Design Considerations for a Space-Variant Visual Sensor with Complex-Logarithmic Geometry Alan S. Rojer Eric L. Schwartz Brain Research Laboratory New York University Medical Center Courant Institute of Mathematical Sciences Department of Computer Science New York University Abstract Human vision is both active and space-variant. Recent interest in exploiting these characteristics in machine vision naturally focuses attention on the design parameters of a space-variant sensor. We consider space-variant sensor design based on the conformal map- ping of the half disk, w = log (z+a), with real aO, which character- izes the anatomical structure of the primate and human visual sys- tems. There are three relevant parameters: the ���circumferential index��� X, which we define as the number of pixels around the peri- phery of the sensor, the ���visual field radius��� R (of the half-disk to be mapped), and the ���map parameter��� a from above, which dis- places the logarithm���s singularity at the origin out of the domain of the mapping. We show that the log sensor requires o(K���1og (R/a)) pixels. The pixel width in the fovea (foveal resolution) LfOv is pro- portional to U/K. If we accept a fixed circumferential index (constant x), the space complexity of the log sensor with respect to foveal resolution goes as O(-logLfov). By contrast, a uniform-resolution sensor has a space complexity that goes as O(Lfov-2). Similarly, when the space complexity of the sensor is considered with respect to the field size with a fixed foveal resolution, we find that the space complexity goes as O(logR), while for the uniform-resolution sen- sor, the space resolution goes as O(R2). Using this analysis, it is possible to directly compare the space complexity of different sensor designs in the complex logarithmic family. In particular, we can obtain rough estimates of the parameters necessary to duplicate the field widthhesolution performance of the human visual system. Introduction Human vision is both active and space-variant. Recent interest in exploiting these characteristics for machine vision naturally focuses attention on the design parameters of a space variant sensor. In the case of conventional space-invariant (or uniform-resolution) sensors, the number of pixels in the sensor provides a single number which characterizes space complexity. Given the number of pixels in a uni- form sensor, the ratio of visual field width and sensor resolution is fully determined. Thus, it is a simple matter to compare conven- tional sensors. In contrast, the space complexity of space-variant sensors depends on the ���architecture��� of the sensor, with potentially enonnous variation, which is beyond the scope of this paper. We focus our attention on a particular space-variant architecture, the conformal map associated with the complex logarithm, which characterizes a prominent part of the anatomical structure of the pri- mate and human visual systems. The complex-log mapping provides an accepted model of the mapping from retina to primary visual cortex in primates at both the local (hypercolumn) and global (retinotopic representation) scales [ 1-31, Traditionally, researchers in computer vision have found motivation in small-scale (cellular) properties of biological vision: the application of the V2G operator for edge enhancement is one example [4]. The complex-log retinotopic representation is a strik- ing example of the large-scale architecture of biological vision for that reason alone it merits study as a potential architecture for com- puter vision. The complex-log mapping of scenes also has favorable compu- tational properties. It embodies a useful isomorphism between mul- tiplication in its domain and addition in its range. This is especially interesting when a two-dimensional scene is considered to be defined with respect to a complex argument then complex multiplication of the argument is equivalent to scaling of the scene (by the modulus of the multiplier) and rotation of the scene around the origin (by the argument of the multiplier). In the range of the log mapping, com- plex multiplication becomes addition, so the mapped image of the scene is shifted horizontally in proportion to the log of the scale change, and vertically in proportion to the angle of rotation. This isomorphism has been exploited for efficient implementation of computer graphic and image processing operations [SI. Models of the perceptual equivalence of scaled and rotated objects have also utilized the complex log mapping [l]. ���The prob- lem of position, scale and rotation-independent template matching for images has been addressed by the use of Fourier-Mellin transforms [6,71. To create a position-, scale- and rotation- independent template for an object, the magnitude of the Fourier transform of the object is first computed. This is invariant to shifts in position of the object. The complex-log mapping converts changes of scale and rotation (which are presexved in the magnitude of the original Fourier transform) to shifts in the range of the log mapping. A second Fourier transform magnitude (on the log scene) standardizes with respect to the shifts in the log scene, producing a template (for correlation) which is independent of the scale, position and rotation of the original object. The combination of log-mapping and Fourier transform is equivalent to the Mellin transform [8]. Other useful properties of the log-mapping have been discovered with regard to computations in a moving visual field. Determination of time-to-collision and depth-from-motion for a moving camera in a stationary world has been considered in [9-111. It is readily apparent that when the camera is moving in a stationary world, the optical flow of the scene is purely radial. In the log- mapped image, this corresponds to pure horizontal flow. Deviations CH2898-5/90/0000/0278$01 .OO 0 1990 IEEE 278