In primate cortex roughly one million afferent axons synapse on area V1, which contains nearly a billion neurons. This over-representation of the retinal signal by almost three orders of magnitude is expressed by a "scattered" topographic map of the visual field---at each locus of V1, there are multiple retinal inputs whose retinal locations are scattered around that specified by the map, with the scale of scatter roughly on the order of the receptive field size. Nevertheless, this apparently over-represented and blurred system supports perceptual signals with sharply delineated boundaries.; In this thesis the issue of neural over-representation and scattered spatial representation is considered in the context of the computational problem of smoothing and interpolation of noisy and possibly incomplete data, without simultaneously blurring valid object boundaries. Early approaches to this problem in computer vision used purely local operators---an isotropic second difference operator (the Laplacian) preceded by a low-pass Gaussian filter. This method, associated with the term scale space, was eventually recognized as being equivalent to isotropic diffusion. In the past fifteen years nonlinear diffusion, in both computational and neural contexts, has been shown to provide superior results. Recently, it has been shown that by displacing the lobes of conventional spatial filters from their topographic locus, it is possible to achieve results that are equivalent to nonlinear diffusion, an algorithm termed offset filtering.; Here, it is shown that offset filtering can be achieved by a scattered and over-represented neural system, motivated by the anatomical structure of V1. The over-representation, coupled with simple models of lateral inhibition, allows rapid computation of the offset filtered input data, in effect trading neural space for rapid response and physiological simplicity. It is shown that this analysis can account for a variety of psychopliysical and physiological results related to the "nonclassical" receptive field structure of neurons in V1. The optimum cortical scatter is shown to be about one-half the local receptive field size, a result which is consistent with recent physiological measurements.; Finally, the data structure provided by the sparse; thresholded offset filter data points is used with a graph-theoretic partitioning algorithm to achieve high quality visual segmentations. These results provide a biologically consistent and computationally effective means of utilizing an over-represented but sparsely sampled cortical representation of the visual field.
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