A note on the gradient of a multi-image
Computer Vision, Graphics, and Image Processing - Lectures notes in computer science, Vol. 201 (G. Goos and J. Hartmanis, Eds.)
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Scale-Space and Edge Detection Using Anisotropic Diffusion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Feature-oriented image enhancement using shock filters
SIAM Journal on Numerical Analysis
Computing oriented texture fields
CVGIP: Graphical Models and Image Processing
Edge detection in multispectral images
CVGIP: Graphical Models and Image Processing
Multidimensional Orientation Estimation with Applications to Texture Analysis and Optical Flow
IEEE Transactions on Pattern Analysis and Machine Intelligence
Image selective smoothing and edge detection by nonlinear diffusion. II
SIAM Journal on Numerical Analysis
Area and Length Preserving Geometric Invariant Scale-Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Geometric heat equation and nonlinear diffusion of shapes and images
Computer Vision and Image Understanding
Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections
Journal of Mathematical Imaging and Vision
Geometric partial differential equations and image analysis
Geometric partial differential equations and image analysis
Signal Processing for Computer Vision
Signal Processing for Computer Vision
Computational Framework for Segmentation and Grouping
Computational Framework for Segmentation and Grouping
Orthonormal Vector Sets Regularization with PDE's and Applications
International Journal of Computer Vision
Regularization of MR Diffusion Tensor Maps for Tracking Brain White Matter Bundles
MICCAI '98 Proceedings of the First International Conference on Medical Image Computing and Computer-Assisted Intervention
Image Processing for Diffusion Tensor Magnetic Resonance Imaging
MICCAI '99 Proceedings of the Second International Conference on Medical Image Computing and Computer-Assisted Intervention
Edge Preserving Regularization and Tracking for Diffusion Tensor Imaging
MICCAI '01 Proceedings of the 4th International Conference on Medical Image Computing and Computer-Assisted Intervention
Nonlinear Matrix Diffusion for Optic Flow Estimation
Proceedings of the 24th DAGM Symposium on Pattern Recognition
The Perceptual Organization of Texture Flow: A Contextual Inference Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fiber Tract Mapping from Diffusion Tensor MRI
VLSM '01 Proceedings of the IEEE Workshop on Variational and Level Set Methods (VLSM'01)
Gradient flows and geometric active contour models
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
Geometric Level Set Methods in Imaging,Vision,and Graphics
Geometric Level Set Methods in Imaging,Vision,and Graphics
Numerical Geometry of Images: Theory, Algorithms, and Applications
Numerical Geometry of Images: Theory, Algorithms, and Applications
Vector-valued image regularization with PDE's: a common framework for different applications
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
Anisotropic diffusion of multivalued images with applications to color filtering
IEEE Transactions on Image Processing
PDE-Driven Adaptive Morphology for Matrix Fields
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
A Directional Rouy-Tourin Scheme for Adaptive Matrix-Valued Morphology
ISMM '09 Proceedings of the 9th International Symposium on Mathematical Morphology and Its Application to Signal and Image Processing
Fast GL(n)-Invariant Framework for Tensors Regularization
International Journal of Computer Vision
New Riemannian techniques for directional and tensorial image data
Pattern Recognition
A generic approach to the filtering of matrix fields with singular PDEs
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
Adaptive Continuous-Scale Morphology for Matrix Fields
International Journal of Computer Vision
Journal of Mathematical Imaging and Vision
Flexible segmentation and smoothing of DT-MRI fields through a customizable structure tensor
ISVC'06 Proceedings of the Second international conference on Advances in Visual Computing - Volume Part I
Harmonic analysis filtering techniques for forced and decaying homogeneous isotropic turbulence
Computers & Mathematics with Applications
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Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo's concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts.