Ganesh Swami

Table of Contents:

Introduction

A major part of my work term at the Medical Image Analysis Lab was focused on understanding a computer vision algorithm called Large Deformation Diffeomorphic Metric Mapping (LDDMM) for non-rigid image registration. After learning sufficiently enough of the technology, I gave a talk at the Canadian Undergraduate Math Conference. Here’s an outline of the talk I gave. To be clear, I didn’t come up with any of the following stuff and all of what I have described has been published. My only contribution so far is to put down about 25 years of innovation onto a single page. The devil is in the details.

Agenda & Concepts

To motivate what we are solving, I showed them a bunch of images that can transformed into one another by a function. Image registration is to come up with the function that can do the mapping.

Registration can be formulated as an optimization problem. The goal is to come up with an optimal displacement vector field that minimizes the L2-norm of the differences between the target and the transformation applied to the template image.

Image registration

Registration is inherently an ill-posed problem. You’d require a regularizer to restrict the space the solutions for the registration problem.

In elastic matching, internal body forces can be used as an regularizer. The problem with elastic matching is that it can only be used for small deformations. The fix for this is to have a time indexed displacement vector field. Inspired by fluid mechanics, the fluid model was proposed by Christensen and group. In fluid matching, viscosity acts as an regularizer.

Infinite-dimensional optimization

The fluid model was solved using a method that only gave locally optimal solutions. Instead, if we compute the Euler-Lagrange equation from the cost function and solve that, we get a globally optimal solution. Moreover, this path also defines a metric along the mapping.

The slide above is the bulk of LDDMM. I won’t attempt to explain it all, read the paper referenced for that. The key point is that the minimization is done by a steepest descent type algorithm for the Euler-Lagrange equation (not shown here.) The norm for the velocity vector field is taken as a Hilbert norm instead of the usual L2 norm. Solving for the velocity vector field at one time step gives us the deformation vector field via integration. This is then used in the consecutive descent step. The sigma term is the weight between the regularizer and the matching.

Some math to do accounting.

Keeping It Real

Most of the math behind these sophisticated registration algorithms was invented by mathematicians. As engineers, by definition we are interested in applications of the technique to real world problems.

LDDMM comes with some interesting features. On a basic level, it can be used to register images. Since the mapping is derived from the Euler-Lagrange equations, it’s a geodesic and thus defines a metric. You can use this to compute distances between images.

LDDMM also has an information compression scheme. After the computation of the geodesic, you only need the velocity vector field at initial timestep. With this you can regenerate the entire trajectory for the mapping. This can be used to generate atlases of many images.

The image above deforms a cube to two types of rectangular prisms. Once the mappings have been computed, they can be averaged and then shot forward to compute the average deformation.

This is only one of the applications, there are many more.

References

• M. F. Beg, M. Miller, A. Trouv’e, and L. Younes. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. International Journal of Computer Vision, Volume 61, Issue 2; February 2005.