Welcome to this first theory lecture on shape modelling. Today, we will be discussing the basic notions of shape modelling.
The most fundamental concept in shape modelling is the notion of a shape. On this slide, we see three different geometrical objects. We would refer to each of them as a square even though they differ in size, position, and rotation. This is reflected by the classical definition of “shape.” A “shape” is defined as all geometric information that remains when location, scale, and rotational effects are filtered out from an object. In this course, we will be talking mainly about anatomical shapes. The hand shapes are our running example. For anatomical shapes, we change this definition slightly. We say a shape is defined as all geometric information that remains when location and rotational effects are filtered out from an object, but not the size.
So we would say that this shape here and this shape here are the same, but this one here is not the same because it is smaller. The reason for that is, that in biology, size and shape are often correlated. The shape we see here is a typical shape of a baby hand, but it would be very atypical if it was the shape of an adult. If we want to model such effects and capture such effects in our model, we would need to introduce size as a parameter of the model and we cannot do that if we filtered it out beforehand. The next important concept is the concept of a shape family. Here, we see the family of triangle shapes.
Triangle shapes, they all have the same mathematical property– that they are defined on three points which are connected by straight lines. We can also think of shape families of anatomical shape. For example, the family of hands. However, it is a little bit more difficult because there is no mathematical definition available anymore that defines ‘when is an object part of this shape family’? But the notion still makes a lot of sense, and we can see that if we consider another shape. For this shape here, this foot, we can actually immediately recognise that it doesn’t belong to the same shape family.
Although we cannot give a mathematical formula to define the family of anatomical shapes, we can still use mathematics to characterise what constitutes, for example, a hand shape. The trick is that we start with example shapes that are normally examples from this shape family,
and then we use statistics to model the characteristics: ‘What makes this shape a hand shape?’, for example. The models that we built in this way, they can, on one hand, tell us for any shape how likely it is that the shape is part of the family. What these models can also do is they can generate new shapes from this same family. Before we can discuss how we model shapes, we need to discuss how we represent the shapes. The most classical representation of shapes is in terms of landmark points. Landmark points are points that have an anatomical meaning, such as, for example, the tip of the thumb. These points we can easily find in all the shapes.
The problem with landmark points is that there are only a few of them. And if we want to have a good faithful shape representation, we actually need many more points. And in this course, we’re using a dense set of points on the contour to represent a shape. In fact, the mathematical theory that we will be developing also works if we take an infinite set of point. So the full contour. In the theory lectures, we’ll illustrate in all the concepts using 2D hand shapes. In Scalismo Lab, when you do the exercises, you will see that we do the exercises on surfaces in three dimensions. The concept is exactly the same. We represent surfaces as a set of points.
What you will see in addition, in surfaces, the points are connected by triangles. But this has nothing at all to do with the mathematical theory of shape modelling. This is only because we need it to visualise the shapes using computer graphics algorithm. Now that we have represented shapes purely in terms of points on the contour, when we model them in shape models, the only possibility we have is to model how these points can move such that the shape remains within the family. These type of models are called “point distribution models.” The main assumption that is behind these models is that the points can move according to a multivariate normal distribution.
In the next article, we have summarised the main properties of multivariate normal distributions that we will need in this course. This is indeed the most important concept of this course. I’m looking forward to welcome you back next week when I can show you how the multivariate normal distribution can be used to model shapes.