Object-based dynamic imaging with level set methods

In this dissertation, we study the reconstruction of geometric objects from various imaging data with a variational approach. More specifically, our work focuses on three particular challenges in this area: modeling the relationship of multiple shapes to each other, overcoming local minima in the optimization of shape functionals, and the real-time numerical implementation of the curve evolution method which underlies the associated optimization process of many object reconstruction problems.; We investigate two different scenarios where prior models of multiple shapes are important for a successful reconstruction. In the first case, we consider the problem of object-based dynamic tomography, where only sparse projection data of a sequence of time varying shapes are available for each time instant. We introduce a temporal object evolution model relating the sequence of shapes. A tractable variational solution is then proposed to incorporate these shape dynamics with a novel distance between shapes. This allows the robust reconstruction of dynamic objects from extremely limited tomographic data. In the second scenario, we consider multiple static shapes in the same image and propose to model their topological relations with the minimin shape distance. To incorporate such models into previous variational frameworks, we propose a differentiable approximation of the minimin distance. The topological priors defined using this distance enables us to segment closely spaced objects in blurry medical images.; In performing object-based optimization, the object boundaries are typically evolved using the level set method with a speed derived from the shape gradient of the energy. A difficult challenge for this method is the frequent presence of local minima, requiring a good initialization for successful reconstruction. To overcome this difficulty, we propose a topological-derivative-driven level-set approach and study its application in object reconstruction from static tomographic projection data. With our approach, interior objects can be detected automatically without special initializations. This leads to more robust convergence for shape-based optimization.; A large impediment to the practical use of existing level-set-based curve evolution methods is their high computational cost despite the superior results they can achieve. To solve this problem, we present a fast implementation of the level set method that only needs to update two lists of grid points simultaneously with a specially designed level set function. With our method, we evolve the curve at pixel accuracy and two orders of magnitude speedups can be achieved compared with previous narrow banding techniques. Our fast algorithm promises to remove the barriers for the adaptation of the level-set techniques into time critical imaging problems. Application of our algorithm in 3D image segmentation and real-time video tracking are presented.