Mathematical Sciences: New Results in Sampling and Wavelet Applications in Tomography

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9500909 Walnut The project consists of two parts. First, the investigator proposes to apply new results in sampling theory to do high-resolution signal processing by combining several low-resolution versions of a given signal. It was observed by Berenstein and others that a mathematical model of this problem (called multisensor deconvolution) is uniquely solvable but ill- posed, and difficult to solve numerically. The investigator has found Shannon-type sampling formulas on unions of regular lattices with incommensurate densities, which provide simple solutions to the multichannel deconvolution problem in special cases. The following applications are envisioned: 1) Sharpening measurements taken by CT scanners by combining measurements from several low-resolution detectors. This could ultimately lead to inexpensive measuring devices with resolution comparable to expensive high-resolution machines. 2) A recently proposed scheme to increase resolution in an industrial tomography problem involves sampling on unions of regular grids of incommensurate size. The investigator's sampling results could lead to a complete solution to this problem. 3) New techniques are required to do "superresolution" in electronic imaging since the theoretical limit of pixel resolution is rapidly being reached. Deconvolution at high-resolution of several low-resolution images is a natural approach to this problem. Second, the investigator and others have successfully shown that wavelets are a natural tool for recovering edge features of an image from local Radon transform data. The investigator proposes to develop these local algorithms with the following goals envisioned: 1) Find ways to recover locally density as well as edge features of an image, and develop an algorithm competitive with existing local tomography algorithms and 2) Identify the wavefront set of an image using wavelets, and ultimately apply the techniques to the at tenuated Radon transform. The first part of the project is concerned with increasing the resolution of remote sensing devices such as electronic cameras. Suppose, for example, that a CCD camera has a resolution of 1 millimeter, that is, it can distinguish features of a scene that are at least 1 mm on a side. Features smaller than that are blurred into their surroundings. Berenstein and others observed that some classical mathematics led to a possible way to increase resolution without designing an expensive high-resolution device. If one took several images of the scene with cameras identical to the first but which had slightly poorer resolutions than the first (this could also be achieved by repositioning the same camera), then in theory one could recover features of the original scene to arbitrary resolution. In practice, arbitrary resolution is not possible, but it seems that real increases in resolution can be achieved. The investigator has formulated an approach to this problem from the point of view of sampling theory. Using this approach, the investigator has been able to produce numerically stable twenty-fold increases in resolution in a one-dimensional model problem. More work is required, but the following applications are envisioned: 1) Producing high-resolution measurements from CT scanners by combining the measurements from several low-resolution scanners and 2) Increasing the resolution in a specific industrial tomography problem by a direct application of the investigator's new sampling results. The second part of the project is concerned with "local" CT scans. In existing CT scanners, an entire slice of a patient's body must be exposed to radiation even if the doctor is interested in looking at only a small region. The investigator proposes to use a new and powerful signal processing technique called wavelets to find efficient algorithms for obtaining an image of a small area of a patient's body while only exposing the area of interest to radiation. It is hoped that these techniques will be competitive with existing so-called local tomography algorithms.