Numerical Harmonic Analysis and Image Processing

Abstract

Signal processing has become an essential part of contemporary scientific and technological activity and even of everyday life. It is used in telecommunications, medical imaging, geophysics, and the transmission and analysis of satellite images. The objectives of signal processing are analysis and diagnostics, coding and compression, and transmission and reconstruction. Common to all these objectives is the extraction of information of a signal, which is present but hidden in its complex representation. Thus, a major issue is to represent the given data as well as possible. Clearly, the optimal representation of a signal has to be tied to an objective goal. A signal representation that is optimal for compression can be disastrous for analysis. A transform that is optimal for one class of signals can yield modest results for a different class of signals. In the last decade a number of new tools have been developed to analyze, compress, transmit, and reconstruct digital signals. In this chapter we present an overview of methods from numerical harmonic analysis that have proven to be useful in digital image processing. In the first part of this chapter we discuss numerical methods designed for the restoration of missing data in digital images and the reconstruction of an image from scattered data. We describe efficient and robust numerical algorithms for the reconstruction of multidimensional (essentially) band-limited signals. We demonstrate the performance of the proposed methods by applying them to reconstruction problems in areas as diverse as medical imaging, exploration geophysics, and digital image restoration. In the second part we focus on image analysis and optimal image representation. Time-frequency methods, such as wavelets and Gabor expansions, have been recognized as powerful tools for various tasks in image processing. We give an overview of recent developments in Gabor theory.