The nexus of conformal geometry, action principles and tau-functions: a pathway to novel constructive methods for shape analysis and imaging

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The study of shapes is a central problem in medical imaging, computer vision and many engineering fields. Suppose we are given the silhouette of a human organ, such as the brain or colon, from an X-ray scan. Human organs change their shape because of disease progression, surgery and other treatments, and simply by natural growth. How are we to tell from such a silhouette, or 2D shape, whether the organ is diseased (as in the brain of an Alzheimer's patient for example) and, if so, the degree of progression of the disease? One way is to compare, or "register", the shape of the scan against templates or averaged shape data from previous patients. This process is known as shape registration and it is a vibrant area of active research, along with shape classification and recognition. All this requires a way to compute how far one shape is from another. For mathematicians, the distance between entities is measured by defining a metric on a suitably defined "space" or set of entities; over the years, many such metrics over different spaces have been proposed for the purposes of shape registration in medical imaging.Conformal geometry is the study of the conformal structure of a shape and, mathematically, it has many favourable attributes that render it particularly suitable for shape registration tasks. Mathematicians give the name "multiply connected'' to a shape that has holes in it. In medical imaging, it is common to identify on any given medical scan certain distinguishing features, attributes or "landmarks'' that have special significance. In the technologically important area of facial recognition, the 2D silhouette of a person's face might be punctuated by "holes'' around the eyes, nose and mouth which are considered in order that the image can impart more accurate information. The mathematical notion of multiple connectivity is therefore deeply relevant to these applications.The conformal geometric approach to shape analysis has not been fully developed in the multiply connected setting, and this is an area where the PI has particular expertise acquired in past research on quite different application areas. The PI also has special interest in constructive methods, and his past work has included both identifying important mathematical tools and bringing these to implementational fruition by the development of computational methods and software. This proposal develops a novel approach to shape analysis by building on surprising connections, found only in the last decade, between several different mathematical subfields.Consider a simple closed curve in the plane. It has an inside and the outside. The Riemann mapping theorem of complex analysis says that there exists a "mapping" from a circular disc to the inside of the curve, and a different mapping from the same circular disc to the outside of the curve. Conformal geometers have considered a clever composition of these two mappings and have come up with an identifier known as a "fingerprint" of the original curve.On the other hand, in a classic free boundary problem called the Hele-Shaw problem it has been recognized since 1972 that it is useful to consider the (Richardson) "moments" of the inside of a curve, as well as the moments of the outside of a curve. In the last decade, mathematical physicists have discovered that the interior moments can be generated by the exterior moments by means of a so-called tau-function. Surprisingly, this same tau-function has also been obtained by yet another community (conformal field theorists) using very different mathematical arguments.Both the "fingerprint" of the curve and the "tau-function" of the curve come about by marrying data about the interior of the curve to data about the exterior of the curve. These objects must be related. But how? This project is about exploring those connections, and examining if novel constructive methods for shape analysis and medical imaging can be found as a result.