Quantitative analysis of error bounds in the recovery of depth from defocused images

Depth from defocus involves estimating the relative blur between a pair of defocused images of a scene captured with different 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. 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 for a pair of defocused images. A new theorem is proposed that proves that the Cramér-Rao bound on the variance of the error in the estimate of blur decreases with an increase in the number of observations. The difference in the bounds turns out to be a function of the relative blurring between the observations. Hence one can indeed get better estimates of depth from multiple defocused images compared with those using only a single image, provided that these images are differently blurred. Results on synthetic as well as real data are given to further validate the claim.