Tim E. Olson

Wavelet localization of the Radon transform

The authors develop an algorithm which significantly reduces radiation exposure in X-ray tomography, when a local region of the body is to be imaged. The algorithm uses the properties of wavelets to essentially localize the Radon transform. This algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image, not the image module the nullspace of the local tomography operator, or the Lambda transform of the image. This is possible because the authors do not truly invert the interior Radon transform, but rather sample the Radon transform sparsely away from the local region of interest. Much attention in the field has been directed towards localized tomography. The authors believe that this technique represents a significant contribution towards this effort. >

A wreath product group approach to signal and image processing .I. Multiresolution analysis

We propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and, in particular, image processing. We pay significant attention to groups constructed as wreath products of cyclic groups. Within this large class of groups, our approach recovers the discrete Fourier transform (DFT), the Haar wavelet transform, various multichannel pyramid filter banks, and other aspects of multiresolution analysis as special cases of a more general phenomenon. In addition, the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals. Our approach relies on a synthesis of ideas found in the early work of Holmes (1987, 1990), Karpovsky and Trachtenberg (1985), and others on noncommutative filtering, as well as Diaconiss (1989) spectral analysis approach to understanding data.

A wreath product group approach to signal and image processing .II. Convolution, correlation, and applications

For pt.I see ibid., vol.48, no.1, p.102-32 (2000). This paper continues the investigation of the use of spectral analysis on certain noncommutative finite groups-wreath product groups-in digital signal processing. We describe the generalization of discrete cyclic convolution in convolution over these groups and show how it reduces to multiplication in the spectral domain. Finite group-based convolution is defined in both the spatial and spectral domains and its properties established. We pay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point in terms of operations with specific signals and filters. Group-based correlation is defined in a natural way, and its properties follow from those of convolution (the detection of similarity of perceptually similar signals) and an application of correlation (the detection of similarity of group-transformed signals). Several examples using images are included to demonstrate the ideas pictorially.

Wavelet localization of the Radon transform in even dimensions

One of the phenomena associated with the Radon transform is the following: in odd dimensions, local values of a function f:R/sup n/ to R can be determined by local measurements of the integrals of f over (n-1)-dimensional hyperplanes; in even dimensions, local values are globally dependent on the integrals over hyperplanes. The wavelet transform is used to essentially localize the Radon transform in even dimensions. It is believed that this will be significant in the field of medical imaging.<<ETX>>

Optimal time-frequency projections for localized tomography.

An algorithm for recovering a function from essentially localized values of its Radon transform and sparse nonlocal values was outlined in Reference 13. That algorithm utilized the time-frequency properties of wavelets, coupled with the range theorems for the Radon transform, to localize essentially the dependence of the Radon transform. In this paper we utilize alternative time-frequency projections which were introduced by Coifman and Meyer (4). We present evidence that these bases are optimal according to our criterion for localized tomography. These bases require significantly less data than the wavelet bases that were used in Reference 13. Finally, we present numerical results supporting this work.

Limited angle tomography via multiresolution analysis and oversampling

A method is presented for reconstructing a function from its line integrals. This problem arises in imaging whenever the physical constraints of the system prohibit the gathering of the line integrals over the angles - pi /2< theta <0. The first portion of the algorithm decomposes the limited angle operator L into a tensor product structure that allows an efficient calculation of its singular value decomposition. The second portion of the algorithm uses multiresolution analysis in order to mollify the problem, which is known to be extremely ill conditioned. It is thought that these methods may lead to a more stable algorithm for limited angle reconstruction.<<ETX>>

Wreath products for image processing

We present a wreath product approach for matched filtering to detect rotated copies of a template in an image. We view the image as a homogeneous space for a wreath product, a noncommutative symmetry group. The corresponding Fourier analysis has a natural multiresolution structure and accompanying efficient algorithm which we explain and illustrate with an example. The associated matched filter is a new example of the use of a noncommutative convolution for image processing. Numerical experiments are described in which this noncommutative approach outperforms standard Fourier-based methods.

Reduced motion artifacts in medical imaging by adaptive spatio-temporal reconstruction

AbstractIn this paper we introduce an algorithm for imaging a time varying object

Stabilized inversion for limited angle tomography

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