Compressive Sensing Approach for Reducing the Number of
Exposures in Multiple View Projection Holography
Yair Rivenson
1
, Adrian Stern
2
and Joseph Rosen
3
1,3
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.
2
Electro-Optics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel.
1
rivenson@ee.bgu.ac.il,
2
stern@bgu.ac.il,
3
rosen@ee.bgu.ac.il
Abstract: Compressive imaging enables the reconstruction of images from far fewer number
of measurements predicted by classical sampling theorem. Here we demonstrate how this
approach can dramatically reduce the number of exposures in multi-view projection holography.
© 2010 Optical Society of America
OCIS Code: (090.1995) Digital Holography; (070.2025) Discrete optical signal processing.
1. Introduction
Multiple view projection holography [1] is a method for obtaining a hologram of a 3-D scene with a simple digital
camera, under regular illumination conditions. The main advantages of such a process is the ability to record a
hologram using white light, thus cancelling unwanted effects that are inherent with recording and reconstruction of
holograms, using coherent (laser) illumination. The process of acquiring an MVP Fourier hologram may be
described as follows: At the first stage, many perspectives of the scene are collected by a CCD camera. According
to [1],[2] we can characterize each projection by a pair of angels ( , )
m n
. Thus, the mn-th projection is denoted by
( , )
mn p p
p x y , where x
p
,y
p
are the coordinates in the projection domain. The next step is that each projection is
multiplied by a complex phase function,
exp 2 sin sin
mn p m p n
f j bx y , where b is some real constant. If we
integrate (perform summation) on the multiplication result, that is, we calculate ( , ) ( , )
mn p p mn p p
hmn p x y f dx dy
obtaining a complex value for every ( , )
m n
. Ordering these values we get a complex matrix h(m,n). It can be
shown [2] that Fourier transforming this matrix gives us a reconstruction of the scene. Hence, we get a digital
Fourier hologram.
If we perform a Fourier transform on h(m,n) we get a reconstruction which only corresponds to the z=0 plane of
the scene. Details from the other planes are out of focus. In order to reconstruct other planes, we multiply the
hologram with a quadratic phase function, which corresponds to the z
i
plane. Formally:
2 2 2 2
0
( , ) exp exp
i
z i x y z
i
j
u hmn j z f f u x y
z
F , (1)
where f
x
,f
y
denotes spatial frequencies, λ denotes the wavelength and F denotes the Fourier transform.
2. Compressive sensing
The recently introduced theory of compressive sensing (CS) [3],[4] gave birth to a new design methodologies for
existing optical applications [5]. CS suggests a new framework for simultaneous sampling and compression of
signals. In a nutshell, CS seeks to capture only the essential signal components, assuming that it could be sparsely
represented in some arbitrary basis. Thus, CS seeks to minimize the collection of redundant data in the acquisition
step. Practically, CS suggests that one can reconstruct a signal with only M=O(KlogN) projections, where K is the
number of non-zero elements in the signal under an arbitrary sparsifying basis (e.g. wavelet, total-variation, DCT),
and N is the number of object pixels. In CS the sensing basis should hold low coherence with the sparsifying basis
[4], the lower the coherence, the smaller the number of acquired projections needed for reconstruction, M. For
example, the Fourier sensing basis holds low coherence with Haar wavelet and total variation (TV) [6]. Extensive
work has been done on sampling using Fourier basis for compressive sensing, which has demonstrated its
effectiveness, especially in MRI applications. The sampling scheme we have used gives a higher weight on the low
frequency, and it was designed using partial Fourier sampling reconstruction results form [6].
a276_1.pdf
FThM2.pdf
© 2010 OSA /FiO/LS 2010
FThM2.pdf