The problem of recognition of objects in images is investigated from the simultaneous viewpoints of image bandwidth compression and automatic target recognition. A scenario is suggested in which recognition is implemented on features in the block cosine transform domain which is useful for data compression as well. While most image frames would be processed by the automatic recognition algorithms in the compressed domain without need for image reconstruction, this still allows for visual image classification of targets with poor recognition rates (by human viewing at the receiving terminal). It has been found that the Mandala sorting of the block cosine domain results in a more effective domain for selecting target identification parameters. Useful features from this Mandala/cosine domain are developed based upon correlation parameters and homogeneity measures which appear to successfully discriminate between natural and man-made objects. The Bhattacharyya feature discriminator is used to provide a 10:1 compression of the feature space for implementation of simple statistical decision surfaces (Gaussian and minimum distance classification). Imagery sensed in the visible spectra with a resolution of approximately 5-10 ft is used to illustrate the success of the technique on targets such as ships to be separated from clouds. A data set of 38 images is used for experimental verification with typical classification results ranging from the high 80's to low 90 percentile regions depending on the options choosen.

This paper presents a degree of freedom or information content analysis of images in the context of digital image processing. As such it represents an attempt to quantify the number of truly independent samples one gathers with imaging devices. The degrees of freedom of a sampled image itself are developed as an approximation problem. Here, bicubic splines with variable knots are employed in an attempt to answer the question as to what extent images are finitely representable in the context of digital sensors and computers. Relatively simple algorithms for good knot placement are given and result in spline approximations that achieve significant parameter reductions at acceptable error levels. The knots themselves are shown to be useful as an indicator of image activity and have potential as an image segmentation device, as well as easy implementation in CCD signal processing and focal plane smart sensor arrays. Both mathematical and experimental results are presented.

This paper presents a theoretical analysis and computational technique for constrained least squares image restoration using spline basis functions. A realistic continuous–discrete physical imaging model has been adopted throughout the formulation. The optical system is assumed to be incoherent, and the general problem of image restoration with space-variant or space-invariant point-spread function degradations has been studied.

This paper is intended as a tutorial review of certain digital image processing transform techniques utilizing the notion of outer product expansions. Examples from Fourier, Walsh, Haar, and other well known transforms are reviewed in the notation of matrix-vector outer products; and implementation of the singular value decomposition (SVD) of large sized images is presented. The use of the SVD as an aid in image restoration utilizing the pseudoinverses is presented. Conditions on the point spread matrix are investigated in the light of singular value decomposition, Kronecker products, and general imaging conditions.

This correspondence presents results concerning the display of interpolated images for cosmetically pleasing effects. Replication, bilinear interpolation, and various cubic spline function interpolators are investigated as to numeric computational difficulty and psychovisually pleasing results. A degree of freedom analysis is provided to demonstrate the fact that apparent image improvement does not necessarily increase the inherent quantitative information within an image.

The numerical techniques of transform image coding are well known in the image bandwidth compression literature. This concise paper presents a new transform method in which the singular values and singular vectors of an image are computed and transmitted instead of transform coefficients. The singular value decomposition (SVD) method is known to be the deterministically optimal transform for energy compaction [2]. A systems implementation is hypothesized, and a variety of coding strategies is developed. Statistical properties of the SVD are discussed and a self adaptive set of experimental results is presented, Imagery compressed to 1, 1.5, and 2.5 bits per pixel with less than 1.6, 1, and 1/3 percent, respective mean-square error is displayed. Finally, additional image coding scenarios are postulated for further consideration.

In pattern recognition, the raw data and dimensionality of the measurement space is usually very large. Therefore, some form of dimensionality reduction has been commonly considered as a practical preprocessing method for feature selection. Based on a method that increases the variance while maintaining local structure, a technique is developed to determine intrinsic dimensionality. A cost function is introduced to guide the maintenance of the rank order and therefore local structure. Two criteria of using the cost function to increase the variance have been introduced. Several methods of defining the local regions are suggested. A program is implemented and tested to find the intrinsic dimensionality of a variety of experimental data.

This paper provides a tutorial exposition of the possible use of sequency filters for the processing of signals that are functions of two space variables. Historically, optical and, more recently, digital techniques have been used for two variable processing of images with the computer providing a domain for other than sinecosine filtering. Sequency filters based on Walsh or similar functions can be implemented with about equal effort for time and space variable signals. They could actually be built with electronic components up to the ultimate case of signals with one time and three space variables. Two kinds of filters have emerged so far: instantaneous filters and sampling filters. In the following sections we briefly discuss other possible transforms for space filters before investigating sequency-oriented filtering techniques. The sequency techniques include instantaneous filters, efficient practical implementation, and some potential circuits for imagery applications.

This paper develops performance criteria for evaluating transform data coding schemes under computational constraints. Computational constraints that conform with the proposed basis-restricted model give rise to suboptimal coding efficiency characterized by a rate-distortion relationR(D)similar in form to the theoretical rate-distortion function. Numerical examples of this performance measure are presented for Fourier, Walsh, Haar, and Karhunen-Loève transforms.

An important aspect in mathematical pattern recognition is the usually noninvertible transformation from the pattern space to a reduced dimensionality feature space that allows a classification process to be implemented on a reasonable number of features. Such feature-selecting transformations range from simple coordinate stretching and shrinking to highly complex nonlinear extraction algorithms. A class of feature-selection transformations to which this note addresses itself is that given by multidimensional rotations. Unitary transformations of particular interest are the Karhunen-Loeve, Fourier, Hadamard or Walsh, and the Haar transforms. A character recognition experiment is selected for exemplary purposes and the use of features in the rotated spaces results in effective minimum distance classification.

An algorithm is presented which enables certain matrix multiplications in a digital computer to be implemented with a considerable savings in storage and computational operations. For NxN matrix vector multiplication (that is a multiplication of a matrix by a vector) a maximum of N Σn-1 r=0 pr, storage words are necessary compared to normal full matrix storage requirements of N2locations. In addition only N Σn-1 r=0 Pr, computer operations are necessary compared to N2operation in a general vector matrix multiplication. N is chosen to be a highly composite number, N= πn-1 r=0 pr, and the pr, are integers. The algorithm takes advantage of the redundancies in the definition of certain matrices and develops a matrix description based on the number of degrees of freedom necessary in defining an NxN matrix. The algorithm is optimal in the sense that it describes a vector matrix multiplication in exactly as many operations as available degrees of freedom N σn-1 r=0 Pr(no greater number of degrees of freedom could be implemented in fewer operations). The matrix transformation formulation is based largely on noting the use of lexicographic positional digit notation in keeping track of the few parameters describing the final N by N matrix. The algorithm includes the generation of the fast Fourier transform, fast Hadamard transform, fast Walsh transform, fast Kronecker matrix transform, and an infinite class of transformations unnamed but potentially useful in generalized spectral analysis as well as coding, bandwidth reduction, and feature selection.

This bibliography contains articles and books on the subject of rate distortion theory. Growing interest in the subject is evident by the large number of recent publications.

A closed product form and a simplified algorithm for efficient matrix operation is presented for a class of generalized Kronecker matrices.Powers of two Kronecker matrices are further described where the closed form representation is easily implemented with parallel binary register operations.Orthogonal and symmetric orthogonal Kronecker matrices are described in the context of generalized spectral analysis.Specific applications are presented which include the Fourier, Hadamard, and other transformations.

The book provides a new and interesting approach to the field of information and communication theory and is a welcome and thought- provoking exposition on a subject too often reviewed by conventional Fourier methods. Dr. Harmuth develops a generalized theory of signal transmission based on the concepts of orthogonal functions, the subject of his first chapter. His presentation is soon directed to the particular class of orthogonal functions known as the system of Walsh functions which have the convenient property of taking on only plus and minus unity values. Thus his subject becomes particularly relevant to the transistor and integrated circuit technology of today. The mathematical theories expounded in the text are unique in that experimental implementation is not far away. The first chapter develops the analytic tools necessary for applying the generalized orthogonal function theory to Walsh function applications in various phases of the communication process. The chapter is exemplary of Dr. Harmuth's historical accuracy and scholarly style in his exacting references of authors of both published and unpublished works contributing to this area. The fifteen pages of references at the end of the text attest to this fact.

A technique is presented to implement a class of orthogonal transformations on the order of pN logpN operations. The technique is due to Good [1] and implements a fast Fourier transform, fast Hadamard transform, and a variety of other orthogonal decompositions. It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method. A generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible. Finally, an eigenvalue presentation is provided as a possible means of characterizing some of the transforms with similar parameters.