Principal component analysis (PCA) is widely used in image analysis and pattern recognition for dimensionality reduction and denoising. In many applications, it is advantageous to include all possible rotations of the images into PCA. The resulting principal components are tensor products of radial functions and angular Fourier modes. Since rotating the principal components is easily achieved by a simple phase shift of their angular part, people use the term “steerable PCAâ€. The principal
components are invariant to any in-plane rotation of the images, therefore finding steerable principal components is equivalent to finding in-plane rotationally invariant principal components.
In this talk, I introduce an algorithm that efficiently and accurately performs steerable PCA on a large set of two-dimensional images. I will also talk about its application in cryo-electron microscopy image processing. The new algorithm computes the expansion coefficients of the images in a Fourier-Bessel basis efficiently using the non-uniform fast Fourier transform, reducing the computational complexity from O(nL^4) to O(nL^3) for n images of size L by L. A special truncation rule is applied to ensure that the transformation from the Cartesian grid to the truncated Fourier-Bessel expansion is nearly unitary.