A comparative study of dimensionality reduction methods for p-type based contour representations

Morphology is paramount important for knowledge discovery in different disciplines, including, biology, archeology, among others. Experts in these domains commonly use computational methods to study different shapes. Beside a precise shape representations, these methods may also require high levels of interpretability to link data and expert knowledge. Most of the methods proposed in the literature to study shape lack of this interpretability. Recently, a related approach has been proposed to study open leaf contours in tropical vegetation. This approach combines a complex Fourier transform (CFT) of the contour and PCA, in the so called P-type representation. The CFT results on a set of complex numbers, over which PCA should be applied. It is not clear how to obtain good a quality dimensionality reduction with high levels of interpretability, because of the complex nature of this representation. In this work, we compare different PCA complex based dimensionality reduction approaches for P-type representations. Our aim was to characterize the behaviour of the different approaches, both quantitatively but also in terms of their level of interpretability. For this study, we used 80 binary images in four different shape related categories (apples, human head, cars and bells). A P-type representation, with a total of 20 harmonics, was used to represent the objects contours. Different PCA dimensionality reductions methods for complex were computed, namely, real part + PCA, imaginary part + PCA, complex norm + PCA and full complex + PCA. To compare methods we computed objective clustering measurements and cumulated variance. When possible, the interpretability level was approached by examining the inverse of the firsts principal components. Our results suggest that studied methods may provide similar levels of clustering quality. Nevertheless, the full complex based PCA provides the highests levels of interpretability.