FRG: The Geometry, Mechanics and Statistics of the Infinite-dimensional Manifold of Shapes

本文重点研究了具有右不变度量的微分同胚群作用于形状空间的结构。该项目包含与这些空间相关的四个组成部分：它们的几何分析，适当统计方法的发展，所需的数值分析，以及将结果应用于医学成像和计算解剖学。一般框架是通过一种新的方法来解决的，这种方法在某种意义上形式化了形状的机制。具有正确不变度量的李群确实是可以证明经典力学定律适用的结构，特别是沿着最小能量路径的动量守恒。事实证明，在这种情况下，动量是表征变形的关键。虽然它很难处理，因为它通常是单一的，作为一种度量，或在单一支撑上的分布。这一点以及由此产生的数字难度可能是我们在研究中要解决的主要挑战之一。其他重要的方面是研究这种方法在形状空间上引起的几何，包括曲率的研究，以及法坐标的存在性和稳定性。这将与形状统计中的开放问题有关，并特别应用于生物医学成像问题。因此，设计这种方法是为了提供描述和分析形状的新工具。尽管形状在外部世界和科学中很流行，但这是一个难题。对于人类的大脑来说，有一种直观的概念，即什么是形状，为什么它们不同或相似，或者什么时候它们与普通观察相比表现出异常。雕塑是渲染现有形状或创造新形状的艺术，艺术家仍然能够通过扭曲或示意图表示提供主题的明确实例，这一事实强烈表明了人类形状识别引擎的鲁棒性。然而，对形状的分析描述就不那么明显了，人类完成这项任务的效率也低得多，就好像对形状的理解和识别没有准确提取其构成成分一样。我们可以通过一个简单的轮廓来识别南瓜、茄子或辣椒，甚至可以提供一系列区分它们的特征，但要对它们中的任何一种进行足够准确的口头描述，比如让画家复制出来，要困难得多。因此，对于数学来说，形状描述在很大程度上仍然是一个挑战，这并不奇怪。然而，有一些非常重要的应用依赖于这一领域的进展，其中之一是生物医学形状的计算机化分析（计算解剖学），它分析疾病对器官形状的影响，从现代非侵入性3D成像技术中获得。在过去的五十年里，计算机视觉的研究已经展示了为这个目的而设计的各种各样的观点和技术：2D或3D集合（通过体积或边界），基于矩的特征，中间轴或表面，多项式的零集，兴趣点（地标）的配置，等等。然而，无论在概念上还是在计算上，这些方法似乎都不是描述形状的理想方法。我们研究的一个重要方面将是用间接的方法来描述形状，从它们可以变形的方式。这不是一个新想法，可以追溯到20世纪初达西·汤普森（D’arcy Thompson）的开创性作品，但它的数学形式化和实用算法的设计是一项全面的任务，仍然存在许多悬而未决的问题，本小组将试图解决这些问题，并向科学界传达这些问题。

This focused study proposes to analyze the structure induced on spaces of shapes by the action of groups of diffeomorphisms equipped with a right invariant metric. The project contains four components related to these spaces: their geometric analysis, the development of appropriate statistical methods, the required numerical analysis, and the application of the results to medical imaging and computational anatomy. The general framework is addressed by a new approach which in some sense formalizes the mechanics of shapes. Lie groups with right invariant metrics indeed are structures on which classical laws of mechanics can be shown to hold, and in particular the conservation of momentum along paths of minimal energy. It turns out that this momentum is a key to the representation and characterization of deformations in this context. It is albeit difficult to handle, because it is usually singular, as a measure, or a distribution on a singular support. This and the numerical difficulty it creates is probably one of the main challenges that we address in our study. Other important aspects are the study of the geometry such an approach induces on shape spaces, including a study of their curvatures, and the existence and stability of normal coordinates. This will be related to open issues in shape statistics, and applied in particular to biomedical imaging problems. This approach is therefore designed to provide new tools for describing and analyzing shapes. Although shapes are prevalent in the outside world and in science, this is a difficult problem. For the human mind, there is an intuitive notion of what shapes are, why they differ or look alike, or when they present abnormalities with respect to ordinary observations. Sculpture is the art of rendering existing shapes, or creating new ones, and the fact that artists are still able to provide unambiguous instances of subjects through distorted or schematic representations is a strong indication of the robustness of the human shape recognition engine. However, an analytical description of a shape is much less obvious, and humans are much less efficient for this task, as if the understanding and recognition of forms work without an accurate extraction of their constituting components. We can recognize a squash from an eggplant or a pepper via a simple outline, and even provide a series of discriminative features which distinguish them, but it is much harder to phrase a verbal description of any of them, accurate enough, say for a painter to reproduce it. It is therefore not surprising that, for mathematics, shape description remains mostly a challenge. There are however very important applications which depend on progresses made in this field, one of them being the computerized analysis of biomedical shapes (computational anatomy), which analyzes the impact of diseases on shapes of organs, obtained from modern techniques of non-invasive 3D imagery. The last fifty years of research in computer vision has shown a amazingly large variety of points of view and techniques designed for this purpose: 2D or 3D sets they delineate (via either volume or boundary), moment-based features, medial axes or surfaces, null sets of polynomials, configurations of points of interest (landmarks), to name but a few. Yet, it does not seem that any of these methods has emerged as ideal, neither conceptually nor computationally, for describing shapes. An important aspect of our study will be to describe shapes with an indirect approach, from the way they can be deformed. This is not a new idea, and can be traced back to the seminal works of D'Arcy Thompson at the beginning of the 20th century, but its mathematical formalization and the design of practical algorithms is a comprehensive task, still offering many open problems, that the present group will try to address and convey to the scientific community.