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The analysis of closed boundaries of arbitrary shapes on a plane is discussed. Specifically, the problems of representation and reconstruction are considered. A one-to-one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up. Univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary. The stochastic model representing the closed boundary is invariant to transformations like sealing, translation, choice of starting point, and rotation over angles that are multiples of 2/N, where N is the number of observations. Methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included. Constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries. The stochastic model enables the reconstruction of a dosed boundary using FFT algorithms. Results of simulations are included and the application to contour coding is discussed.
Kashyap et al. (1981) studied this question.