Tremendous advances have been made in medical imaging technology during the last decade and the need for automated techniques for the difficult problems involved in the measurement of structure has greatly increased. Reliable, valid, and efficient methods are needed to quantify image information using all available prior information. Boundary finding in medical images has been a critical problem addressed by many in recent years. The two principal sources of image-derived information that are used by most segmentation methods are region-based and boundary-based. In addition, extrinsic information from prior knowledge of shape or shape characteristics can be brought to bear on the problem in the form of shape models, constraints, atlases, etc.
Region-based methods use tissue classification to assign voxels to classes based on the gray level of the voxel, the gray level distribution of the image, neighboring voxels, or other measures of homogeneity. Markov random fields [16,23] have been used to model probabilistic constraints on image gray levels to aid in classification. Wells et al.  proposed estimating brain tissue classes (gray, white, CSF) while simultaneously estimating the underlying magnetic resonance bias field using an expectation-maximization (EM) strategy. Other work includes that of Cline et al. , who use a multispectral voxel classification method in conjunction with connectivity to segment the brain into different tissue types from 3D magnetic resonance (MR) images. Although all of
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these methods work to some extent, region-based methods typically require further processing to group segmented regions into coherent structure(s), and hence constraints based on topological or geometric coherence cannot be easily built into the processing. This inability makes such techniques limited when applied to medical images where such constraints are a necessity. A number of techniques use ad hoc postprocessing after intensity classification using a sequence of arithmetic, logic or morphologic operations [4,5,17] with varying success.
A large amount of work has been carried out using deformable surfaces for 3D boundary finding particularly using the snakes approach ofKass etal. . Close initialization is needed in order to achieve good results; outward balloon forces can help with this problem. Deformable surface models using the finite-element method have also been used . However, the need to override local smoothness to allow for the significant protrusions that a shape may possess (which is highly desirable in order to capture, for example, the folds ofthe cortex) remains a problem. Another type of deformable model involves level set methods [25,26,32], which are powerful techniques for analyzing and computing interface motion. The essential idea is to represent the boundary of interest as a propagating wavefront. Equations governing the motion are developed so that the boundary propagates along its normal direction with a speed controlled as a function of surface characteristics (e.g., curvature, normal direction) and image characteristics (e.g., gray level, gradient). This powerful approach can be extended, as will be described later, to include generic shape constraints in order to control the behavior of the standard level set algorithms, which may be unsuccessful when the image information is not strong enough.
A number of groups have attempted to develop variations of deformable models specifically for solving the problem of segmenting cortical gray matter, a particular challenge to such methods and an important application area. Davatzikos and Bryan used a ribbon for modeling the outer cortex and proposed an active contour algorithm for determining the spine of such a ribbon . However, close initialization and human interaction are still needed to force the ribbon into the sulcal folds. Xu et al.  used gradient vector flow fields in conjunction with tissue membership functions as a way of better controlling the deformation of snakelike active contour models for finding the central cortical layer halfway between the gray/white and gray/CSF boundaries. Teo et al.  used a system that exploited knowledge of cortical anatomy, in which white matter and CSF regions were first segmented, then the connectivity of the white matter was verified in regions of interest. Finally, a connected representation of the gray matter was created by growing out from the white matter boundary. MacDonald et al.  designed an iterative algorithm for simultaneous deformation of multiple surfaces to segment MR brain images with intersurface constraints and self-intersection avoidance using cost function minimization. This approach takes advantage of the information of the interrelation between the surfaces of interest, but is computationally expensive, and it requires tuning a number of weighting factors in the cost function. Kapur et al.  also use a snake approach, in conjunction with EM segmentation and mathematical morphology. Although volume measurement may be reliable with this approach, the shape of the outer surface is poorly representative of the true surface.
Statistical models can be powerful tools for directly capturing the variability of structures being modeled. Such techniques are a necessity for the segmentation of structure that is consistent in shape but poorly defined by image features, as is often the case in medical images. Atlas registration for the purposes of segmentation [10,14] is one way ofusing prior shape information. Collins et al. , for example, segment the brain using an elastic registration to an average brain, based on a hierarchical local correlation. The average brain provides strong prior information about the expected image data and can be used to form probabilistic brain atlases . A variety of specific parametric models for prior shape have been used successfully in our laboratory [34,35,40] and by other groups [12,36,39] for segmentation. The statistics of a sample of images can be used to guide the deformation in a way governed by the measured variation of individuals. Staib and Duncan use prior probabilities to model deformable objects with a Fourier representation for segmentation [34,35]. The prior is computed based on statistics gathered from a sample of image-derived shapes. Point-based models and their associated statistics are used in medical images for segmentation [12,40]. They represent objects using a principal-component analysis of a sample of shapes. The use ofprincipal components has also been adapted for the Fourier representations by Szekely etal. . Others have used moment-based constraints  to incorporate shape information. Region-based information can be combined with prior models in a single-objective [7, 33] or game-theoretic manner  in order to enhance the robustness of the approach.
We categorize segmentation problems into two types: the delineation of structure of known shape whose form is a continuous variation around a mean shape and the finding of more complex structure that varies significantly between individuals. In the first case, we have developed an integrated method using parametric shape models applied to structures such as the caudate in the subcortex of the brain and the left ventricle of the heart. In the second, we have developed a coupled level-set approach, applied primarily to the cortex of the brain. These methods have been tested on a variety of synthetic and real images.
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