Several state-of-the-art mathematical models useful in image processing are considered. These models include the traditional fast unitary transforms, autoregessive and state variable models as well as two-dimensional linear prediction models. These models introduced earlier [51], [52] as low-order finite difference approximations of partial differential equations are generalized and extended to higher order in the framework of linear prediction theory. Applications in several image Processing problems, including image restoration, smoothing, enhancement, data compression, spectral estimation, and filter design, are discussed and examples given.

Advances in mathematical models for image processing

Two-dimensional signals are normally processed as rectangularly sampled arrays; i.e., they are periodically sampled in each of two orthogonal independent variables. Another form of periodic sampling, hexagonal sampling, offers substantial savings in machine storage and arithmetic computations for many signal processing operations. In this paper, methods for the processing of two-dimensional signals which have been sampled as two-dimensional hexagonal arrays are presented. Included are methods for signal representation, linear system implementation, frequency response calculation, DFT calculation, filter design, and filter implementation. These algorithms bear strong resemblances to the corresponding results for rectangular arrays; however, there are also many important differences. Some comparisons between the two methods for representing planar data will also be presented.

The processing of hexagonally sampled two-dimensional signals

This paper attempts to demonstrate that a modem optimization methodology known as semidefinite programming (SDP) can serve as the algorithmic core of a unified design tool for a variety of two-dimensional (2-D) digital filters Representative SDP-based designs presented in the paper include minimax and weighted least-squares designs of FIR filters with continuous and discrete coefficients, and minimax design of stable separable-denominator IIR filters Our studies are motivated by the fact that SDP as a subclass of convex programming can be solved efficiently using recently developed interior-point methods and, more importantly, constraints on amplitude/phase responses in certain frequency regions and on stability (for IIR filters), that are often encountered in many filter design problems, can be formulated in a natural way as linear matrix inequalities (LMI) which allow SDP to apply Design examples for each class of filters are included to demonstrate that SDP-based methods can in many cases be useful in producing optimal or near-optimal 2-D filters with reduced computational complexity

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A unified approach for the design of 2-D digital filters via semidefinite programming

Singular-value decomposition (SVD) is applied to the design of two-dimensional (2-D) FIR (finite-impulse-response) digital filters. It is shown that three realizations are possible, namely, a direct realization, a modified version of direct realization, and a realization based on the combined application of SV and LU decomposition. Each of the realizations consists of a parallel arrangement of cascaded pairs of 1-D filters; hence extensive parallel processing and pipelining can be applied. The 1-D FIR filters can be designed using standard methods, and linear-phase causal 2-D filters that are suitable for real-time or quasi-real-time applications can also be designed. The three realizations are compared, and it is shown that the realization based on combined SV and LU decomposition leads to the lowest approximation error and involves the smallest number of multiplications. SVD, the McClellan transformation, and 2-D window design methods are used to design a bandpass and a fan filter, and the results are compared. >

Design of two-dimensional FIR digital filters by using the singular-value decomposition

This paper concerns development of an efficient method for the design of two-dimensional (2-D) recursive digital filters. The specific design problem addressed is that of obtaining half-plane recursive filters which satisfy prescribed frequency response characteristics. A novel design procedure is presented which incorporates a spectral factorization algorithm into a constrained, nonlinear optimization approach. A computational implementation of the design algorithm is described and its design capabilities demonstrated with several examples.

Two-dimensional recursive filter design--A spectral factorization approach

A procedure for the design of separable two-dimensional digital filters is presented. The computation involved in both the filter design and implementation is shown to be efficient. Several examples are presented which illustrate the application of the technique.

Computer-aided design of separable two-dimensional digital filters

The sectioning approach to the implementation of two-dimensional convolution operations is reviewed. An optimal sectioning algorithm is presented and its computational requirements (CPU and I/O) completely characterized. Procedures are then given to enable the optimal choice of sectioning parameters for an arbitrary computer system. This procedure is Illustrated for a large scientific compUter (CDC 7600), and the sectioning implementation compared with the traditional FFT approach. Computer programs are included for both the sectioning algorithm and the determination of optimal sectioning parameters.

Optimal sectioning procedure for the implementation of 2-D digital filters

The sectioning approach to the implementation of two-dimensional convolution operations is described. Two computational algorithms are presented for sectioning, and the performance of the most efficient one is described in terms of both CPU and I/O requirements. 1 figure, 1 table.

R.E. Twogood

Papers

Computer-aided design of separable two-dimensional digital filters

Optimal sectioning procedure for the implementation of 2-D digital filters

Implementation of two-dimensional FIR digital filters by sectioning