Aggregated Pairwise Classification of Statistical Shapes
This work addresses shape classification problems in fields like medical imaging and computer vision, but it is incremental as it builds on existing statistical frameworks with a novel approach tailored to shape data properties.
The paper tackles the classification of statistical shapes, which are infinite-dimensional and non-Euclidean, by developing a method that uses square-root velocity functions, tangent spaces, and principal components to reduce dimensionality and combine pairwise classifiers, resulting in an illustration of how projection points and subspace choices affect misclassification rates.
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as a vector in a relatively low-dimensional Euclidean space. Statistical shape data have two main properties that suggest a need for a novel approach: (i) shapes are inherently infinite dimensional with strong dependence among the positions of nearby points, and (ii) shape space is not Euclidean, but is fundamentally curved. To accommodate these features of the data, we work with the square-root velocity function of the curves to provide a useful formal description of the shape, pass to tangent spaces of the manifold of shapes at different projection points which effectively separate shapes for pairwise classification in the training data, and use principal components within these tangent spaces to reduce dimensionality. We illustrate the impact of the projection point and choice of subspace on the misclassification rate with a novel method of combining pairwise classifiers.