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 manmade 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 timechanging 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 realworld cases. Success will therefore have a transformative effect, opening up exciting new vistas in both the theory and applications of the mathematics of shape.
