A geometric approach to blind deconvolution with application to shape from defocus
We propose a solution to the generic "bilinear calibration-estimation problem" when using a quadratic cost function and restricting to (locally) translation-invariant imaging models. We apply the solution to the problem of reconstructing the three-dimensional shape and radiance of a scene from a number of defocused images. Since the imaging process maps the continuum of three-dimensional space onto the discrete pixel grid, rather than discretizing the continuum we exploit the structure of maps between (finite-and infinite-dimensional) Hilbert spaces and arrive at a principled algorithm that does not involve any choice of basis or discretization. Rather, these are uniquely determined by the data, and exploited in a functional singular value decomposition in order to obtain a regularized solution.