Generalized PCA via the Backward Stepwise Approach in Image Analysis

Principal component analysis (PCA) for various types of image data is analyzed in terms of the forward and backward stepwise viewpoints. In the traditional forward view, PCA and approximating subspaces are constructed from lower dimension to higher dimension. The backward approach builds PCA in the reverse order from higher dimension to lower dimension. We see that for manifold data the backward view gives much more natural and accessible generalizations of PCA. As a backward stepwise approach, composite Principal Nested Spheres, which generalizes PCA, is proposed. In an example describing the motion of the lung based on CT images, we show that composite Principal Nested Spheres captures landmark data more succinctly than forward PCA methods.