Multiscale medial shape-based analysis of image objects

Medial representation of a three-dimensional (3-D) object or an ensemble of 3-D objects involves capturing the object interior as a locus of medial atoms, each atom being two vectors of equal length joined at the tail at the medial point. Medial representation has a variety of beneficial properties, among the most important of which are 1) its inherent geometry, provides an object-intrinsic coordinate system and thus provides correspondence between instances of the object in and near the object(s); 2) it captures the object interior and is, thus, very suitable for deformation; and 3) it provides the basis for an intuitive object-based multiscale sequence leading to efficiency of segmentation algorithms and trainability of statistical characterizations with limited training sets. As a result of these properties, medial representation is particularly suitable for the following image analysis tasks; how each operates will be described and will be illustrated by results: segmentation of objects and object complexes via deformable models; segmentation of tubular trees, e.g., of blood vessels, by following height ridges of measures of fit of medial atoms to target images; object-based image registration via medial loci of such blood vessel trees; statistical characterization of shape differences between control and pathological classes of structures. These analysis tasks are made possible by a new form of medial representation called m-reps, which is described.

I. Medial Representations

Fig. 1. 2-D figure shown in terms of its boundary, then in terms of bitangent circles wholly interior to the figure, and finally, in terms of medial atoms (see also Fig. 5 for the 3-D case). Individual positions, circles, and medial atoms are shown at sample positions, but in each case, the locus is continuous. The part of this figure that forms the representation is shown in each case with bolder lines. Fig. 2. Medial atom. In this paper, we are concerned with object representations for use in three-dimensional (3-D) image analysis methods such as segmentation, registration, and statistical characterization of geometric differences between classes. To support these uses with regard to a population of “object(s),” i.e., single objects, such as kidneys or blood-vessel trees, or object ensembles, such as a liver or kidney pair, it is necessary that each member of the population be effectively captured by the form of representation. For the image analysis objectives described in this paper, we desire that the resulting models be useful for the following:

statistical characterization of the geometry of a class of object(s) [18] (see Section V);

segmentation by deforming a model into image intensity data (see Section III-A);

segmentation by measuring the fit of the local primitive from which the representation is formed to image data either so that •

ridges of this measure can be used to define the object (cf. Canny edges) (see Section III-B);

•

local statistics of this measure can be used to locate an object section and find its geometric type (slab, tube, sphere) (see [26]).