11Continuous Medial Representations for GeometricObject Modeling in 2D and 3DPaul Yushkevich, P. Thomas Fletcher, Sarang Joshi, Andrew Thall, Stephen M. PizerMedical Image Display and Analysis GroupUniversity of North Carolina at Chapel Hill, USAE-mail: [email protected]— We describe a novel continuous medial repre-sentation for describing object geometry and a deformabletemplates method for fitting the representation to images.Our representation simultaneously describes the boundaryand medial loci of geometrical objects, always maintainingBlum’s symmetric axis transform (SAT) relationship. Cu-bic b-splines define the continuous medial locus and the as-sociated thickness field, which in turn generate the objectboundary. We present geometrical properties of the rep-resentation and derive a set of constraints on the b-splineparameters. The 2D representation encompasses branchingmedial loci; the 3D version can model objects with a sin-gle medial surface, and the extension to branching medialsurfaces is a subject of ongoing research. We present prelim-inary results of segmenting 2D and 3D medical images. Therepresentation is ultimately intended for use in statisticalshape analysis.I. IntroductionMedial loci, or skeletons, have enjoyed wide use in com-puter vision and medical image analysis because they pro-vide important intuition about shape and formation of bio-logical and anatomical objects. Medial loci naturally divideobjects into a hierarchy of simple figures and describe theinherent symmetry and local thickness of each figure.Medial loci of objects have traditionally been computedfrom discrete boundary-based descriptions by skeletoniza-tion algorithms. Such boundary descriptions, however,yield medial loci with a complex branching structure; forinstance, each edge in a 3D polygonal boundary contributesa branch to the medial locus. Methods that simplify andregularize skeletons can eliminate unstable branches andyield object-relevant medial loci [18], [28], [23], [22]. Nev-ertheless, the boundary-to-medial transformation is inher-ently unstable; the resulting branching topology is sensitiveto slight boundary perturbations, especially at the regionsknown as ligatures [4], [1].Whereas the above methods start with a boundary de-scription and yield the medial locus, synthetic medial rep-resentations, such as the one presented in this paper, usethe medial loci themselves as a model for object represen-tation. The model describes the medial branching topol-ogy and defines each branch of the medial locus using afew parameters. The model defines a smooth parameter-ized thickness field over the entire medial locus. A two-dimensional medial locus is a set of smooth curve segmentsjoined at endpoints. A three-dimensional medial locus isa set of surface patches connected along curves. (See Fig.1.)ABγγγγ(a) (b)Fig. 1. Branching medial loci in in 2D and 3D. (a) In 2D, threebranches generically join at shared endpoints. Approaching theshared endpoint, the thickness (radius) values associated with eachbranch converge to a common value. (b) In this 3D example, twomedial surfaces are connected along a seam γ. The seam forms acrease in A and is part of the edge of B.The medial locus and the associated thickness field syn-thesize a stable object boundary by inverting the skele-tonization process. The generated boundary is equivalentto the envelope of spheres (or disks) placed at each pointin the medial locus with the radius prescribed by the asso-ciated thickness value. The model establishes a correspon-dence between each point on the medial locus and a pair ofpoints on the generated boundary. Synthetic medial repre-sentations enforce a fixed medial branching topology andprovide a simultaneous description of the medial locus andthe object boundary.In this paper we present a continuous representation thatuses cubic b-splines to model both the medial manifoldsand the associated thickness field. We develop constraintson the parametric definition of the medial model that guar-antee that the generated boundary surface is a closed, con-nected, non-singular, manifold with curvature continuity.The current state of the method allows representation of2D objects with branching medial topology and 3D objectswith a single medial surface. The extension to branchingmedial surfaces is the subject of ongoing research.Our representation is essentially a continuous extensionof m-reps, a discrete medial representation developed byPizer et al. [26]. Discrete m-reps describe medial loci andboundaries of objects using sparse discrete samples calledmedial atoms. Each medial atom encapsulates local sur-face and thickness properties of the medial locus and im-plies local surface properties of the object boundary. M-reps have been shown effective for object representation,modeling, and image segmentation using deformable tem-plates because natural operations such as bending, widen-12ing and elongation are easily implemented [17], [34], [36].Styner automatically computed m-rep templates with fixedbranching topology that through deformation accurately fitpopulations of hippocampi, amygdalae and other subcorti-cal organs [32].In this paper we show that the continuous medial repre-sentation can be applied to segment anatomical objects inmedical images, following the same deformable templatesframework used for discrete m-reps in work of Joshi, et al[17]. As an example, we automatically segment a vertebralimage from a CT slice using a 2D model with branchingmedial topology. In 3D, we deform a template model ofthe hippocampus to fit manually segmented magnetic res-onance images of the brain.We developed continuous m-reps with the ultimate goalof improving our present methods in statistical shape anal-ysis. The methods previously developed in our lab use dis-crete m-reps with a fixed branching topology to describea population of objects [31], [39]. These methods estimateprobability distributions of the statistical features derivedfrom medial atoms. These distributions are used to gener-ate new instances of m-reps, to visualize the primary modesof variability in the population in terms of bending or thick-ening of objects, to perform classification on the basis ofshape and to pinpoint locations in objects where shape vari-ability is most pronounced. Continuous m-reps augmentshape analysis methods by allowing arbitrary sampling ofmedial loci. The continuous medial representation makes itpossible to elastically model and