Steerable $e$PCA: Rotationally Invariant Exponential Family PCA

In photon-limited imaging, the pixel intensities are affected by photon count noise. Many applications, such as 3-D reconstruction using correlation analysis in X-ray free electron laser (XFEL) single molecule imaging, require an accurate estimation of the covariance of the underlying 2-D clean images. Accurate estimation of the covariance from low-photon count images must take into account that pixel intensities are Poisson distributed, rendering the sub-optimality of the classical sample covariance estimator. Moreover, in single molecule imaging, including in-plane rotated copies of all images could further improve the accuracy of covariance estimation. In this paper we introduce an efficient and accurate algorithm for covariance matrix estimation of count noise 2-D images, including their uniform planar rotations and possibly reflection. Our procedure, steerable $e$PCA, combines in a novel way two recently introduced innovations. The first is a methodology for principal component analysis (PCA) for Poisson distributions, and more generally, exponential family distributions, called $e$PCA. The second is steerable PCA, a fast and accurate procedure for including all planar rotations for PCA. The resulting principal components are invariant to the rotation and reflection of the input images. We demonstrate the efficiency and accuracy of steerable $e$PCA in numerical experiments involving simulated XFEL datasets.