*The content linked on this page refer to the journal paper, "**Compressive Holography of Diffuse Object**", Appl. Opt. 49, H1-H10 (2010).*

The field scattered from a diffuse object can be modeled as complex circular Gaussian. This allows for building a statistical model between the incoherent scattering density of a diffuse object and the scattered field detected on the CCD, by integrating the statistical model into the propagation system. We show that the mapping often forms an illconditioned linear system and, thus, effectively results in measurement incompleteness when small singular values are ignored to produce numerically stable solutions. We resolve such measurement incompleteness by a constrained optimization technique enforcing sparsity constraints, which is inspired by compressive sensing theory. This sparsity constraint is incorporated by enforcing minimum total variation (TV) of the estimate of the image.

Figures show photographs of the experimental setup and a 3D diffuse object DISP, respectively. The CCD has 1624*1224 resolution with 4.4 micron pixel pitch with 16 bit quantization. The beams are created with an He-Ne laser with wavelength 632.8 nm. The spatial filter is applied in the Fourier domain via 4f optics. The spatial filter is inserted between the two Fourier transform lenses to ensure complete separation of the object Fourier spectra, the autocorrelation Fourier spectra, and the twin-image Fourier spectra. Note that the beam in the object arm illuminates the object through a stationary diffuser. The purposes of the diffuser are to fully develop the speckles in each speckle field and also to minimize the correlations between the speckle fields. We induce a random phase by introducing the effect of a stationary diffuser into each speckle field. To change the random phase patterns in each speckle field, a goniometer is introduced to illuminate the diffuser from different angles for each speckle field measurement.

Figure (a) shows a backpropagation reconstruction with a single speckle field. Figure (b) shows the average of the backpropagation reconstruction intensities of 50 speckle fields. As expected, the reconstruction is smoother and has a better contrast when multiple intensities are averaged compared to the backpropagation reconstruction intensity created from a single speckle field. Nonetheless, speckle artifacts still manifest as rough surface features. Moreover, the intensity average reconstruction shows poor axial resolution, as is clear from blurred features in the axial direction. For example, although the letter S becomes in focus and presents clear features on the third plane, the estimated intensity of the S is strongly blurred into the second and fourth planes. For reference, Fig. (c) shows a speckle reduction result obtained by applying a 5*5 median filter to the estimate in Fig. (b).

Figure (a) shows an estimate obtained by solving the constrained optimization problem when d is constructed with 50 speckle fields. Note that speckle artifacts have been significantly reduced, as indicated by the smooth surfaces of the letters D, I, S, and P. In addition, the estimate shows improved axial resolution, by suppressing the letters not in focus. Figure (b) shows a Tikhonov pseudoinverse estimate from 50 speckle fields. In this reconstruction, the speckle artifacts are still manifest as the rough surfaces of the letters in focus. Figure 4 shows an estimate obtained by solving the constrained optimization problem from the same d. The estimate shows remarkably suppressed speckle artifacts and improved axial resolution. For comparison, Fig. (c) demonstrates an estimate obtained by solving the constrained optimization problem with a single speckle field. The letters S and P in the estimate created with 50 speckle fields are clearer compared to the single estimate created with a single speckle field.

Figure shows another experiment. In the experiment, we placed the letters LADYBUG directly behind a real ladybug in the axial direction. Figure (b) shows the average intensity of the backpropagation reconstructions created with 20 object speckle fields. The wings of the real ladybug show speckle artifacts, and the letters LADYBUG are disguised in the blurs of the real ladybug. Figure (c) shows an estimate obtained by solving the constrained optimization problem. The surfaces of the wings become smoother. In particular, image contrast has been improved, as illustrated by the black spots on the wings. Also, the blurs placed on the letters have been reduced, while the letters become smoother.

The reconstruction was performed on a digital computer with Intel Core2 Quad CPU Q9300 at 2.5 GHz and 8 GB of RAM. The codes were written in Matlab 7.7.