The Applications Of Geometric Variational Theory In Image Registration

This thesis focuses on studying the non-rigid image registration problem, which uses the principles on geometrical variational methods, the partial differential equations and information theories. We establish the new models for three non-rigid image registration problems. Meanwhile, we prove the existence, uniqueness and stability for some of the new models. Finally, combining the variational method with the fast algorithm, we obtain the numerical solution and experimental results. The main research results are as follows:1. Modified mono-modality non-rigid image registration model.Generally, the mono-modality image registration problem is mainly to find the op-timal transformation, which can make two or more images be very similar on the gray value. For finding the optimal transformation, we must choose the proper regularization constraint. By discussing the characteristics of the transformation, we consider the de-formation as a sum of two components:a piece-wise constant component and a smooth component. So we introduce a novel regularization constraint, which consists of the to-tal variation regularization for the piece-wise constant component and the fourth-order diffusion regularization for smooth component. Additionally, the local-global similarity measure is used in our method to improve the accuracy and robustness for matching. We also give the existence and uniqueness of solution about the new model.2. Modified multi-modality non-rigid image registration model.For the multi-modality image registration problem, the mutual information is more important and popular similarity measure. However, using the conventional mutual in-formation measure, it is supposed that the gray value of the pixels in the image are the independent and identically distribute variable. Furthermore, the size of the overlapping part in the images influences the mutual information measure. In fact, there is different utility for the different pixels in the image for the image processing. So we present a novel image similarity measure, named as quantitative-qualitative measure of normalized mu-tual information (QNMI). Based on the ONMI, we present two new models for non-rigid multi-modality image registration problem. Numerical results show the effectiveness.3. Modified inverse consistent non-rigid image registration model.In some applications, especially for guiding the medical treatment, the optimal trans-formation for image registration must be one-to-one and differentiable. This problem is referred to the inverse consistent image registration, since the image function is complex and the dimension is high, and there are a lot of local extremum for the common image registration model. It is difficult to make the transformation be one-to-one. We propose two novel variational model for the inverse consistent image registration problem, and give some existence theory about the models. Some fast algorithms bring forth the good numerical results.