| Biological shapes such as the brain are difficult to register due to their complicated geometry. To deal with this, registration methods often rely on a transformation model consisting of a dense regular grid such as a free form deformation or B-spline grid. However, very dense grids or meshes are usually needed to register images with convoluted shapes, and a regular mesh structure is not well suited for the irregular structure of the brain. What is therefore needed is a meshfree approach such as a radial basis function transformation model. Unfortunately, because radial basis functions are typically non-compact, using them with large numbers of points is fraught with numerical difficulties and, as a result, their use in image registration is not prevalent.; The goal of this work is to overcome these computational difficulties so that radial basis function transformations can be used efficiently, even with large numbers of points. To achieve this, a new registration framework was developed based on automatic differentiation and the fast multipole method. Automatic differentiation is useful since an important component of registration is computing the gradient of the similarity metric which is to be optimized. Automatic differentiation allows one to efficiently calculate gradients without having to write any gradient code explicitly. Although the technique of automatic differentiation is well established, it does not appear to be used for image registration. The fast multipole method was developed to efficiently evaluate large sums such as radial basis functions but its use in image registration is still minimal. With the integration of these algorithms within a complete registration framework, it should be possible to obtain a truly meshfree registration. |
