Variational methods for shape and image registrations

One of most important image analysis tools that greatly benefits from the process of registration, and which will be addressed in this dissertation, is the image segmentation. This dissertation addresses the registration problem from a variational point of view, with more focus on shape registration. First, a novel variational framework for global-to-local shape registration is proposed. The input shapes are implicitly represented through their signed distance maps. A new Sum-of-Squared-Differences (SSD) criterion which measures the disparity between the implicit representations of the input shapes, is introduced to recover the global alignment parameters. This new criteria has the advantages over some existing ones in accurately handling scale variations. In addition, the proposed alignment model is less expensive computationally. Complementary to the global registration field, the local deformation field is explicitly established between the two globally aligned shapes, by minimizing a new energy functional. This functional incrementally and simultaneously updates the displacement field while keeping the corresponding implicit representation of the globally warped source shape as close to a signed distance function as possible. This is done under some regularization constraints that enforce the smoothness of the recovered deformations. The overall process leads to a set of coupled set of equations that are simultaneously solved through a gradient descent scheme. Several applications, where the developed tools play a major role, are addressed throughout this dissertation. For instance, some insight is given as to how one can solve the challenging problem of three dimensional face recognition in the presence of facial expressions. Statistical modelling of shapes will be presented as a way of benefiting from the proposed shape registration framework. Second, this dissertation will visit the shape-based segmentation problem. The piece-wise constant Chan and Vese segmentation models [1] are chosen as the underlying segmentation models and it will be shown how the proposed global shape registration technique can serve in enhancing the segmentation results of an input image when some prior knowledge of shapes is integrated in the underlying segmentation framework. The resulting paradigm allows the recovery of a segmentation map that is in accordance with the shape prior model as well as an affine transformation between this map and the model. Furthermore, it can deal with noisy, occluded and missing or corrupted data. The classical way of solving the shape-based segmentation problems within a level set framework is by directly solving the underlying Euler-Lagrange equations using a gradient descent scheme. This is very computationally expensive given the non-linear parabolic nature of the corresponding PDE's. To overcome these difficulties, a fast algorithm is designed and implemented to solve both the two-phase and the multi-phase shape-based segmentation problem. This algorithm exploits the fact that only the sign of the level set function, not its value, is needed to evolve the segmenting interface. The integration of multiple selective shape priors and the segmentation into multiple regions has never been addressed before. Third, a new image/volume non-rigid registration approach based on scale space and level set theories, will be introduced. This contribution is the fruit of a collaborative effort with two other members of the CVIP Lab. New feature descriptors are built as voxel signatures using scale space theory. These descriptors are used to capture the global motion of the imaged object. Local deformations are modelled through an evolution process of equi-spaced closed curves/surfaces (iso-contours/surfaces) which are generated using fast marching level sets and are matched based on a cross correlation measure between neighboring voxels. A novel Finite Element (FE)-based approach is developed to validate the performance of the proposed image registration techniques. Both two and three dimensional tissue deformations are simulated using the FE method. The registration accuracy with respect to the FE simulations is assessed by co-registering the deformed images with the original ones and comparing the recovered displacement field with the bio-mechanically simulated ones.