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The calculus of two-dimensional mappings is reviewed. Currently available data suggests that a conformal (isotropic) mapping is sufficient to model the global topography of primate striate cortex. The addition of a shear component to this mapping is outlined. A similar analysis is applied to the local map structure of striate cortex. Models based on a local reiteration of the complex logarithm and on Radon (and backprojection) mappings are presented. A variety of computational functions associated with the novel architectures for image processing suggested by striate cortex neuroanatomy are discussed. Applications to segmentation, perceptual invariances, shape analysis, and visual data compression are included. Finally, recent experimental results on shape analysis by neurons of infero-temporal cortex are presented.