Digital tools for analyzing nondiffeomorphic shapes

What is it about?

Shapes are not numbers, and at such it is hard to quantify how much they vary. One way to solve this is to pick points on the shapes and look at their coordinates, these are numbers and we can quantify their variation with statistics. However, picking the points is somewhat arbitrary and subjective, and doing so ignores some of the information that the shape carries. Another way is to represent a shape as a surface, and then measure how much what mathematicians call 'energy' is needed to stretch one surface to another, this gives a principled high-fidelity representation of the shape. But some shapes cannot be stretched to each other, for example, if they have different number of holes. For example, the two suborders of primates, the haplorhini and strepsirrhini have different number of cavities in their skulls, so we cannot stretch them to one another. One interesting and theoretically pleasing way is to analyze such heterogeneous data makes use of Euler Characteristic, which is a generalization of counting how many connected components the shape has. An important and useful way to use the Euler characteristic in analyzing shapes is to record the Euler characteristic of the shape when we restrict it to a certain direction at certain height. This is called the Euler Characteristic Transform, or ECT. The ECT has been very successfully applied to analyze shape data that would be hard to analyze otherwise. An important theoretical result is that the ECT is a lossless summary of the shape, it contains all the information in the original one, if we scan all directions and all the heights. In practice, when we use the ECT in applications, we pick finitely many scanning directions and finitely many scanning heights. Mathematically speaking, we discretize the transform. The main reason for doing this is that it does not seem possible to scan at all directions and heights, because there are infinitely many of them. While the discretization seems to work in practice, it undermines the full exploitation of some of the nice theoretical properties that the transform enjoys. Also, from an end user perspective, if we use a discretization, we also need to choose discretization parameters. While there are some rules of thumb on how to choose these, this is not always so easy. In this work we showed that the ECT can be computed digitally, in other words, exactly up to computer precision. We showed that this is not only theoretically possible, but also practically feasible and beneficial. This builds on previous work on the transform. One of the main observations of in our paper is visualized in the Infographic. In this visualization, we compute the distance between two solid octahedram visualized on the left. We can represent their ECTs as height functions supported on spherical polygons, which we dubbed the 'proto-transform format' in our paper. These proto-transforms are visualized in the middle column. Inside each polygon, the height function is mathematically tractable (you can see the contours). The integral of the height function can be easily computed as we have shown in the paper. The representations in this visual look quite simple, this is because our meshes are convex. For real life data, the representation contains many overlapping spherical polygons, and managing how these overlaps gets more complicated. Keeping track of the overlapping parts in a systematic way is another main result in our paper. Unfortunately, this is bit harder to visualize. The third column shows a superposition of the two spherical triangulations, this represents the integrand, i.e. the squared difference between the two transforms in the middle. Here again, the mathematical expression of the integrand is, from a mathematical perspective, quite simple, and we can decompose the spherical integral into manageable pieces, which we can easily compute with a computer. The value of this integral is a quantity that we can use to quantify variation in shapes. Read the paper for the details.

Featured Image

Photo by Shubham Dhage on Unsplash

Photo by Shubham Dhage on Unsplash

Why is it important?

First, the new representation allows us to analyze the data in a truly lossless way, analogous (pun intended) to digital vs analog signal, fulfilling the promise of digitally analyzing heterogeneous shape data. Secondly, by removing the discretization, we also remove the need to choose the discretization parameters. This means that whoever wants to use this method to analyze their data does not have to worry about selecting parameters, because there are none. You could say that the computation algorithm got more complicated, but the end product got simplified more approachable. Thirdly, from a researcher's perspective, the digital representation allows us to use mathematically more sophisticated and principled tools to manipulate the transform, and hopefully in the future also to ask more complicated scientific questions with it. For example, the digital representation allows for a seamless interplay between the shape and the transform, a result on builds on deep mathematical theory that we can perhaps exploit more in the future. This would be very difficult with the discretized transform. Another, immediately actionable example of a mathematical tool is gradient descent, which we can put on top of the digital transform to align shapes, as we have demonstrated in the paper. More generally, the functions used in our ectoplasm algorithm are auto-differentiable, so they also fit very well general artificial intelligence and machine learning pipelines.

Perspectives