Error Bounds in Depth from Defocus

Depth from defocus (DFD) involves estimating the relative blur between a pair of defocused images of a scene captured with di erent lens settings. When a priori information about the scene is available, it is possible to estimate the depth even from a single image. However, experimental studies indicate that the depth estimate improves with multiple observations. In this paper, we provide a mathematical underpinning to this evidence by deriving and comparing the theoretical bounds for the error in the estimate of blur corresponding to the case of a single image, and a pair of defocused images, respectively. A new theorem is proposed which proves that the Cram er-Rao bound on the variance of the error in the estimate of blur, decreases with an increase in the number of observations. The di erence in the bounds turns out to be a function of the relative blurring between the observations. Results on synthetic as well as real data are given to validate the claim.