Two-Dimensional Functional Principal Component Analysis for Image Feature Extraction

Methodologies for functional principal component analysis are well established in the one-dimensional setting. However, for two-dimensional surfaces, for example, images, conducting functional principal component analysis is complicated and challenging, because the conventional eigendecomposition approach would require the estimation of a four-dimensional covariance function, which may incur high cost in terms of time and machine memory. To circumvent such computational difficulties, we propose a novel two-dimensional functional principal component analysis for extracting functional principal components and achieving dimensionality reduction for images. Different from the conventional eigendecomposition approach, our proposed method is based on the direct estimation of the optimal two-dimensional functional principal components via tensor product B-spline, which opens up a new avenue for estimating functional principal components. We present theoretical results that prove the consistency of the proposed approach. Our method is illustrated by analyzing brain images of subjects with the Alzheimer’s Disease and the handwritten digits images. The finite sample performance of our method is further assessed with some simulation studies. Supplementary materials for this article are available online.