# Compressive Holography

*The content linked on this page refer to the journal paper recently published by David J. Brady et al.,
"Compressive Holography", Opt. Express 17, 13040-13049 (2009).
*

Gabor
holography is an amazingly simple and effective encoder for compressive sampling.
Decompressive inference improves holographic systems by increasing
the number of pixels or voxels one can infer from a single hologram and by
resolving reconstruction ambiguities. Holography is a comparatively effective
encoder for compressive imaging because holographic multiplex measurement
weights are complex valued.

We illuminated two seed parachutes of common
dandelions (taraxacum) with a collimated, spatially filtered Helium-Neon
laser of 632.8 nm wavelength. One object is placed 1.5 cm away from the
detector array, and the other dandelion is placed 5.5 cm away from the detector
array.

The illumination and scattered fields were captured in the Gabor hologram
shown in Figure (a).
Figures (b) and (c) are photographs of the two seed parachutes.
Figure (d) is the 3D datacube estimated from the Gabor
recording by the TV-minimization algorithm. As the reconstruction shows,
the stem and the petals, representing the high-frequency features in the image,
are reconstructed well. In addition, the distance between the detector
plane and the first parachute and the distance between the two parachutes
are also accurately estimated. The reconstruction error in the plane of z = 0
is explained in the online supporting material.
Figure (d) is a 512*512*9 datacube of voxels with 5.2 micron transverse
resolution and 0.8 cm axial resolution reconstructed from a single 512*512
hologram. This demonstrates the main advantages of compressive holography,
i.e. that holograms naturally encode high quality multiplex data and
that decompressive inference can infer multidimensional objects from lower
dimensional data. Extensions of compressive holography may use off-axis
encoding to filter nonlinear terms and multispectral illumination to increase
the band volume and improve axial resolution.

Figure is the 3D datacube estimated from the Gabor recording by l1 minimization
algorithm. The optimization problem is defined in the main paper,
and we again adapt the two-step iterative shrinkage/thresholding algorithm
(TwIST) to solve the l1 minimization problem. While the two object
planes show reasonably good reconstructions, the errors in the other planes
are larger in the l1-minimization estimate.

Figure (a)
shows the reconstruction from a simulated squared field e, a rectangle with
no diffraction patterns. As expected, all the signal e remains in the plane of
z = 0 since there is no information pertaining to which plane it diffracts
from, meaning that the correlation of e with any object plane is small. In
figure (b) , we simulated a 3D object that has a rectangle in some object planes
from which the simulated squared field e is generated; the reconstruction is
produced from only e without the field E. As the reconstruction illustrates,
the reconstruction contains a small error in some object planes induced by
little correlation between e and the interference patterns that the estimated
object would produce.

*[The operation code below the result figure is downloadable by clicking the blue-labeled caption.]*
## Computer Specification

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.