CONFORMAL IMAGE WARPING

This report describes numerical and computer-graphic methods for conformai image mapping between two simply connected regions. The immediate motivation for this application is that the visual field is represented in the brain by mappings which are, at least approximately, con-formal.1-4 Thus, to simulate the imaging properties of the human visual system (and perhaps other sensory systems), conformai image mapping is a necessary technique. There are two distinct aspects to this problem. First, one must implement a numerical or analytic method which allows for the computation of a given conformai mapping, constrained by the shape of the two simply connected regions (hereafter known simply as regions) to be mapped, and by a single point and orientation correspondence between them. Second, it is necessary to apply a space-variant texture-mapping algorithm to warp the image, once the mapping itself has been specified. for generating the conformai map, we show a method for analytic mappings, and also an implementation of the Symm algorithm for numerical conformai mapping.5 The first method evaluates the inverse mapping function at each pixel of the range, with antialiasing via multiresolution texture prefiltering and bilinear interpolation.6 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. We illustrate these algorithms with examples of conformai mappings constructed analytically from elementary mappings, such as the linear fractional map, the complex algorithm, etc. We also show applications of numerically generated maps between highly irregular regions, and also an example of the visual field mapping that motivates this work. In addition to providing a necessary tool for simulation of cortical architectures, these illustrations may be of pedagogical use for students attempting to visualize the geometric properties of elementary conformai mappings; they may also find applications in such areas as fluid mechanics and electrostatics, where conformai mapping is a natural and basic tool. © 1990 IEEE