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

形状的研究是医学成像、计算机视觉和许多工程领域的核心问题。假设我们从x射线扫描中得到一个人体器官的轮廓，比如大脑或结肠。人体器官会因为疾病进展、手术和其他治疗而改变形状，或者仅仅是自然生长。我们如何从这样的轮廓或二维形状判断器官是否患病（例如在阿尔茨海默病患者的大脑中），如果患病，疾病的进展程度如何？一种方法是将扫描的形状与模板或以前患者的平均形状数据进行比较或“注册”。这个过程被称为形状配准，它是一个活跃的研究领域，与形状分类和识别。所有这些都需要一种方法来计算一个形状与另一个形状之间的距离。对于数学家来说，实体之间的距离是通过在一个适当定义的“空间”或实体集合上定义一个度量来测量的；多年来，许多这样的度量在不同的空间已经提出了形状注册在医学成像的目的。保形几何是研究形状的保形结构，在数学上，它有许多有利的属性，使它特别适合形状配准任务。数学家把有孔的形状称为“多连通”。在医学成像中，通常在任何给定的医学扫描中识别具有特殊意义的某些显著特征、属性或“地标”。在面部识别这一技术重要领域，人脸的二维轮廓可能会被眼睛、鼻子和嘴巴周围的“洞”打断，这被认为是为了让图像传递更准确的信息。因此，多重连接的数学概念与这些应用程序密切相关。形状分析的保形几何方法尚未在多连接环境中得到充分发展，这是PI在过去对不同应用领域的研究中获得的特殊专业知识的领域。PI还对建设性方法特别感兴趣，他过去的工作包括确定重要的数学工具，并通过开发计算方法和软件将这些工具实现。这一建议发展了一种新颖的方法，通过建立在几个不同数学子领域之间的惊人联系来进行形状分析，这种联系仅在过去十年中才被发现。考虑平面上的一条简单的闭合曲线。它有内部和外部。复分析中的黎曼映射定理说，存在一个从圆圆盘到曲线内部的“映射”，以及一个从同一圆圆盘到曲线外部的不同映射。共形几何学者考虑了这两种映射的巧妙组合，并提出了一种被称为原始曲线“指纹”的标识符。另一方面，在被称为Hele-Shaw问题的经典自由边界问题中，自1972年以来人们已经认识到，考虑曲线内部的（Richardson）“力矩”以及曲线外部的力矩是有用的。在过去的十年里，数学物理学家发现内部矩可以通过所谓的tau函数由外部矩产生。令人惊讶的是，同样的tau函数也被另一个团体（共形场理论家）用非常不同的数学论证得到了。曲线的“指纹”和曲线的“头函数”都是通过将曲线内部的数据与曲线外部的数据相结合而产生的。这些物体一定是相关的。但如何?这个项目是关于探索这些联系，并检查是否可以找到新的建设性的方法来进行形状分析和医学成像。

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.