Orthonormal Vector Sets Regularization with PDE's and Applications
International Journal of Computer Vision
From High Energy Physics to Low Level Vision
SCALE-SPACE '97 Proceedings of the First International Conference on Scale-Space Theory in Computer Vision
Regularizing Flows for Constrained Matrix-Valued Images
Journal of Mathematical Imaging and Vision
A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
SIAM Journal on Matrix Analysis and Applications
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
Curvature-Driven PDE Methods for Matrix-Valued Images
International Journal of Computer Vision
Visualization and Processing of Tensor Fields (Mathematics and Visualization)
Visualization and Processing of Tensor Fields (Mathematics and Visualization)
Coordinate-free diffusion over compact Lie-groups
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Riemannian curvature-driven flows for tensor-valued data
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Regularization of mappings between implicit manifolds of arbitrary dimension and codimension
VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
Color TV: total variation methods for restoration of vector-valued images
IEEE Transactions on Image Processing
A general framework for low level vision
IEEE Transactions on Image Processing
SIAM Journal on Imaging Sciences
A Class of Generalized Laplacians on Vector Bundles Devoted to Multi-Channel Image Processing
Journal of Mathematical Imaging and Vision
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We propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach. The manifold of symmetric positive-definite (SPD) matrices, P n , is parameterized via the Iwasawa coordinate system. In this framework distances on P n are measured in terms of a natural GL(n)-invariant metric. Via the mathematical concept of fiber bundles, we describe the tensor-valued image as a section where the metric over the section is induced by the metric over P n . Then, a functional over the sections accompanied by a suitable data fitting term is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of $\frac{1}{2}n(n+1)$ coupled equations of motion. By means of the gradient descent method, these equations of motion define a Beltrami flow over P n . It turns out that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics that leads to fast convergence of the algorithm. Regularization results as well as results of fibers tractography for DTI are presented.