Depth from Defocus as a Special Case of the Transport of Intensity Equation

The Transport of Intensity Equation (TIE) in microscopy and the Depth from Differential Defocus (DfDD) method in photography both describe the effect of a small change in defocus on image intensity. They are based on different assumptions and may appear to contradict each other. Using the Wigner Distribution Function, we show that DfDD can be interpreted as a special case of the TIE, well-suited to applications where the generalized phase measurements recovered by the TIE are connected to depth rather than phase, such as photography and fluorescence microscopy. The level of spatial coherence is identified as the driving factor in the trade-off between the usefulness of each technique. Specifically, the generalized phase corresponds to the sample's phase under high-coherence illumination and reveals scene depth in low-coherence settings. When coherence varies spatially, as in multi-modal phase and fluorescence microscopy, we show that complementary information is available in different regions of the image.