Related papers: An Algebraic Perspective on Multivariate Tight Wav…
Research on refinable functions in wavelet theory is mostly focused to localized functions. However it is known, that polynomial functions are refinable, too. In our paper we investigate on conversions between refinement masks and…
Finding efficient representations is one of the most challenging and heavily sought problems in mathematics. Representation using shearlets recently receives a lot of attention due to their desirable properties in both theory and…
In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame…
The purpose of this paper is to present new classes of function systems as part of multiresolution analyses. Our approach is representation theoretic, and it makes use of generalized multiresolution function systems (MRSs). It further…
Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called…
Generalizing wavelets by adding desired redundancy and flexibility,framelets are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing…
Framelets (a.k.a. wavelet frames) are of interest in both theory and applications. Quite often, tight or dual framelets with high vanishing moments are constructed through the popular oblique extension principle (OEP). Though OEP can…
By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…
We provide constructive necessary and sufficient conditions for a family of periodic wavelets to be a Parseval wavelet frame. The criterion generalizes unitary and oblique extension principles. The case of one wavelet generator and…
We proved that for any matrix dilation and for any positive integer $n$, there exists a compactly supported tight wavelet frame with approximation order $n$. Explicit methods for construction of dual and tight wavelet frames with a given…
The wavelet scattering transform creates geometric invariants and deformation stability. In multiple signal domains, it has been shown to yield more discriminative representations compared to other non-learned representations and to…
We study the properties of a set of vectors called tight frames that obtained as the orthogonal projection of some orthonormal basis of $\R^n$ onto $\R^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross…
We investigate the passivity constraints on the relations between transmission, reflection, and absorption eigenvalues in linear time-invariant systems. Using techniques from matrix analysis, we derive necessary and sufficient conditions…
We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures;…
In analogy with steerable wavelets, we present a general construction of adaptable tight wavelet frames, with an emphasis on scaling operations. In particular, the derived wavelets can be "dilated" by a procedure comparable to the operation…
Multivalency is a common biological mechanism of formation of strong reversible and selective bonds by grouping weak bonds. Polymers often act as a scaffold to which multiple binding groups are attached. Here I present an analytical theory…
Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We…
Understanding the mechanisms underlying deep neural networks remains a fundamental challenge in machine learning and computer vision. One promising, yet only preliminarily explored approach, is feature inversion, which attempts to…
Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go,…