The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
International Journal of Computer Vision
Geometric computing with Clifford algebras: theoretical foundations and applications in computer vision and robotics
Orthonormal Vector Sets Regularization with PDE's and Applications
International Journal of Computer Vision
Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces
IEEE Transactions on Visualization and Computer Graphics
Regularizing Flows for Constrained Matrix-Valued Images
Journal of Mathematical Imaging and Vision
Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
Regularizing Flows over Lie Groups
Journal of Mathematical Imaging and Vision
A Metric Approach to nD Images Edge Detection with Clifford Algebras
Journal of Mathematical Imaging and Vision
The Clifford-Hodge Flow: An Extension of the Beltrami Flow
CAIP '09 Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns
Fast GL(n)-Invariant Framework for Tensors Regularization
International Journal of Computer Vision
Heat kernels of generalized Laplacians-application to color image smoothing
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
A general framework for low level vision
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
A Short- Time Beltrami Kernel for Smoothing Images and Manifolds
IEEE Transactions on Image Processing
Heat Equations on Vector Bundles--Application to Color Image Regularization
Journal of Mathematical Imaging and Vision
A Class of Generalized Laplacians on Vector Bundles Devoted to Multi-Channel Image Processing
Journal of Mathematical Imaging and Vision
Hi-index | 0.00 |
The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in what follows can be used to smooth multichannel images, multichannel image sequences (videos), vector fields, and orthonormal frame fields in any dimension. From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated with generalized Laplacians, the solutions being approximated by convolutions with kernels. Then, the behavior of the diffusion is determined by the geometry of the vector bundle, i.e., by the metric of the base manifold and by a connection on the vector bundle. For instance, the heat equation associated with the Laplace-Beltrami operator can be considered from this point of view for applications to images and video regularization. The main topic of this paper is to show that this approach can be extended in several ways to vector fields and orthonormal frame fields by considering the context of Clifford algebras. We introduce Clifford-Beltrami and Clifford-Hodge operators as generalized Laplacians on Clifford bundles over Riemannian manifolds. Laplace-Beltrami diffusion appears as a particular case of diffusion for degree 0 sections (functions). Dealing with base manifolds of dimension 2, applications to multichannel image, two-dimensional vector field, and orientation field regularization are presented.