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
Image selective smoothing and edge detection by nonlinear diffusion. II
SIAM Journal on Numerical Analysis
Image selective smoothing and edge detection by nonlinear diffusion
SIAM Journal on Numerical Analysis
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Signal and image restoration using shock filters and anisotropic diffusion
SIAM Journal on Numerical Analysis
Coherence-Enhancing Diffusion Filtering
International Journal of 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
Tracking With Sobolev Active Contours
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences)
Generalized Gradients: Priors on Minimization Flows
International Journal of Computer Vision
International Journal of Computer Vision
Image Compression with Anisotropic Diffusion
Journal of Mathematical Imaging and Vision
Graph spectral image smoothing using the heat kernel
Pattern Recognition
Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Two new nonlinear nonlocal diffusions for noise reduction
Journal of Mathematical Imaging and Vision
IJCAI'83 Proceedings of the Eighth international joint conference on Artificial intelligence - Volume 2
New possibilities with Sobolev active contours
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Anisotropic diffusion of multivalued images with applications to color filtering
IEEE Transactions on Image Processing
Modified curvature motion for image smoothing and enhancement
IEEE Transactions on Image Processing
Fourth-order partial differential equations for noise removal
IEEE Transactions on Image Processing
Forward-and-backward diffusion processes for adaptive image enhancement and denoising
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
Fractional-Order Anisotropic Diffusion for Image Denoising
IEEE Transactions on Image Processing
New Possibilities in Image Diffusion and Sharpening via High-Order Sobolev Gradient Flows
Journal of Mathematical Imaging and Vision
A fully implicit framework for Sobolev active contours and surfaces
DAGM'11 Proceedings of the 33rd international conference on Pattern recognition
A Gauss-Newton Method for the Integration of Spatial Normal Fields in Shape Space
Journal of Mathematical Imaging and Vision
Nonlinear least squares and Sobolev gradients
Applied Numerical Mathematics
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Motivated by some recent work in active contour applications, we study the use of Sobolev gradients for PDE-based image diffusion and sharpening. We begin by studying, for the case of isotropic diffusion, the gradient descent/ascent equation obtained by modifying the usual metric on the space of images, which is the $L^2$ metric, to a Sobolev metric. We present existence and uniqueness results for the Sobolev isotropic diffusion, derive a number of maximum principles, and show that the differential equations are stable and well-posed both in the forward and backward directions. This allows us to apply the Sobolev flow in the backward direction for sharpening. Favorable comparisons to the well-known shock filter for sharpening are demonstrated. Finally, we continue to exploit this same well-posed behavior both forward and backward in order to formulate new constrained gradient flows on higher order energy functionals which preserve the first order energy of the original image for interesting combined smoothing and sharpening effects.