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
SIAM Journal on Numerical Analysis
Image Sharpening by Flows Based on Triple Well Potentials
Journal of Mathematical Imaging and Vision
Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising
International Journal of Computer Vision
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences)
Theoretical foundations for spatially discrete 1-D shock filtering
Image and Vision Computing
Staircasing in semidiscrete stabilised inverse linear diffusion algorithms
Journal of Computational and Applied Mathematics
Image segmentation and edge enhancement with stabilized inverse diffusion equations
IEEE Transactions on Image Processing
Forward-and-backward diffusion processes for adaptive image enhancement and denoising
IEEE Transactions on Image Processing
Image Sharpening via Sobolev Gradient Flows
SIAM Journal on Imaging Sciences
Novel schemes for hyperbolic PDEs using osmosis filters from visual computing
SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
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Forward-and-backward (FAB) diffusion is a method for sharpening blurry images (Gilboa et al. 2002). It combines forward diffusion with a positive diffusivity and backward diffusion where negative diffusivities are used. The well-posedness properties of FAB diffusion are unknown, and it has been observed that standard discretisations can violate a maximum-minimum principle. We show that for a novel nonstandard space discretisation which pays specific attention to image extrema, one can apply a modification of the space-discrete well-posedness and scale-space framework of Weickert (1998). This allows to establish well-posedness and a maximum-minimum principle for the resulting dynamical system. In the fully discrete 1-D case with an explicit time discretisation, a maximum-minimum principle and total variation reduction are proven in spite of the fact that negative diffusivities may appear. This provides a theoretical justification for applying FAB diffusion to digital images.