Observing shape from defocused images

Accommodation cues are measurable properties of an image that are associated with a change in the geometry of the imaging device. To what extent can three-dimensional shape be reconstructed using accommodation cues alone? This question is fundamental to the problem of reconstructing shape from focus (SFF) and shape from defocus (SFD) for applications in inspection, microscopy, image restoration and visualization. We address it by studying the "observability" of accommodation cues in an analytical framework that reveals under what conditions shape can be reconstructed from defocused images. We do so in three steps: (1) we characterize the observability of any surface in the presence of a controlled radiance ("weak observability"), (2) we conjecture the existence of a radiance that allows distinguishing any two surfaces ("sufficient excitation") and (3) we show that in the absence of any prior knowledge on the radiance, two surfaces can be distinguished up to the degree of resolution determined by the complexity of the radiance ("strong observability"). We formulate the problem of reconstructing the shape and radiance of a scene as the minimization of the information divergence between blurred images, and propose an algorithm that is provably convergent and guarantees that the solution is admissible, in the sense of corresponding to a positive radiance and imaging kernel.