Abstract:
The work presented in this thesis examines several aspects of two-dimensional recursive digital filter stability in the context of image coding via linear prediction. Unstable prediction filters in image coding models have a catastrophic effect on image quality. Namely, they render the resynthesised image unintelligible. Experimental results show that a significant proportion of analysis frames taken from natural images produce unstable prediction filters. The physical cause of the high rate of instability is examined, and is shown to be due to two separate mechanisms. Firstly, analysis frames which have high inter-pixel correlation produce a disproportionate number of unstable prediction filters. Secondly, an analysis frame provides a truncated view of the image data and may inadvertantly 'deceive' the Linear prediction analysis, creating the impression that the image is unbounded globally. Four related techniques, (based on the Fourier and Hadamard orthogonal transforms), are presented for reducing the intra-frame correlation and therefore reducing the instability rate. It is shown that the probability of deriving a stable prediction filter is increased if the dominant components in either Fourier or Hadamard space are removed. The performance of each method is evaluated and a cost analysis is presented, based on algorithm complexity and transmission bit-rate overhead. It is concluded that removing the dominant sequency components, (when ordered on a magnitude basis), performs most satisfactorily. Unstable prediction filters underlie the more general problem of determining the stability margin of a two-dimensional infinite-extent impulse response (ΠR) filter. While the mathematical formalisms describing the familiar concepts of poles and zeros extend readily to multiple dimensions, the simple and reassuring geometric interpretations do not. As a graphical aid and practical tool, the rootmap is a much-ignored device for visualising the stability condition of two-dimensional recursive filters. An algorithm for its calculation is presented, together with refinements which increase the speed of computation, as well as improve the robustness of the algorithm. frι order to rapidly detect unstable filters, (a basic requirement of real-time image coding systems), a suite of necessary (but insufficient) tests are developed. The new tests successfully detect a large fraction of unstable filters, and are shown to perform well when compared with another necessary (but insufficient) test, as well as against 'sure-fire' methods. The simplicity of the tests is such that they can be implemented in real-time, and the structure of the tests suggests a stabilization strategy which depends on which test (or tests) fail.