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).
We used a standard PSI holographic setup as seen in Fig 1. The targets were a 30mm miniature, an archer, and a similarly sized cartoon-like plastic cat. To increase the intensity of the light scattered off the object and homogenize the surface, photo stop spray was applied to the surface, resulting in a coating of fine, white dust. The recorded holograms were stored losslessly as images with further processing applied in MatLab.
Fig 2.shows the basic structure of the algorithm. The wavefront was reconstructed in focus in the object plane by applying a Fresnel-Transformation. The resulting complex wavefront was split into two 2D sets of real-valued data, one representing the amplitude, the other the phase of the wavefront. Each of these two sets of data was subjected to a real-valued, 2d wavelet decomposition to a depth of five cascades using the Haar-wavelet in the Mallat base. This approach is very close to Fresnelets, which combine Fresnel propagation of the wavefront with the wavelet analysis into one convolution. To determine the applicability of compression algorithms, the resulting subbands were analyzed statistically, predicting compression performance.
Complementing this theoretical approach, existing implementations of JPEG and JPEG2000 were applied, both to the originally recorded holograms, to the PSI reconstructed complex wavefronts and to the Fresnel-propagated wavefront in the object plane.
Current results of the statistical analysis have shown, that the statistics of the wavelet coefficients for the amplitude of the reconstructed wavefront in the object plane show a distinct two-component behavior (see Fig. 3). One component, with the coefficient distributed Gaussian, represents the speckle field, while the other, with an approximately Laplacian distribution, correspond to the macroscopic shape of the object. The wavelet coefficients for the phase are Gaussian distributed, although the distribution is very noisy. This noise is the result of numerical instabilities in calculating the phase for amplitudes close to zero. We have shown that the noise can be suppressed using a mask based on the amplitude of the wavefront. These results indicate, that standard compression algorithms can be applied successfully to Fresnel propagated and wavelet analyzed PSI holograms, especially to the amplitude coefficients which are Laplace distributed. The results also indicate, that a separation of the wavefront into a speckle field and a remainder representing the macroscopic shape would be advantageous.
The test performed using JPEG2000 and JPEG clearly suggest compression in the object plane using a wavelet-based, rate-distortion optimized algorithm like JPEG2000.
- 1. Marc Wilke, Alok K. Singh, Ahmad Faridian, Thomas Richter, Giancarlo Pedrini, Wolfgang Osten (2012). Statistics of Fresnelet-Coefficients in PSI Holograms. Proceedings of SPIE, Vol. 8499.