Particle methods for diffeomorphic registration

Matching objects in 2D and 3D plays an important role in many tasks in computer vision like object recognition, tracking, clustering and also medical imaging problems. Computational anatomy is the discipline which emerged 13 years ago, it is based on the quantitative comparison of various biological shapes to determine their variability among populations of healthy and diseased subjects. This thesis studies a topology preserving deformation based matching technique called Large Deformation Diffeomorphic Metric Matching (LDDMM) for curves, surfaces and images. Diffeomorphisms are differentiable functions with differentiable inverse, hence they preserve local smoothness and connectivity properties and the topology of the object they are acting upon, which allows us to study local variations in anatomical shapes and images. A deformation-based method finds the minimum-cost transformation which deforms one object to the other. The deformation cost, also called the deformation energy functional (defined on the two objects being matched), is designed such that its optimal value over the set of diffeomorphic deformations can be used to define a metric between the two objects being compared. The deformation energy consists of a weighted sum of a regularization term, which is a measure of smoothness of the registration and a data attachment term quantizing the discrepancy measure between the solution being considered and the target object.;In this thesis we have analyzed and extended LDDMM based registration techniques for registering curves, surfaces and images using particle based methods. In particle based methods the objects to be registered (curves/surfaces or images) are discretized into points (called particles) whose displacement governs the deformation in the object. In [42, 90, 96] such particle based methods for diffeomorphic registration have been discussed. In our work we have analyzed some variations in these techniques and some methods to make these algorithms faster.;In the first part of this thesis three different data attachment terms are formulated. Also quantitative and qualitative comparisons of the results of matching curves and surfaces using three different data attachment terms are done. We provide a series of experiments which show that the three criteria address high curvature regions differently and describe specific properties that one must be aware of while designing the application.;Extracting boundaries (curves or surfaces) from images disregards some information in the form of intensity values. When this information is important, one needs to use image matching. For matching images, optimizing the deformation energy over diffeomorphisms acting on the image is computationally intensive because the deformations are now defined over the whole image instead of the boundary only (as was the case for curves and surfaces). In this thesis we study a technique which is based on the fact that the deformation in the image can be parameterized around those regions in the image which have high intensity gradient. We have used one such parameteriation to reduce the dimensionality of the problem.;The deformations we work with are generated by integrating a set of ODEs defined by a velocity field on the points on the object (curves, surfaces or images). We only consider the velocity vector fields winch belong to a Reproducing Kernel Hilbert Space (RKHS) of functions. This is one of the necessary conditions which ensures that the deformation generated by these velocity fields are diffeomorphic. Solving the set of equations which generate these deformations involves computing weighted kernel (the kernel associated to the RKHS) sums over a set of points. This leads to a huge computational overhead if the number of points in the discretization of the object is large. In this thesis we introduce and establish properties of a finitely generated kernel class in which the kernel is defined using a double interpolation from a discrete kernel supported by a regular grid covering the domain of the system of particles under consideration. This construction not only speeds up the calculations by utilizing standard algorithms for faster computations over regular grids, but also maintains the exactness and consistency of the system. We provide experimental results in support of this, comparing in particular the computation time and accuracy to similar competing methods.