar 

  • Proesmans, M., Pauwels, E., and Van Gool, L. 1994. Coupled Geometry-Driven Diffusion Equations for Low-Level Vision Geometry-Driven Diffusion in Computer Vision, B.M. ter Haar Romeny (Ed.). Kluwer Academic Publishers: Boston, MA, pp. 191–228.

    Google Scholar 

  • RealviZ. 1999. Web-site: http://www.realviz.com.

  • Rudin, L. and Osher, S. 1994. Total variation based image restoration with free local constraints. In Proceedings of the International Conference on Image Procedings, vol. I, pp. 31–35.

    Google Scholar 

  • Rudin, L., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268.

    Google Scholar 

  • Sapiro, G. 1997. Color snakes. Computer Vision and Image Understanding, 68(2).

  • Sapiro, G. 1996. Vector-valued active contours. In Proceedings of the International Conference on Computer Vision and Pattern Recognition, San Francisco, CA. IEEE, pp. 680–685.

    Google Scholar 

  • Sapiro, G. 2001. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press.

  • Sapiro, G. and Ringach, D.L. 1996. Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Transactions on Image Processing, 5(11):1582–1585.

    Google Scholar 

  • Shah, J. 1996. A common framework for curve evolution, segmentation and anisotropic diffusion. In International Conference on Computer Vision and Pattern Recognition.

  • SHFJ CEA. 2000. Web page, http://wwwdsv.cea.fr/thema/shfj/.

  • Sochen, N., Kimmel, R., and Malladi, R. 1997. From high energy physics to low level vision. In Scale-Space Theories in Computer Vision. pp. 236–247.

  • Sternberg, P. 1991. Vector-valued local minimizers of nonconvex variational problems. Rocky Mt.J.Math., 21(2):799–807.

    Google Scholar 

  • Strong, D.M. and Chan, T.F. 1996a. Relation of regularization parameter and scale in total variation based image denoising. In Proc.IEEE Workshop on Mathematical Methods in Biomedical Image Analysis.

  • Strong, D.M. and Chan, T.F. 1996b. Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing. UCLA, Technical Report 46.

  • Tang, B., Sapiro, G., and Caselles, V. 2000. Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. The International Journal of Computer Vision, 36(2):149–161.

    Google Scholar 

  • Teboul, S., Blanc-Féraud, L., Aubert, G., and Barlaud, M. 1998. Vari-ational approach for edge-preserving regularization using coupled PDE's. IEEE Transaction on Image Processing, Special Issue on PDE based Image Processing, 7(3):387–397.

    Google Scholar 

  • ter Haar Romeny, B.M. 1994. Geometry-Driven Diffusion in Computer Vision. Computational Imaging and Vision. Kluwer Academic Publishers: Boston, MA.

    Google Scholar 

  • Tikhonov, A.N. 1963. Regularization of incorrectly posed problems. Soviet.Math.Dokl., 4:1624–1627.

    Google Scholar 

  • Tschumperlé, D. and Deriche, R. 2001a. Constrained and uncon-strained pde's for vector image restoration. In Proceedings of the 10th Scandinavian Conference on Image Analysis, Ivar Austvoll, (Ed.), Bergen, Norway, pp. 153–160.

  • Tschumperlé´e, D. and Deriche, R. 2001b. Diffusion tensor regularization with constraints preservation. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Kauai Marriott, Hawaii.

  • Tschumperlé, D. and Deriche, R. 2001c. Regularization of orthonor-mal vector sets using coupled PDE's. In IEEE Workshop on Variational and Level Set Methods, Vancouver, Canada, pp. 3–10.

  • Tschumperlé D. and Deriche, R. 2002. Diffusion pde's on vector-valued images: Local approach and geometric viewpoint. IEEE Signal Processing Magazine.

  • Vemuri, B., Chen, Y., Rao, M., McGraw, T., Mareci, T., and Wang, Z. 2001. Fiber tract mapping from diffusion tensor mri. In 1st IEEE Workshop on Variational and Level Set Methods in Computer Vision (VLSM'01).

  • Vese, L.A. and Osher, S. 2002. Numerical methods for pharmonic flows and applications to image processing. In SIAM 50th Anniversary and Annual Meeting.

  • Weickert, J. 1997. A review of nonlinear diffusion of filtering. In Scale-Space Theory in Computer Vision, vol. 1252 of Lecture Notes in Comp. Science. Springer; Berlin, pp. 3–28. Invited paper.

    Google Scholar 

  • Weickert, J. 1998. Anisotropic Diffusion in Image Processing. Teubner-Verlag; Stuttgart.

    Google Scholar 

  • Weickert, J. 1999. Linear scale space has first been proposed in Japan. Journal of Mathematical Imaging and Vision, 10(3):237–252.

    Google Scholar 

  • Weickert, J. and Schnörr, C. 2001. A theoretical framework for convex regularizers in pde-based computation of image motion. The International Journal of Computer Vision, 45(3):245–264.

    Google Scholar 

  • You, Y.L., Kaveh, M., Xu, W.Y., and Tannenbaum, A. 1994. Analysis and Design of Anisotropic Diffusion for Image Processing. In Proceedings of the International Conference on Image Processing, vol. II, pp. 497–501.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odyssee Lab, I.N.R.I.A, Sophia Antipolis, 06902, Sophia Antipolis, France

    David Tschumperlé & Rachid Deriche

Authors
  1. David Tschumperlé
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rachid Deriche
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tschumperlé, D., Deriche, R. Orthonormal Vector Sets Regularization with PDE's and Applications. International Journal of Computer Vision 50, 237–252 (2002). https://doi.org/10.1023/A:1020870207168

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020870207168

Search

Navigation