How do we see? This simple question has challenged many thinkers since ancient times. Despite a rich and broad spectrum of contributions from many artists, philosophers, physicians and scientists, its solution is nowhere in sight! Much research is still needed to explain the processes as early as when the light rays reach the eyes, to the final stages of the visual task such as object recognition and visual attention. The complexity of the problem stems from having more than one-half of one hundred billion brain neurons, and with an average of ten thousand synaptic contacts per neuron, forming circuits and networks primarily dedicated to vision. Nonetheless, vision scientists have undertaken the ambitious task of building an artificial eye!
How could differential topology contribute to understanding vision? The visual system solves numerous geometric problems, such as estimating the shape and spatial position of objects in a scene. Two levels of information processing are distinguished: the bottom-up processes in Early (or Low Level) Vision, and the top-down processes in High Level Vision. While Low Level Vision is primarily concerned with local information, High Level Vision focuses on global phenomena. Where and how does the local-to- global transition in information processing occur?
We propose a model that lies intermediate to these two levels. The mathematical counterpart attempts to construct geometric structures that take into account the statistical nature of visual perception and our ability to estimate global shapes of objects. Differential topology has the conceptual framework to integrate the global geometry from given local information, and together with mathematical analysis, estimate geometric features such as curvature, orientation and relative spatial position of surfaces.
In the first half of this expository lecture, I outline some key facts from neuroscience and visual perception that form the cognitive basis for our mathematical model. Next, I describe the model and some of its consequences (progress report on joint research with Steve Palmer.) The geometric structure in the model consists of families of surfaces together with additional geometric and statistical structures to accommodate the possibilities for visual perception by a viewer in a variety of circumstances. Based on this model, we provide estimates for curvature and other geometric attributes of (possibly non-smooth) surfaces with texture, e.g. as in a golf field or a mountain at a distance. Based on this model, we propose an additional stage in visual perception corresponding to transition from the local to the global information processing.