The article introduces digital image restoration to the reader who is just beginning in this field, and provides a review and analysis for the reader who may already be well-versed in image restoration. The perspective on the topic is one that comes primarily from work done in the field of signal processing. Thus, many of the techniques and works cited relate to classical signal processing approaches to estimation theory, filtering, and numerical analysis. In particular, the emphasis is placed primarily on digital image restoration algorithms that grow out of an area known as "regularized least squares" methods. It should be noted, however, that digital image restoration is a very broad field, as we discuss, and thus contains many other successful approaches that have been developed from different perspectives, such as optics, astronomy, and medical imaging, just to name a few. In the process of reviewing this topic, we address a number of very important issues in this field that are not typically discussed in the technical literature.

Digital Image Restoration

This paper is a pragmatic tutorial review of the cepstrum literature focusing on data processing. The power, complex, and phase cepstra are shown to be easily related to one another. Problems associated with phase unwrapping, linear phase components, spectrum notching, aliasing, oversampling, and extending the data sequence with zeros are discussed. The advantages and disadvantages of windowing the sampled data sequence, the log spectrum, and the complex cepstrum are presented. The influence of noise upon the data processing procedures is discussed throughout the paper, but is not thoroughly analyzed. The effects of various forms of liftering the cepstrum are described. The results obtained by applying whitening and trend removal techniques to the spectrum prior to the calculation of the cepstrum are discussed. We have attempted to synthesize the results, procedures, and information peculiar to the many fields that are finding cepstrum analysis useful. In particular we discuss the interpretation and processing of data in such areas as speech, seismology, and hydroacoustics. But we must caution the reader that the paper is heavily influenced by our own experiences; specific procedures that have been found useful in one field should not be considered as totally general to other fields. It is hoped that this review will be of value to those familiar with the field and reduce the time required for those wishing to become so.

The cepstrum: A guide to processing

Constrained least squares estimation is a technique for solution of integral equations of the first kind The problem of image restoration requires the solution of an integral equation of the first kind However, application of constrained least squares estimation to image restoration requires the solution of extremely large linear systems of equations In this paper we demonstrate that, for convolution-type models of image restoration, special properties of the linear system of equations can be used to reduce the computational requirements The necessary computations can be carried out by the fast Fourier transform, and the constrained least squares estimate can be constructed in the discrete frequency domain A practical procedure for constrained least squares estimation is presented, and two examples are shown as output from a program for the CDC 7600 computer which performs the constrained least squares restoration of digital images

The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer

The rapid rate at which the field of digital picture processing has grown in the past five years had necessitated extensive revisions and the introduction of topics not found in the original edition.

Table of Contents

Introduction. Mathematical Preliminaries. Visual Perception. Digitization. Compression. Enhancement. Restoration. Reconstruction. Index.

Digital Picture Processing, Volume 1

Blind Image Deconvolution

In this paper we prove the discrete convolution theorem by means of matrix theory. The proof makes use of the diagonalization of a circulant matrix to show that a circular convolution is diagonalized by the discrete Fourier transform. The diagonalization of the circular convolution shows that the eigenvalues of a circular convolution operator are identical with the discrete Fourier frequency spectrum.

A matrix theory proof of the discrete convolution theorem

The difficulty of numerical deconvolution is explained by examining the matrix of the linear equations generated by approximating the linear system superposition integral. It is shown that any row of the matrix is approximately a linear combination of other rows.

B. Hunt

Papers

A matrix theory proof of the discrete convolution theorem

A theorem on the difficulty of numerical deconvolution