Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections
Journal of Mathematical Imaging and Vision
Diffusions and Confusions in Signal and Image Processing
Journal of Mathematical Imaging and Vision
Using Beltrami Framework for Orientation Diffusion in Image Processing
IWVF-4 Proceedings of the 4th International Workshop on Visual Form
A Continuous Shape Descriptor by Orientation Diffusion
EMMCVPR '01 Proceedings of the Third International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition
Combing a Porcupine via Stereographic Direction Diffusion
Scale-Space '01 Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision
Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE's
International Journal of Computer Vision
Regularizing Flows over Lie Groups
Journal of Mathematical Imaging and Vision
A volume-based heat-diffusion classifier
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
Coordinate-free diffusion over compact Lie-groups
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Tensor-based brain surface modeling and analysis
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
An improved representation of junctions through asymmetric tensor diffusion
ISVC'06 Proceedings of the Second international conference on Advances in Visual Computing - Volume Part I
Denoising tensors via lie group flows
VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
Curvature-Preserving regularization of multi-valued images using PDE's
ECCV'06 Proceedings of the 9th European conference on Computer Vision - Volume Part II
PSIVT'11 Proceedings of the 5th Pacific Rim conference on Advances in Image and Video Technology - Volume Part I
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In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to regularize directional data and to obtain multi-scale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps in liquid crystals.This theory deals with the regularization of vectorial data, while satisfying the unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L1 norm.In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.