This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as...

Originally, mathematical morphology was a theory of signal transformations which are invariant under Euclidean translations. An interest in the extension of mathematical morphology to spatially-variant (SV) operators has emerged due to the requirements imposed by numerous applications in adaptive signal (image) processing. This paper presents a general theory of spatially-variant mathematical m...

In this paper, we investigate the role of square functions defined for a d-tuple commuting Ritt operators $$(T_1,\ldots ,T_d)$$ acting on general Banach space X. Firstly, prove that if admits $$H^\infty $$ joint functional calculus, then it verifies various function estimates. Then study converse when every $$T_k$$ is R-Ritt operator. Under last hypothesis, and X K-convex space, show estimates ...

Mathematical morphology is a successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of supremum and infimum with respect to an order. From dilation and erosion one can build readily other useful elementary morphological operators and filters, such as opening, closing, morphological top-ha...

Nonlinear multiscale algorithms often involve nonlinear perturbations of linear coarse-to-fine prediction operators S (also called subdivision operators). In order to control these perturbations, estimates of the “commutator” SF − FS of S with a sufficiently smooth map F are needed. Such estimates in terms of bounds on higher-order differences of the underlying mesh sequences have already appea...

Many useful morphological filters are built as long concatenations of erosions and dilations: openings, closings, size distributions, sequential filters, etc. This paper proposes a new algorithm implementing morphological dilation and erosion of functions. It supports rectangular structuring element, runs in linear time w.r.t. the image size and constant time w.r.t. the structuring element size...

Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standar...

Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another extension concerns bipolar fuzzy sets, where ...

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionist...

The fractional Laplacian (−�)γ/2 commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential Iγ which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (contin...