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.

SummaryElectron crystallography of membrane proteins determines the structure of membrane-reconstituted and two-dimensionally (2D) crystallized membrane proteins by low-dose imaging with the transmission electron microscope, and computer image processing. We have previously presented the software system 2dx, for user-friendly image processing of 2D crystal images. Its central component 2dx_image is based on the MRC program suite, and allows the optionally fully automatic processing of one 2D crystal image. We present here the program 2dx_merge, which assists the user in the management of a 2D crystal image-processing project, and facilitates the merging of the data from multiple images. The merged dataset can be used as a reference to re-process all images, which usually improves the resolution of the final reconstruction. Image processing and merging can be applied iteratively, until convergence is reached. 2dx is available under the GNU General Public License at http://2dx.org.

Abstract-Solutions of least squares support vector machines (LS-SVMs) are typically nonsparse. The sparseness is imposed by subsequently omitting data that introduce the smallest training errors and retraining the remaining data. Iterative retraining requires more intensive computations than training a single nonsparse LS-SVM. In this paper, we propose a new pruning algorithm for sparse LS-SVMs: the sequential minimal optimization (SMO) method is introduced into pruning process; in addition, instead of determining the pruning points by errors, we omit the data points that will introduce minimum changes to a dual objective function. This new criterion is computationally efficient. The effectiveness of the proposed method in terms of computational cost and classification accuracy is demonstrated by numerical experiments.Index Terms-Least squares support vector machine, pruning, sequential minimal optimization (SMO), sparseness.

Maximum Likelihood (ML) processing of transmission electron microscopy images of protein particles can produce reconstructions of superior resolution due to a reduced reference bias. We have investigated a ML processing approach to images centered on the unit cells of two-dimensional (2D) crystal images. The implemented software makes use of the predictive lattice node tracking in the MRC software, which is used to window particle stacks. These are then noise-whitened and subjected to ML processing. Resulting ML maps are translated into amplitudes and phases for further processing within the 2dx software package. Compared with ML processing for randomly oriented single particles, the required computational costs are greatly reduced as the 2D crystals restrict the parameter search space. The software was applied to images of negatively stained or frozen hydrated 2D crystals of different crystal order. We find that the ML algorithm is not free from reference bias, even though its sensitivity to noise correlation is lower than for pure cross-correlation alignment. Compared with crystallographic processing, the newly developed software yields better resolution for 2D crystal images of lower crystal quality, and it performs equally well for well-ordered crystal images.