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On the measurement of the curvature of the boundaries of two-dimensional quantized shapes Bennett, John Reavely

Abstract

The use of the mathematical concept of curvature as a practical descriptor of shape for pattern recognition and image processing application is investigated. A mathematical analogy between the construction of a two-dimensional curve from a given curvature function and the frequency modulation of a sinusoidal carrier with a message, is drawn. It is shown that the curve describing the boundary of a two-dimensional shape is in fact a circle, frequency modulated with appropriate curvature information. The accuracy to which the curvature of the boundary of a binary shape can be estimated from a given quantized version of that shape, depends upon two factors in the estimation process: - the contour tracing algorithm by which boundary points on the quantized shape are defined and the method whereby the curvature function is smoothed to partially remove the quantization noise resulting from the digitization of the original shape. In this thesis, six contour tracing algorithms are described and used to extract curvature functions from quantized test shapes in the absence of smoothing of any kind. Models of the average quantization noise characteristics in the frequency domain are then developed for the curvature functions corresponding to each of the six contour tracing algorithms. The models are used to compare the performance of the six algorithms on the basis of the quantization noise characteristics of each. It is found that the useful bandwidth of curvature functions obtained from quantized shapes is somewhat less than the theoretical limit imposed by the Nyquist Sampling Theorem. The models developed serve to demonstrate the variation in the useful bandwidth of the curvature functions corresponding to each, contour tracing algorithm. The variation in bandwidth- with- quantizing array resolution can also be predicted with the models. The relationship between the bandwidth of a curvature function and the amount of detail representable in the corresponding curve in the plane is then qualitatively explored. Finally, three approaches to the problem of smoothing curvature functions to eliminate quantization noise are studied and methods are developed to compare their effectiveness on sample shapes.

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