Space-variant computer vision with a complex-logarithmic sensor geometry

Candidate: Rojer,Alan S.

The complex logarithm as a conformal mapping has drawn interest as a sensor architecture for computer vision due to its psuedo-invariance with respect to rotation and scaling, its high ratio of field width to resolution for a given number of pixels, and its utilization in biological vision as the topographic mapping from the retina to primary visual cortex. This thesis extends the computer vision applications of the complex-logarithmic geometry. Sensor design is based on the complex log mapping w = log (z + a), with real a $>$ 0, which smoothly removes the singularity in the log at the origin. Previous applications of the complex-logarithmic geometry to computer vision, graphics and sensory neuroscience are surveyed. A quantitative analysis of the space complexity of a complex-logarithmic sensor as a function of map geometry, field width and angular resolution is presented. The computer-graphic problems of warping uniform scenes according to the complex logarithm and inversion of log-mapping scenes to recover the original uniform scene are considered, as is the problem of blending the resulting inverse log maps to reconstruct the original (uniform) scene. A series of simple algorithms for segmentation of log scenes by contour completion and region filling are presented. A heuristic algorithm for figure/ground segmentation using the log geometry is also shown. The problem of fixation-point selection (visual attention) is considered. Random selection of fixation points, inhibition around previous fixations, spatial and temporal derivatives in the sensor periphery, and regions found by segmentation are all examined as heuristic attentional algorithms. For the special case where targets can be parametrically defined, a theory of model-based attention based on the Hough transform is introduced. A priori knowledge about the consistency between potential objects in the scene and measured features in the scene is used to select fixation points. The exponential storage requirements of the usual Hough transform are avoided.