The class of geometric deformable models, also known as level sets, has brought tremendous impact to medical imagery due to its capability of topology preservation and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful state-of-the-art applied mathematical tools, this paper is an attempt to explore these geometric methods, their implementations and integration of regularizers to improve the robustness of these topologically independent propagating curves/surfaces. This paper first presents the origination of level sets, followed by the taxonomy of level sets. We then derive the fundamental equation of curve/surface evolution and zero-level curves/surfaces. The paper then focuses on the first core class of level sets, known as "level sets without regularizers." This class presents five prototypes: gradient, edge, area-minimization, curvature-dependent and application driven. The next section is devoted to second core class of level sets, known as "level sets with regularizers." In this class, we present four kinds: clustering-based, Bayesian bidirectional classifier-based, shape-based and coupled constrained-based. An entire section is dedicated to optimization and quantification techniques for shape recovery when used in the level set framework. Finally, the paper concludes with 22 general merits and four demerits on level sets and the future of level sets in medical image segmentation. We present applications of level sets to complex shapes like the human cortex acquired via MRI for neurological image analysis.

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In this paper, we describe ongoing work in the Image Processing and Analysis Group (IPAG) at Yale University specifically aimed at the analysis of structural information as represented within magnetic resonance images (MRI) of the human brain. Specifically, we will describe our applied mathematical approaches to the segmentation of cortical and subcortical structure, the analysis of white matter fiber tracks using diffusion tensor imaging (DTI), and the intersubject registration of neuroanatomical (aMRI) data sets. Many of our methods rally around the use of geometric constraints, statistical (MAP) estimation, and the use of level set evolution strategies. The analysis of gray matter structure and connecting white matter paths combined with the ability to bring all information into a common space via intersubject registration should provide us with a rich set of data to investigate structure and variation in the human brain in neuropsychiatric disorders, as well as provide a basis for current work in the development of integrated brain function-structure analysis.

Abstract. The cortex is the outermost thin layer of gray matter in the brain; geometric measurement of the cortex helps in understanding brain anatomy and function. In the quantitative analysis of the cortex from MR images, extracting the structure and obtaining a representation for various measurements are key steps. While manual segmentation is tedious and labor intensive, automatic, reliable and efficient segmentation and measurement of the cortex remain challenging problems due to its convoluted nature. A new approach of coupled surfaces propagation using level set methods is presented here for the problem of the segmentation and measurement of the cortex. Our method is motivated by the nearly constant thickness of the cortical mantle and takes this tight coupling as an important constraint. By evolving two embedded surfaces simultaneously, each driven by its own image-derived information while maintaining the coupling, a final representation of the cortical bounding surfaces and an automatic segmentation of the cortex are achieved. Characteristics of the cortex such as cortical surface area, surface curvature and thickness are then evaluated. The level set implementation of surface propagation offers the advantage of easy initialization, computational efficiency and the ability to capture deep folds of the sulci. Results and validation from various experiments on simulated and real 3D MR images are provided.

The gating ring of cyclic nucleotide-modulated channels is proposed to be either a two-fold symmetric dimer of dimers or a four-fold symmetric tetramer based on high-resolution structure data of soluble cyclic nucleotide-binding domains and functional data on intact channels. We addressed this controversy by obtaining structural data on an intact, full-length, cyclic nucleotide-modulated potassium channel, MloK1, from Mesorhizobium loti, which also features a putative voltage-sensor. We present here the 3D single-particle structure by transmission electron microscopy and the projection map of membrane-reconstituted 2D crystals of MloK1 in the presence of cAMP. Our data show a four-fold symmetric arrangement of the CNBDs, separated by discrete gaps. A homology model for full-length MloK1 suggests a vertical orientation for the CNBDs. The 2D crystal packing in the membrane-embedded state is compatible with the S1-S4 domains in the vertical "up" state.