Theory of shape-invariant imaging systems

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Abstract
The human visual system exhibits two properties that may be useful in other pattern-recognition systems: (1) images do not change their shape with changes in image size and (2) resolution declines rapidly with distance from the center of fixation. We show here that, in general, shape invariance requires inhomogeneous resolution over image space in a manner similar to that of the human visual system. Thus shape-invariant systems must process less information when compared with uniform-resolution systems. Although shape-invariant systems can be rotationally invariant, they cannot, in general, be translationally invariant. The properties of shape-invariant systems are explored in the spatial-frequency domain using a modified Fourier transform called a scaled transform. The features of scaled transforms are discussed and their behavior illustrated in the image domain by using them to filter various images, including the dot, the line, and the edge. It is shown that the filtered profile of an edge is preserved when it passes through the origin of a scaled transform. This result suggests that scaled transforms may be useful in edge-detection algorithms.