Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
Regularization, Scale-Space, and Edge Detection Filters
Journal of Mathematical Imaging and Vision
Relations Between Regularization and Diffusion Filtering
Journal of Mathematical Imaging and Vision
Scale-Space Theory in Computer Vision
Scale-Space Theory in Computer Vision
Gaussian Scale-Space Theory
An Extended Class of Scale-Invariant and Recursive Scale Space Filters
IEEE Transactions on Pattern Analysis and Machine Intelligence
Scale Adaptive Filtering Derived from the Laplace Equation
Proceedings of the 23rd DAGM-Symposium on Pattern Recognition
On the Axioms of Scale Space Theory
Journal of Mathematical Imaging and Vision
IJCAI'83 Proceedings of the Eighth international joint conference on Artificial intelligence - Volume 2
Image restoration combining a total variational filter and a fourth-order filter
Journal of Visual Communication and Image Representation
Blind Deconvolution Models Regularized by Fractional Powers of the Laplacian
Journal of Mathematical Imaging and Vision
Properties of Higher Order Nonlinear Diffusion Filtering
Journal of Mathematical Imaging and Vision
Two Enhanced Fourth Order Diffusion Models for Image Denoising
Journal of Mathematical Imaging and Vision
Anisotropic $\alpha$-Kernels and Associated Flows
SIAM Journal on Imaging Sciences
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We investigate the use of fractional powers of the Laplacian for signal and image simplification. We focus both on their corresponding variational techniques and parabolic pseudodifferential equations. We perform a detailed study of the regularisation properties of energy functionals, where the smoothness term consists of various linear combinations of fractional derivatives. The associated parabolic pseudodifferential equations with constant coefficients are providing the link to linear scale-space theory. These encompass the well-known α-scale-spaces, even those with parameter values α 1 known to violate common maximum-minimum principles. Nevertheless, we show that it is possible to construct positivity-preserving combinations of high and low-order filters. Numerical experiments in this direction indicate that non-integral orders play an essential role in this construction. The paper reveals the close relation between continuous and semi-discrete filters, and by that helps to facilitate efficient implementations. In additional numerical experiments we compare the variance decay rates for white noise and edge signals through the action of different filter classes.