Gilbert G. Walter

General sampling theorems for functions in reproducing kernel Hilbert spaces

In this paper we prove general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space. We present details of this approach as applied to the standard sampling theory and its extension to nonuniform sampling. The general theory for orthogonal sampling sequences and nonorthogonal sampling sequences is developed. Our approach includes as concrete cases many recent extensions, for example, those based on the Sturm-Liouville transforms, Jacobi transforms, Laguerre transforms, Hankel transforms, prolate spherical transforms, etc., finite-order sampling theorems, as well as new sampling theorems obtained by specific choices of the RKHS. In particular, our setting includes nonorthogonal sampling sequences based on the theory of frames. The setting and approach enable us to consider various types of errors (truncation, aliasing, jitter, and amplitude error) in the same general context.

Object representation and recognition in shape spaces

In this paper, we describe a shape space based approach for invariant object representation and recognition. In this approach, an object and all its similarity transformed versions are identified with a single point in a high-dimensional manifold called the shape space. Object recognition is achieved by measuring the geodesic distance between an observed object and a model in the shape space. This approach produced promising results in 2D object recognition experiments: it is invariant to similarity transformations and is relatively insensitive to noise and occlusion. Potentially, it can also be used for 3D object recognition.

Approximation of the delta function by wavelets

Abstract Approximation of the delta distribution by functions associated with wavelets is studied. These functions, the reproducing kernels of the dilation subspaces, are shown to converge to δ(x − y) in the Sobolev space H−α. The error rate is found and three applications are given.

Sampling Bandlimited Functions of Polynomial Growth

Two new versions of the sampling theorem extended to functions whose Fourier transform is a generalized function are given. One involves a correction by means of an arbitrary polynomial and the other involves

A sampling expansion for nonbandlimited signals in chromatic derivatives

(C,\alpha )

Differential operators which commute with characteristic functions with applications to a lucky accident

-summability. The best approximation to nonbandlimited functions of polynomial growth by functions whose transform has compact support in the Sobolev norm is found.

Wavelet subspaces with an oversampling property

In this paper, we construct infinite-band filterbanks for perfect reconstruction (PR) using Hermite polynomials and Hermite functions. The analysis filters are linear combinations of derivative operators based on these polynomials-the so-called chromatic derivative filters. Together with the synthesis filterbanks, they give PR for a large class of signals that may have infinite bandwidth. Several other related filterbanks are discussed as well. The error in reconstruction for finite-channel chromatic derivative filterbanks is calculated. Examples are given to demonstrate the use of these chromatic filterbanks.

On complex eigenvalues of compartmental models

The differential operator associated with the prolate spheroidal functions commutes with a certain integral operator. Its kernel is the reproducing kernel of the Paley-Wiener space restricted to a bounded interval. This “lucky accident” occurs because both the differential operator and its Fourier transform commute with multiplication by the characteristic function of an interval. It is the only operator of its type to do so.

Periodic Wavelets from Scratch

Abstract In the classical Shannon sampling theorem, the same sequence of functions is both orthonormal and a sampling sequence. This is not true for most wavelet subspaces in which the sampling functions and the orthonormal bases are different. However by oversampling at double the rate the property of the Shannon wavelets is extended to a much larger class which includes the Meyer wavelets. In fact together with another condition, it characterizez them.

Bayes estimation of the binomial parameter n

Abstract Conditions under which the eigenvalues of the matrix of a compartmental model have a nonzero imaginary part are studied. Inequalities for the total imaginary part are obtained. The effect of excretions and combinations of cycles on this imaginary part are studied.