A New Foundation for Statistical Shape Analysis

形状无处不在：它是我们这个世界的基本属性。我们用它来区分物体(狗还是猫？)它对工具和其他人造物品(锤子、椅子等)的功能至关重要。它对于科学(分子的形状与它们的化学密切相关，不同类型的细胞通常通过它们的形状来区分等等)、对于工程(例如流线型)、对于医学(疾病通常改变细胞或器官的形状、检测肿瘤等)、对于考古学(例如对物体的识别和重建)等等都是非常重要的。对于统计学、机器学习和人工智能在所有这些领域的应用，能够将形状相互比较，看看它们有多不同或相似，并能够描述形状如何随时间变化是至关重要的。这一研究领域被称为形状分析，有着悠久的历史。尽管如此，仍然存在一些从未克服的根本性困难。克服这些困难是本研究的目的。它们是什么？以前所有比较形状的方法都分为三类。一种是通过变形来比较形状：使一个形状平滑地转换为另一个形状；需要的变形越多，形状就越不同。另一些则更抽象，通过将形状转换为其他数学对象来实现。在这样做的过程中，它们要么丢失了有关形状的信息，使一些形状无法区分，要么它们太笼统，失去了我们与单词“Shape”联系在一起的几何意义。那么变形方法有什么问题呢？问题是，当你平滑地变形一个形状时，有些地方你不能去。例如，您不能将一个形状变成两个形状，或者在一个形状上打一个洞，因为这将涉及撕裂。但在现实世界中，许多形状确实存在这种差异，我们需要能够对它们进行比较。例如，两辆自行车，一辆有横杆，另一辆没有，有不同数量的“洞”，不能比较。器官中血管的不同配置可能有各种各样的片段和环路，因此无法使用变形进行比较。当变形用于表示形状随时间变化时，同样的问题也会出现：例如，细胞分裂涉及一个形状变成两个形状。本研究提出了一种新的方法来表示这种变化。这是最容易可视化使用2D形状，你可以画在一张纸上。想象一下，一个圆圈逐渐伸展成哑铃形状，然后分裂成两个圆圈，就像细胞分裂一样。现在来看一条运动服的内裤。一端是腰部，呈圆形。另一端是脚踝的洞，这是两个圆圈。这条裤子代表着一个圆圈变成两个圆圈，没有任何撕裂，技巧是让我们感兴趣的2D形状成为3D形状的“末端”。然后，可以使用3D形状来测量2D形状之间的相似性，或者表示随时间变化的2D形状，即使这些形状不能彼此变形。对于时变的形状，还会发生一些有趣的事情：我们可以把裤子剪到中间，代表某个时间点的形状。我们可以以任何我们喜欢的波浪式方式来做这件事；我们不必直接切开。通过这种方式，3D形状可以表示变化的2D形状，但不同的部分以不同的速度移动。变形不能做到这一点，但这显然是最有用的情况。形状的无处不在的本质意味着形状分析的任何重大进展都会产生广泛而重要的后果。这项研究试图克服的困难阻碍了当前的技术解决一些最典型的现实世界案例。因此，成功将产生变革性的影响，在形状数学的理论和应用方面开辟令人兴奋的新前景。

Shape is all around us: it is a basic property of our world. We use it to tell objects apart (dog or cat?) and it is crucial to the functioning of tools and other man-made objects (hammers, chairs, ...). It is also very important to science (the shape of molecules is closely related to their chemistry, different types of cells are often distinguished by their shape, and so on); to engineering (e.g. streamlining); to medicine (disease often changes the shape of cells or organs, detection of tumours, ...); to archaeology (e.g. recognition and reconstruction of objects); and on and on. For applications of statistics, machine learning, and artificial intelligence to all these areas, it is vital to be able to compare shapes to each other, to see how different or alike they are, and to be able to describe how shapes change in time. This area of study, known as shape analysis, has a long history. Despite this, there are still fundamental difficulties that have never been overcome. It is the purpose of this research to overcome these difficulties. What are they?All previous methods of comparing shapes fall into three categories. One compares shapes by morphing: making one shape change smoothly into another; the more morphing required, the more different are the shapes. Others work more abstractly, by transforming the shape into some other mathematical object. In doing so, they either lose information about the shape, making some shapes impossible to distinguish, or they are too general, losing the sense of geometry we associate with the word 'shape'.What then is wrong with the morphing methods? The problem is that when you smoothly morph a shape, there are some places you cannot go. You cannot, for example, turn one shape into two, or make a hole in a shape, because this would involve tearing. But in the real world, many shapes do differ in this way, and we need to be able to compare them. For example, two bicycles, one with a crossbar and one without, have different numbers of 'holes', and cannot be compared. Different configurations of blood vessels in an organ may have a wide variety of pieces and loops, and so cannot be compared using morphing. The same problem arises when the morphing is used to represent shapes changing in time: cell division involves one shape changing into two, for example.This research proposes a new way to represent this kind of change. It is easiest to visualize using 2d shapes which you can draw on a piece of paper. Think of one circle gradually stretching into a dumbbell shape and then splitting into two circles, like a cell dividing. Now consider a pair of tracksuit bottoms. At one end is the waist, which is a circular shape. At the other end are the ankle holes, which are two circles. The pair of trousers represent one circle turning to two without any tearing, by the trick of making the 2d shapes in which we are interested be the 'ends' of a 3d shape. The 3d shape can then be used to measure the similarity between 2d shapes, or to represent a 2d shape changing in time, even if those shapes cannot be morphed into one another. For time-changing shapes, something else interesting happens: we can cut the trousers across the middle, representing the shape at a certain point in time. We can do this in any wavy way we like; we do not have to cut straight across. In this way, the 3d shape can represent 2d shapes changing, but with different parts moving at different speeds. Morphing cannot do this, and yet this is clearly the most useful case. The ubiquitous nature of shape means that any major advance in shape analysis has widespread and important consequences. The difficulties that this research attempts to overcome prevent current techniques from addressing some of the most typical real-world cases. Success will therefore have a transformative effect, opening up exciting new vistas in both the theory and applications of the mathematics of shape.