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Statistics of Fresnelet coefficients in PSI holograms

Advances in computer technology are moving real-time capable, digital holography into the realm of near future feasibility. The small pixel size required in the recording of even small objects and the large detector area (high numerical aperture in a lenseless recording setup) required for high resolution reconstruction results in large amounts of data, especially considering real-time video applications. The special requirements posed by digital holographic microscopy using lasers operating in the UV range are another application generating large quantities of data that suggest the use of compression for transmission and storage. Holograms differ significantly from natural images, as both the intensity and the phase of the incoming wavefront are recorded. The information about the recorded object is non-localized in the detector plane and in many applications the phase is far more important than the intensity as it provides information about different optical path length (e.g. distance and thus shape in metrology, presence of transparent structures in microscopy). This paper examines the statistical properties of PSI holograms. The holograms are transformed using Fres- nelets, a wavelet analysis of the reconstructed wavefront in the object plane. Since the wavefront is complex valued, the complex amplitude has been separated into real-valued phase and amplitude before wavelet trans- formation. The results show that while the phase can be statistically modeled using a Generalized Gaussian Distribution (GGD) with exponent α ≈ 1.5, the statistics of the amplitude seem to be the result of two separable components, each corresponding to GGD. These are identified as the speckle field caused by sub-wavelength surface roughness with α ≈ 2 and the actual object with α ≈ 1. These result suggest the separate application of classical image compression based on GGD statistics in the subbands to the phase, the speckle amplitude and the object amplitude.