1998 - Draper Laboratory

of age-dependent decrease in caudate volume and diminishedincreases in ventricular volume (Ref. [6]). Patients with earlyonsetschizophrenia showed greater caudate and lateralventricular volumes than normal subjects (Ref. [7]). Patientswith Alzheimer’s disease showed enlarged lateral ventricles (Ref.[8]). Patients with chronic alcoholism also displayed anenlargement of the ventricular volume, while abstinence fromalcohol appears to reverse this trend (Ref. [9]). A study ofautistic male adults showed that these subjects have significantlygreater total brain tissue as well as lateral ventricle volumes thanthe control subjects (Ref. [10]). Collectively, these studies pointto the usefulness of quantifying ventricular and caudate volumesfor diagnostic purposes.Inherent characteristics of the human brain, as well as MRmachines, conspire to make automatic segmentation a difficultproblem. These characteristics include the partial volume effect,magnetic field bias effect, and low contrast between tissueclasses. Although past research has sought to remedy some ofthese effects (see, for example, Ref. [11]), we believe thecombined effects of these artifacts can substantially degrade theperformance of an automated segmentation system, and as such,we have turned instead to the exploitation of prior informationand semi-automated means in order to overcome these problems.More specifically, the additional information we use is in theform of prior segmentations -- an atlas, a manual segmentation(say of the same patient from another time) or a segmentationfrom a previous slice. The purpose of this paper is todemonstrate the ability to incorporate this prior information in asystematic and robust fashion in order to improve and expeditethe segmentation process.Conceptually, our approach revolves around several keyconcepts. First, as discussed above, we believe there is a need toincorporate prior information in order to alleviate the manyartifacts that are present in the data. Such prior information (orits equivalent in the form of operator input) is also necessary forfocusing the attention of the algorithm on the appropriateanatomical structure. Second, to perform volumetricmeasurements, it is necessary to develop a technique thatproduces closed curves. Third, for robustness and repeatabilityreasons, we believe the final segmentation must generally besmooth and well behaved. Finally, to be useful, the approachmust require only minimal human interactions. Collectively,these issues prompted us to devise an approach that usesnonlinear diffusion, incorporates curve evolution, and exploitsprior information. A key concept in our approach is the notionof an edge strength function -- a smoothly varying functionindicating the likelihood that a particular pixel is an edge pixel.As we show later in the paper, the ridges of the edge strengthfunction also serve as an attractor when the atlas is evolved to fitthe observed data. As such, the process of obtaining a robustedge strength function is a major element of this paper.This paper contains the following sections:• A discussion of the variational formulation, both forsegmentation and for curve evolution.• A description of our processing steps.• The results from processing both phantom and actualMR data.• Concluding remarks.Variational Segmentationand Curve EvolutionShah recently introduced a new approach to segmentation thatcombines various aspects of earlier formulations (discussed inthe Edge Strength Function section and in the Curve Evolutionsection). In order to provide the reader with a perspective ofwhere the following discussions are headed, we simply mentionhere that this new formulation is given by2ρνE( u, ν) = ∫∫ α( 1 2 2−ν) || ∇ u || + β | u − g | + || ∇ ν || + dxdyΩ 2 2ρ(1)where g denotes the observed intensity image, u denotes thepiecewise smooth approximation of g, ∇ denotes the gradientoperator, and ν denotes the estimated degree to which each pixellocation contains an edge (Ref. [12]). This new formulationcombines elements of both nonlinear diffusion as well as curveevolution -- elements necessary for well-behaved segmentation aswell as well-behaved regions that resemble the prior information.In this section, we first discuss the estimation of the edgestrength function, followed by a discussion of curve evolution,and finally, the relationship between Shah’s new formulation andthese earlier approaches is made explicit.The Edge Strength FunctionOne general approach to many image processing problems is touse the creation of an energy functional, the optimization of whichprovides a trade-off between fidelity to the observed data andvarious desirable constraints (e.g., smoothness, segmentation,etc.). Using this approach, Mumford and Shah devised acontinuous segmentation functional of the form (Ref. [13])2 2E = ∫∫ ( u( x, y ) − g( x, y )) dxdy + λ ∫∫ ∇ f( x, y ) dxdy + µ B( x, y )ΩΩ − B(2)in which g denotes the data, u the piecewise-smoothapproximate of g, B the binary edge process, and Ω the domainof integration (i.e., the image). More recently, a continuousapproximation to this binary edge term was suggested byAmbrosio and Tortorelli (Ref. [14]), in which |B| is replaced bySegmentation of MR Images Using Curve Evolution and Prior3