Detection, classification and estimation of individual shapes in 2D and 3D point clouds

Robust Functional Manifold Clustering

In machine learning, it is common to interpret each data sample as a multivariate vector disregarding the correlations among covariates. However, the data may actually be functional, i.e., each data point is a function of some variable, such as time, and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance due to the correlations. In this article, we focus on subspace clustering for functional data or curves and propose a new method robust to shift and rotation. The idea is to define a function or curve and all its versions generated by shift and rotation as an equivalent class and then to find the subspace structure among all equivalent classes as the surrogate for all curves. Experimental evaluation on synthetic and real data reveals that this method massively outperforms prior clustering methods in both speed and accuracy when clustering functional data.

Statistical regression analysis of functional and shape data

We develop a multivariate regression model when responses or predictors are on nonlinear manifolds, rather than on Euclidean spaces. The nonlinear constraint makes the problem challenging and needs to be studied carefully. By performing principal component analysis (PCA) on tangent space of manifold, we use principal directions instead in the model. Then, the ordinary regression tools can be utilized. We apply the framework to both shape data (ozone hole contours) and functional data (spectrums of absorbance of meat in Tecator dataset). Specifically, we adopt the square-root velocity function representation and parametrization-invariant metric. Experimental results have shown that we can not only perform powerful regression analysis on the non-Euclidean data but also achieve high prediction accuracy by the constructed model.