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Ian Jermyn: Shape modelling with contours and fields |
| The study and modelling of shape has a long history, in biology, statistics,
and more recently, in computer vision and image processing. Why do we study
and model shape? To summarize the geometric properties of entities in the
real world around us, to be used for a variety of purposes; for example, the
segmentation of an entity from an image. What is a shape? A subset of some
manifold, usually Euclidean space, 'corresponding' in some way to the
entity. But shape really refers not to a single such subset, but to a
probability distribution on the set of subsets describing our knowledge of
the shape corresponding to the entity under certain circumstances.
To capture all except the most trivial geometric information, such a probability distribution must involve long-range dependencies between points of the subset. The most popular way of introducing such long-range dependencies is the use of a reference or template shape around which variations are authorized. The shapes in such a 'shape family' therefore consist of perturbations of a given reference shape. There are applications, however, where the family of shapes involved does not have such a constrained behaviour. Cases where the number of individual objects is unknown a priori, or where the topology of the shape may be otherwise complex (for example network shapes), require new techniques. In this lecture, after a brief survey of other techniques, I will describe an alternative framework in which long-range dependencies are introduced explicitly as interactions between shape boundary points, thereby allowing these more complex cases to be addressed. The framework permits the modelling of shapes of unknown and potentially arbitrary topology, in particular the case of an arbitrary number of instances of a given shape. The shapes involved may be represented by their bounding contour ('higher-order active contour'), a smoothed version of their characteristic function ('phase field'), or by a binary Markov random field, each with their own characteristic advantages, which I will discuss. I will present several models, including models of network shapes and of a 'gas of circles, and their application to the segmentation of road networks and tree crowns from satellite and aerial images, and preliminary results on microscope images. I will also describe how the transformation of models between different shape representations (boundaries, phase fields, binary fields) allows the analytical and algorithmic advantages of each to be exploited, in particular for fixing model parameters. I will briefly outline current and future work aimed at generalizing the method to arbitrary shapes. 
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