The data model for image representation in terms of projective Fourier transform (PFT) is well adapted to both image
perspective transformations and the retinotopic mappings of the brain visual pathways. Here we model first aspects of
the human visual process in which the understanding of a scene is built up in a sequence of fixations for visual
information acquisition followed by fast saccadic eye movements that reposition the fovea on the next target. We make
about three saccades per second with an eyeball's maximum speed of 700 deg/sec. The visual sensitivity is markedly
reduced during saccadic eye movements such that, three times per second, there are instant large changes in the retinal
images without almost any information consciously carried across fixations. Inverse Projective Fourier transform is
computable by FFT in coordinates given by a complex logarithm that also approximates the retinotopy. Thus, it gives the
cortical image representation, and a simple translation in log coordinates brings the presaccadic scene into the
postsaccadic reference frame, eliminating the need for starting processing anew three times per second at each fixation.
Identifying the projective group for patterns by developing the camera model, the projective Fourier transform and its inverse are obtained in analogy with the classical, that is, Euclidean Fourier analysis. Projectively adapted properties are demonstrated in a numerical test. Using the expression of the projective Fourier integral by a standard Fourier integral in the coordinates given by the complex principal logarithm, the discrete projective Fourier transform and its inverse are constructed showing that FFT algorithms can be adapted for their computations.
Projectively invariant classification of patterns is constructed in terms of orbits of the group SL(2,C) acting on an extended complex line (image plane with complex coordinates) by Mobius transformations. It provides projectively adapted noncommutative harmonic analysis for patterns by decomposing a pattern into irreducible representations of the unitary principal series of SL(2,C). It is the projective analog of the classical (Euclidean) Fourier decomposition, well suited for the analysis of projectively distorted images such as aerial images of the same scene when taken from different vantage points.
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