Shape from defocus is the problem of reconstructing the three-dimensional (3D) geometry of a scene from a collection of blurred images captured by a finite aperture lens. The geometry of the scene is described by a depth map, that is the graph of a continuous function with the image plane as domain and the positive reals as co-domain.; This problem is ill-posed since the solution is not unique given the data (the collection of defocused images). This motivates us to first study the conditions under which shape reconstruction is possible and to what degree.; Then, we propose a number of algorithms to optimally reconstruct 3D shape from blurred images. We do so by exploiting different aspects of the structure of the problem. The solutions we propose can be divided into four groups. In the first we exploit the linearity of the problem with respect to some unknowns to arrive at a very general solution which only entails a minimization with respect to shape. In the second, we exploit the nonnegativity of the unknowns to derive a provably convergent minimization algorithm. In the third group, we cast the problem in the context of partial differential equations. We formulate the problem of inferring shape from blurred images as that of inferring the diffusion coefficient of a parabolic differential equation. These solutions are based on the common assumption that the scene is made of a single surface (a depth map). This assumption is often violated in real images, especially when we are in the presence of occlusions between different objects in the scene. We address this issue in the fourth group.
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