A theory of defocus via Fourier analysis

In this paper we present a novel theory to analyze defocused images of a volume density by exploiting well-known results in Fourier analysis and the singular value decomposition. This analysis is fundamental in two respects: First, it gives a deep insight into the basic mechanisms of image formation of defocused images, and second, it shows how to incorporate additional a-priori knowledge about the geometry and photometry of the scene in restoration algorithms. For instance, we show that the case of a scene made of a single surface results in a simple constraint in the Fourier domain. We derive two basic types of algorithms for volumetric reconstruction: One based on a dense set of defocused images, and one based on a sparse set of defocused images. While the first one excels in simplicity, the second one is of more practical use. Both algorithms are tested on real and synthetic data.