The Non-parametric Sub-pixel Local Point Spread Function Estimation Is a Well Posed Problem

Most medium to high quality digital cameras (dslrs) acquire images at a spatial rate which is several times below the ideal Nyquist rate. For this reason only aliased versions of the cameral point-spread function (psf) can be directly observed. Yet, it can be recovered, at a sub-pixel resolution, by a numerical method. Since the acquisition system is only locally stationary, this psf estimation must be local. This paper presents a theoretical study proving that the sub-pixel psf estimation problem is well-posed even with a single well chosen observation. Indeed, theoretical bounds show that a near-optimal accuracy can be achieved with a calibration pattern mimicking a Bernoulli(0.5) random noise. The physical realization of this psf estimation method is demonstrated in many comparative experiments. We use an algorithm to accurately estimate the pattern position and its illumination conditions. Once this accurate registration is obtained, the local psf can be directly computed by inverting a well conditioned linear system. The psf estimates reach stringent accuracy levels with a relative error of the order of 2% to 5%. To the best of our knowledge, such a regularization-free and model-free sub-pixel psf estimation scheme is the first of its kind.