Toward discrete geometric models for early vision

Early vision is usually considered to involve the description of geometric structure in an image or sequence of images. Whether biological or artificial, the behavioural constraints on real-time visual systems typically require that this first stage of visual processing be fast, reliable, general and automatic. The design of a visual system which is general enough to handle a wide variety of tasks is thus most likely to be highly parallel, and involve distributed representations of geometric objects. In this work, we investigate some of these general principles and propose both general methodology and specific applications. We build on a general theory of distributed, local representations which we call thick traces. Thick trace descriptions of continuous graphs preserve topological properties such as connectivity, and allow for the descriptions of multi-valued mappings. Local operators for extracting image curves have been a focus of machine vision research for twenty years. Considered in the context of thick traces, however, we can reassess the goals of these operators and provide a clear description of when they should respond positively and when they should not. In order to achieve this behaviour, we develop an algebra, the Logical/Linear algebra, which incorporates features of both Boolean and linear algebra into a set of non-linear combinators. This algebra is then used to design a family of local operators which explicitly test the logical preconditions underlying the definition of an image curve. Relaxation labelling is a highly parallel, distributed method of extracting consistent structures from a set of labels. There is a natural match between the representations used in relaxation labelling and thick traces. We exploit this connection by developing a general method for relaxing a set of potentially noisy initial estimates of thick traces (as produced by image operators) into descriptions which are thick traces of geometric models. Furthermore we show how such a system can interpolate into gaps in the traces while simultaneously respecting legitimate discontinuities and boundaries. Finally, we apply these methods to two problems in early vision: the description of curves and texture flow fields. For image curves, the resulting descriptions of piecewise smooth curves include both local orientation and curvature information. The entire process accurately describes end-points, corners, junctions and bifurcations by allowing many consistent traces to be incident on a single point in the image. The term texture flow is used to describe a class of static textures with locally parallel dense orientation structure (e.g. Glass or hair patterns). We derive a geometric model of these textures from a smooth non-deforming velocity field. Initial operators and a relaxation network are then defined to interpolate dense, piecewise smooth flow from sparse inputs. The resulting system produces accurate descriptions even in the presence of discontinuities, holes, and overlapping textures.