arXiv subjects

Jeffrey A. Hogan

Publications and source records attributed to Jeffrey A. Hogan.

12 recordsLinked to original sources

arXiv2026

Quaternion-Valued Wavelets on the Plane: A Construction via the Douglas-Rachford Approach

This paper presents a reformulation of the construction of nonseparable multiresolution quaternion-valued wavelets on the plane as a feasibility problem. The constraint sets in the feasibility problem are derived from the standard conditions of smoothness, compact support, and orthonormality. To solve the resulting feasibility problems, we employ a product space formulation of the Douglas-Rachford algorithm. This approach yields novel examples of nonseparable, multiresolution, compactly supported, smooth, and orthonormal quaternion-valued wavelets on the plane. Additionally, by introducing a symmetry-promoting constraint, we construct symmetric quaternion-valued scaling functions on the plane.

math.CA↗

arXiv2025

Deconvolution of inclined channel elutriation data to infer particle size distribution

In this paper we investigate the application of optimisation techniques in the deconvolution of mineral fractionation data obtained from a mathematical model for the operation of a fluidised bed with a set of inclined parallel channels mounted above. The model involved the transport equation with a stochastic source function and a linearly increasing fluidisation rate, with the overflow solids being collected in a finite number of increments (bags). Deconvolution of this data is an ill-posed problem and regularisation is required to provide feasible solutions. Deconvolution with regularisation is applied to a synthetic feed consisting of particles of constant density that vary in size only. It was found that the feed size distribution could be successfully deconvolved from the bag weights, with an accuracy that improved as the rate acceleration of the fluidisation rate was decreased. The deconvolution error only grew linearly with error in the measured bag masses. It was also shown that combining data from two different liquids can improve the accuracy.

math.NA↗

arXiv2024

Constructing $3$-Dimensional Monogenic Homogeneous Functions

This paper is dedicated to the construction of multidimensional spherical monogenics. Firstly, we investigate the construction of monogenic functions in dimension $3$ by applying the Dirac operator to the orthonormal bases of spherical harmonics, resulting in orthogonal spherical monogenics. Additionally, we employ the reproducing kernel for monogenic functions and a specialized optimization method to derive various types of $3$-dimensional spherical harmonics and spherical monogenics.

math.AP↗

arXiv2024

Holistic Processing of Colour Images Using Novel Quaternion-Valued Wavelets on the Plane

Recently, novel quaternion-valued wavelets on the plane were constructed using an optimisation approach. These wavelets are compactly supported, smooth, orthonormal, non-separable and truly quaternionic. However, they have not been tested in application. In this paper, we introduce a methodology for decomposing and reconstructing colour images using quaternionic wavelet filters associated to recently developed quaternion-valued wavelets on the plane. We investigate its applicability in compression, enhancement, segmentation, and denoising of colour images. Our results demonstrate these wavelets as promising tools for an end-to-end quaternion processing of colour images.

cs.CV↗

arXiv2021

Clifford Prolate Spheroidal wave Functions

In the present paper, we introduce the multidimensional Clifford prolate spheroidal wave functions (CPSWFs) defined on the unit ball as eigenfunctions of a Clifford differential operator and provide a Galerkin method for their computation as linear combinations of Clifford-Legendre polynomials. We show that these functions are eigenfunctions of the truncated Fourier transformation. Then we investigate the role of the CPSWFs in the spectral concentration problem associated with balls in the space and frequency domains, the behaviour of the eigenvalues of the time-frequency limiting operator and their spectral accumulation property.

math.CA↗

arXiv2021

Sampling low-spectrum signals on graphs via cluster-concentrated modes: examples

We establish frame inequalities for signals in Paley--Wiener spaces on two specific families of graphs consisting of combinations of cubes and cycles. The frame elements are localizations to cubes, regarded as clusters in the graphs, of vertex functions that are eigenvectors of certain spatio--spectral limiting operators on graph signals.

math.SP↗

arXiv2020

Some adjacency-invariant spaces on products of short cycles

We study certain spaces of vertex functions on the Cayley graphs corresponding to N-fold products of the group of integers modulo m, where m=3, 4, or 5, that are invariant under the adjacency operator that maps a value at a given vertex to each of its neighbors. An application to spatio-spectral limiting, an analogue of time and band limiting, is also discussed.

math.CO↗

arXiv2020

Properties of Clifford Legendre Polynomials

Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra ${\mathbb R}_m$. New recurrence and Bonnet type formulae for these polynomials are proved, as their Fourier transforms are computed. Explicit representations in terms of spherical monogenics and Jacobi polynomials are given, with consequences including the interlacing of the zeros. In the case $m=2$ we describe a degeneracy between the even- and odd-indexed polynomials.

math.CA↗

arXiv2020

A Douglas-Rachford construction of non-separable continuous compactly supported multidimensional wavelets

After re-casting the $n$-dimensional wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas--Rachford algorithm is employed in the search for one- and two-dimensional wavelets. New one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets in the case where the dilation on the plane is the standard $D_af(t)=a^{-1}f(t/a)$ $(t\in{\mathbb R}^n, a>0)$.

math.CA↗

arXiv2019

Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

B-splines $B_{q}$, $\Sc q > 1$, of quaternionic order $q$, for short quaternionic B-splines, are quaternion-valued piecewise Müntz polynomials whose scalar parts interpolate the classical Schoenberg splines $B_{n}$, $n\in\N$, with respect to degree and smoothness. As the Schoenberg splines of order $\geq 3$, they in general do not satisfy the interpolation property $B_{q}(n-k) = δ_{n,k}$, $n,k\in\Z$. However, the application of the interpolation filter $1/\sum\limits_{k\in\Z} \widehat{B}_{q}(ξ+2 πk)$---if well-defined---in the frequency domain yields a cardinal fundamental spline of quaternionic order that does satisfy the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer's Lemma and allow for a family of sampling, respectively, interpolation series.

math.FA↗

arXiv2018

Spatio-spectral limiting on hypercubes: eigenspaces

The operator that first truncates to a neighborhood of the origin in the spectral domain then truncates to a neighborhood of the origin in the spatial domain is investigated in the case of Boolean cubes. This operator is self adjoint on a space of bandlimited signals. The eigenspaces of this iterated projection operator are studied and are shown to depend fundamentally on the neighborhood structure of the cube when regarded as a metric graph with path distance equal to Hamming distance.

math.FA↗

arXiv2016

Quaternionic B-Splines

We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the solutions of distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between $B_q$ and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of $B_q$ is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and $L^p$ convergence of the quaternionic B-splines to quarternionic Gaussian functions.

math.FA↗