Root-mean square error in passive autofocusing and 3D shape recovery

Image focus analysis is an important technique for passive autofocusing and 3D shape measurement. Electronic noise in digital images introduces errors in this techniques. It is therefore important to derive robust focus measures that minimize error. In our earlier research, we have developed a method for noise sensitivity analysis of focus measures. In this paper we derive explicit expressions for the root-mean square (RMS) error in autofocusing based on image focus analysis. This is motivated by the autofocusing uncertainty measure (AUM) defined earlier by us as a metric for comparing the noise sensitivity of different focus measures in autofocusing and 3D shape-from-focus. The RMS error we derive by us has the same advantage as AUM in that it can be computed in only one trial of autofocusing. We validate our theory on RMS error and AUM through experiments. It is shown that the theoretically estimated and experimentally measured values of the standard deviation of a set of focus measures are in agreement. Our results are based on a theoretical noise sensitivity analysis of focus measures, and they show that for a given camera the optimally accurate focus measure may change from one object to the other depending on their focused images.