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A transform coding scheme for closed image boundaries on a plane is described. The given boundary is approximated by a series of straight line segments. Depending on the shape, the boundary is represented by the (x-y) coordinates of the endpoints of the line segments or by the magnitude of the successive radii vectors that are equispaced in angle around the given boundary. Due to the circularity present in the data, the discrete Fourier transform is used to exactly decorrelate the finite boundary data. By fitting a Gaussian circular autoregressive model to represent the boundary data, estimates of the variances of the Fourier coefficients are obtained. Using the variances of the Fourier coefficients and the MAX quantizer, the coding scheme is implemented. The scheme is illustrated by an example.