This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery. The segmentation procedure is based on a Bayesian deformable templates methodology in which the prior information about the geometry and shape of anatomical objects is incorporated via the construction of exemplary templates. The anatomical variability is accommodated in the Bayesian framework by defining probabilistic transformations on these templates. The transformations, thus, defined are parameterized directly in terms of natural shape operations, such as growth and bending, and their locations. A preliminary validation study of the segmentation procedure is presented. We also present a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localized variability at the fine scale. We show that the method can be used to statistically describe shape variability in intuitive terms such as growing and bending.

We consider the average outward flux through a Jordan curve of the gradient vector field of the Euclidean distance function to the boundary of a 2D shape. Using an alternate form of the divergence theorem, we show that in the limit as the area of the region enclosed by such a curve shrinks to zero, this measure has very different behaviours at medial points than at non-medial ones, providing a theoretical justification for its use in the Hamilton-Jacobi skeletonization algorithm of [7]. We then specialize to the case of shrinking circular neighborhoods and show that the average outward flux measure also reveals the object angle at skeletal points. Hence, formulae for obtaining the boundary curves, their curvatures, and other geometric quantities of interest, can be written in terms of the average outward flux limit values at skeletal points. Thus this measure can be viewed as a Euclidean invariant for shape description: it can be used to both detect the skeleton from the Euclidean distance function, as well as to explicitly reconstruct the boundary from it. We illustrate our results with several numerical simulations. 1.

We describe a novel continuous medial representation for object geometry and a deformable templates method for fitting the representation to images. Our representation simultaneously describes the boundary and medial loci of geometrical objects, always maintaining Blum’s symmetric axis transform (SAT) relationship. Cubic b-splines define the continuous medial locus and the associated thickness field, which in turn generate the object boundary. We present geometrical properties of the representation and derive a set of constraints on the b-spline parameters. The 2D representation encompasses branching medial loci; the 3D version can model objects with a single medial surface, and the extension to branching medial surfaces is a subject of ongoing research. We present preliminary results of segmenting 2D and 3D medical images. The representation is ultimately intended for use in statistical shape analysis.

Abstract. For a compact region Ω in R n+1 with smooth generic boundary B, the Blum medial axis M is the locus of centers of spheres in Ω whch are tangent to B at two or more points. The geometry of Ω is encoded by M, which is a Whitney stratified set, and U, the multivalued vector field from points on M to the points of tangency. We give general formulas for integrals of functions over B or Ω in terms of integrals over M. These integral formulas involve a radial shape operator which captures the radial geometry of U on M, an intrinsic medial measure on M, and a radial flow from M to B. For integrals over Ω the formulas remain valid when we relax the conditions on (M, U), yielding a more general skeletal structure. These integral formulas are applied to yield: an extension of Weyl’s volume of tubes formula where we replace tubes by general regions; a medial version of the generalized Gauss-Bonnet formula for B, valid even for odd dimensional B; versions of Crofton-type formulas and Steiner formulas for subregions of Ω; and a version of the divergence theorem over subregions in Ω for vector fields with discontinuities across the medial axis. This last result leads to a justification of an algorithm for finding the medial axis, using an invariant equivalent to a local medial density for singularities introduced elsewhere.

Abstract-- Object descriptions used for 3D segmentation by deformable models and for statistical characterization of 3D object classes benefit from having intrinsic correspondences over deformation of the objects or multiple instances in the same object class. These correspondences apply over a variety of spatial scale levels and consequently lead to efficient segmentation and probability distributions of geometry that are trainable with an achievable number of training instances. This paper describes a figural coordinate system provided by m-reps models and shows how such coordinates not only provide the required positional correspondences, but also are intuitive and provide orientational and metric correspondences. Examples are given for the segmentation of kidneys from CT and for the statistical characterization of schizophrenia and control classes of cerebral ventricles and of hippocampus pairs. I.

The discrete m-rep, a medial representation of anatomical objects made from one or more meshes of medial atoms, has many attractive properties for biomedical image analysis. The nonlinear nature of the m-rep parameters captures nonlinear deformations in anatomical objects such as twisting and bending. Most uses of m-reps require extending them to one or more continuous sheets of medial atoms. To guarantee that a continuous sheet of medial atoms does not fold, we propose an interpolation method on the medial atoms based on two medial shape operators. The radial shape operator describes the rate of swing of a medial atom’s spoke. The edge shape operator describes the rate of swing of a planar curve defining the object crest. The two shape operators generate interpolated internal and end atoms, and they join together smoothly. We show the application of our interpolation method on m-reps of synthetic and real-world objects. 1.

Abstract. In deformable model segmentation, the geometric training process plays a crucial role in providing shape statistical priors and appearance statistics that are used as likelihoods. Also, the geometric training process plays a crucial role in providing shape probability distributions in methods finding significant differences between classes. The quality of the training seriously affects the final results of segmentation or of significant difference finding between classes. However, the lack of shape priors in the training stage itself makes it difficult to enforce shape legality, i.e., making the model free of local self-intersection or creases. Shape legality not only yields proper shape statistics but also increases the consistency of parameterization of the object volume and thus proper appearance statistics. In this paper we propose a method incorporating explicit legality constraints in training process. The method is mathematically sound and has proved in practice to lead to shape probability distributions over only proper objects and most importantly to better segmentation results. 1

Abstract. A crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size. 1

Abstract. We present a new, continuously defined three-dimensional medial shape representation based on subdivision surfaces. The shape is modeled via its medial axis, and the associated boundary is computed directly from this axis at every point. Our model is parameterized over a fixed domain, so comparison among different shapes is possible. It is the first such model to support branch curves, which allows it to represent complex medial axes with more than one medial sheet. 1

The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations will be presented. A novel geometric seeding algorithm for Newton’s method was used in experiments to determine feasible support for these visualization applications. 1 Computing the pipe surface radius Parametric curves have been shown to have a particular neighborhood whose boundary is non-self-intersecting [5]. It has also been shown that specified movements of the curve within this neighborhood preserve the topology of the curve [9, 8], as is desired in visualization. This neighborhood is defined by a single value, which is the radius of a pipe surface, where that radius depends on curvature and the minimum length over those line segments which are normal to the curve at both endpoints of the line segment [5]. Since computation of curvature is a well-treated problem, the focus of this paper is efficient and accurate floating point techniques to compute the other dependeancy for that radius.