Binocular Imaging with The Conformal Eyes

AbstractThe human eye possesses a natural asymmetry of its optical components and an inhomogeneous distribution of photoreceptors causing optical aberrations and providing high acuity only at a 2-degree visual angle. Although these features greatly impact the visual system functions, they have not been supported by the geometric formulation of the fundamental binocular concepts. The author recently constructed a geometric theory of the binocular system with asymmetric eyes (AEs) integrated with the eyes’ movement to address these problems. This theory suggests that a symmetric framework can fully represent the asymmetric properties of this binocular system with the AEs. Pursuing this idea leads first to the conformal eye model furnished by the Riemann sphere for the special case of AE corresponding to the reduced eye with axial symmetry, referred to as the symmetric eye (SE) model. The conformal geometry of the Riemann sphere establishes efficient image representation by the projective Fourier transform (PFT)—the Fourier transform on the group of image projective transformations representing images covariantly to these transformations. The PFT is fast computable by an FFT algorithm in log-polar coordinates known to approximate the retina-cortical mapping of the human brain’s visual pathways. The retinotopy modeling here with PFT is compared to Schwartz’s modeling with the exponential chirp transform showing clear advantages of PFT in physiological conformity and numerical efficacy. Finally, the conformal eye model initially developed for the SE is extended to the AE. Then, in the binocular system with AEs, the PFT becomes available for image processing. The PFT combined with the conformal eye model allows binocular extensions of the previous monocular algorithms for modeling visual stability during the saccade and smooth pursuit eye movements needed to offset the eye’s acuity limitation.