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
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
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
Image Sharpening via Sobolev Gradient Flows
SIAM Journal on Imaging Sciences
Fourth-order partial differential equations for noise removal
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
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We study the use of high-order Sobolev gradients for PDE-based image smoothing and sharpening, extending our previous work on this problem. In particular, we study the gradient descent equation on the heat equation energy functional obtained by modifying the usual metric on the space of images, which is the L 2 metric, to a weighted H k Sobolev metric. We present existence and uniqueness results which show that the Sobolev diffusion PDE are well-posed both in the forward and backward direction. Furthermore, we perform a Fourier analysis on the scale space generated by the Sobolev PDE and show that as the order of the Sobolev metric tends to infinity, the Sobolev gradients converge to a Gaussian smoothed L 2 gradient. We then present experimental results which exploit the theoretical stability results by applying the various Sobolev gradient flows in the backward direction for image sharpening effects. Furthermore, we show that as the Sobolev order is increased, the sharpening effects become more global in nature and more immune to noise.