Non-Euclidean classification of medically imaged objects via s-reps

Abstract

Classifying medically imaged objects, e.g., into diseased and normal classes, has been one of the important goals in medical imaging. We propose a novel classification scheme that uses a skeletal representation to provide rich non-Euclidean geometric object properties. Our statistical method combines distance weighted discrimination (DWD) with a carefully chosen Euclideanization which takes full advantage of the geometry of the manifold on which these non-Euclidean geometric object properties (GOPs) live. Our method is evaluated via the task of classifying 3D hippocampi between schizophrenics and healthy controls. We address three central questions. 1) Does adding shape features increase discriminative power over the more standard classification based only on global volume? 2) If so, does our skeletal representation provide greater discriminative power than a conventional boundary point distribution model (PDM)? 3) Especially, is Euclideanization of non-Euclidean shape properties important in achieving high discriminative power? Measuring the capability of a method in terms of area under the receiver operator characteristic (ROC) curve, we show that our proposed method achieves strongly better classification than both the classification method based on global volume alone and the s-rep-based classification method without proper Euclideanization of non-Euclidean GOPs. We show classification using Euclideanized s-reps is also superior to classification using PDMs, whether the PDMs are first Euclideanized or not. We also show improved performance with Euclideanized boundary PDMs over non-linear boundary PDMs. This demonstrates the benefit that proper Euclideanization of non-Euclidean GOPs brings not only to s-rep-based classification but also to PDM-based classification.

Keywords: Pattern classification; Shape analysis; Statistical analysis.

Figure 1

(a) Skeletal model of a hippocampus s-rep; (b) solid model implied by that s-rep. Yellow spheres are sample points along the skeletal surface. Solid lines extending from these sample points are spoke vectors, which are approximately normal to the boundary surface. Interpolation of a discrete s-rep into a continuous skeleton with a continuous field of spokes forms a continuous s-rep whose spokes completely fill the interior of the object they are representing.

Figure 2

Visualizations of (left) the class likelihoods and (right) the probability mapping function overlaid on top of the distributions. The empirical histogram of the scalar projection of the control cases in the training set onto the separation direction is plotted in the blue dotted lines; then the Gaussian probability distribution for the controls is plotted in the blue solid curve. The histogram for the schizophrenic class is plotted in the green dotted lines, and the corresponding Gaussian probability distribution for the schizophrenic class is plotted in the solid green curve. The function on the right that maps from the scalar projection onto the direction to the probability of being schizophrenic is plotted as solid and dashed curves respectively for two different values of p(schizo).

Figure 3

The ROCs for s-rep based classifcation methods with and without PNS based Euclideanization. The classification method of s-reps without Euclideanization of spherical GOPs in s-reps yields AUC of 0.5617. Our proposed method that uses s-reps as the object representation and uses DWD as the classification method with Euclideanization of s-rep's spherical GOPs via PNS yields the AUC of 0.6550.

Figure 4

The ROCs for aforementioned classifcation methods with and without PNS based Euclideanization. Our proposed method that uses s-reps as the object representation and uses DWD as the classification method with Euclideanization of s-rep's spherical GOPs via PNS yields the AUC of 0.6550.

Figure 5

Selected frames from the sequence of the s-reps while walking along the separation direction through the pooled backwards mean from the schizophrenic class to the control class. Viewing the sequence as a looping movie makes the local shape changes between the two classes more noticeable.