Conformal image warping

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Abstract
Numerical and computer-graphic methods for conformal image mapping between two simply connected regions are described. The immediate motivation for this application is that the visual field is represented in the brain by mappings which are, at least approximately, conformal. Thus, to simulate the imaging properties of the human visual system (and perhaps other sensory systems), conformal image mapping is a necessary technique. For generating the conformal map, a method for analytic mappings and an implementation of the Symm algorithm for numerical conformal mapping are shown. The first method evaluates the inverse mapping function at each pixel of the range, with antialiasing by multiresolution texture prefiltering and bilinear interpolation. The second method is based on constructing a piecewise affine approximation of the mapping in the form of a joint triangulation, or triangulation map, in which only the nodes of the triangulation are conformally mapped. The texture is then mapped by a local affine transformation on each pixel of the range triangulation with the same antialiasing used in the first method. The algorithms are illustrated with examples of conformal mappings constructed analytically from elementary mappings, such as the linear fractional map, the complex algorithm, etc. Applications of numerically generated maps between highly irregular regions and an example of the visual field mapping that motivates this work are also shown.

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