Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization

Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows acting on constrained datasets. We focus our interest on flows of matrix-valued functions undergoing orthogonal and spectral constraints. The corresponding evolution PDE's are found by minimization of cost functionals, and depend on the natural metrics of the underlying constrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable numerical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of diffusion tensor volumes (DTMRI).

[1]  Li-Tien Cheng,et al.  Variational Problems and Partial Differential Equations on Implicit Surfaces: The Framework and Exam , 2000 .

[2]  Guillermo Sapiro,et al.  Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case , 2000, International Journal of Computer Vision.

[3]  Simon R. Arridge,et al.  A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images , 2001, IPMI.

[4]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[5]  Rachid Deriche,et al.  Diffusion tensor regularization with constraints preservation , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[6]  Alain Trouvé,et al.  Diffeomorphisms Groups and Pattern Matching in Image Analysis , 1998, International Journal of Computer Vision.

[7]  Rachid Deriche,et al.  Dense Disparity Map Estimation Respecting Image Discontinuities: A PDE and Scale-Space BasedApproach , 2002, MVA.

[8]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[9]  B. Vemuri,et al.  Fiber tract mapping from diffusion tensor MRI , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[10]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Michel Barlaud,et al.  Two deterministic half-quadratic regularization algorithms for computed imaging , 1994, Proceedings of 1st International Conference on Image Processing.

[12]  Pietro Perona.  Orientation diffusions , 1998, IEEE Trans. Image Process..

[13]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[14]  Guillermo Sapiro,et al.  Image inpainting , 2000, SIGGRAPH.

[15]  O. Faugeras,et al.  Variational principles, surface evolution, PDE's, level set methods and the stereo problem , 1998, 5th IEEE EMBS International Summer School on Biomedical Imaging, 2002..

[16]  R. Deriche,et al.  Regularization of orthonormal vector sets using coupled PDE's , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[17]  Rachid Deriche,et al.  Optical-Flow Estimation while Preserving Its Discontinuities: A Variational Approach , 1995, ACCV.

[18]  Victor G. Kac,et al.  Infinite dimensional groups with applications , 1985 .

[19]  Hans Knutsson,et al.  Signal processing for computer vision , 1994 .

[20]  Rachid Deriche,et al.  Nonlinear operators in image restoration , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[21]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[22]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[23]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[24]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[25]  Yali Amit,et al.  A Nonlinear Variational Problem for Image Matching , 1994, SIAM J. Sci. Comput..

[26]  T. J. Willmore,et al.  RIEMANNIAN GEOMETRY (de Gruyter Studies in Mathematics, 1) , 1983 .

[27]  Rachid Deriche,et al.  CONSTRAINED AND UNCONSTRAINED PDE'S FOR VECTOR IMAGE RESTO RATION , 2001 .

[28]  A. Bloch.  Hamiltonian and Gradient Flows, Algorithms and Control , 1995 .

[29]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[30]  Ron Kimmel,et al.  Orientation Diffusion or How to Comb a Porcupine , 2002, J. Vis. Commun. Image Represent..

[31]  Olivier D. Faugeras,et al.  Variational principles, surface evolution, PDEs, level set methods, and the stereo problem , 1998, IEEE Trans. Image Process..

[32]  Carl-Fredrik Westin,et al.  Image Processing for Diffusion Tensor Magnetic Resonance Imaging , 1999, MICCAI.