Experimental verification of compressive reconstruction of correlation functions in
Lei Tian,1,∗Justin Lee,1Se Baek Oh,1and George Barbastathis1,2
1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2Singapore-MIT Alliance for Research and Technology (SMART) Centre, Singapore 117543, Singapore∗
(Dated: November 8, 2011)
Phase space tomography estimates correlation functions entirely from multiple snapshots in the
evolution of the system, rather than traditional interferometric methods requiring measurement
of multiple two–point correlations. However, as in every tomographic formulation, undersampling
poses a severe limitation. In the context of quantum correlation function estimation, a new theory
utilizing compressed sensing was recently established [D. Gross et al. Phys. Rev. Lett. 105, 150401
(2010)] whereby both measurement and post–processing dimensionality are reduced without giving
up reconstruction fidelity. Here we present the first, to our knowledge, experimental demonstration
of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity
function which is of course the analogue to a conservative quantum state. Our compressive algo-
rithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.)
Since the source was directly accessible in our classical experiment, we were able to compare the
compressive estimate of the mutual intensity to an independent ground–truth estimate from the van
Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.
Correlation functions provide complete characteriza-
tion of wave fields in several branches of physics, e.g.
the mutual intensity of stationary quasi-monochromatic
partially coherent classical light , and the density ma-
trix of conservative quantum systems (i.e., with time–
independent Hamiltonian) . The classical mutual in-
tensity expresses the joint statistics between two points
on a wavefront, and it is traditionally measured us-
ing interferometry: two sheared versions of a field are
overlapped in a Young, Mach–Zehnder or rotational
shear [3, 4] arrangement and two–point ensemble statis-
tics are estimated as time averages by a slow detector
under the assumption of ergodicity [1, 5].
As an alternative to interferometry, phase space to-
mography (PST) is an elegant method to measure corre-
lation functions. In classical optics, PST involves mea-
suring the intensity under spatial propagation [6–8] or
time evolution .In quantum mechanics, analogous
techniques apply, commonly known as quantum state to-
mography (QST) [10–13]. However, a fundamental dif-
ficulty in performing tomography in both cases is the
large dimensionality of the unknown state. For example,
to recover a n × n correlation matrix, a standard imple-
mentation would require at least n2data points.
A theory based on low-rank assumption for density ma-
trix recovery in QST was recently developed . It was
shown that a nearly pure quantum state of dimension n
and rank r requires only O(rnlogn)  data points.
This treatment is directly applicable to classical PST
thanks to the analogy between correlation functions; in
this Letter, we present the first, to our knowledge, exper-
imental measurement and verification of the correlation
function of a classical partially coherent field using low-
rank matrix recovery (LRMR).
Compressed sensing [16–18] exploits sparsity priors to
recover missing data with high confidence from a few
linear measurements. Here, sparsity means that the un-
known vector only contains a small number of nonzero
entries in some specified basis. LRMR [19, 20] is a gener-
alization of compressed sensing from vectors to matrices:
one attempts from very few linear measurements to still
reconstruct a high-fidelity and low-rank description of the
The low-rank assumption for classical partially coher-
ent light anticipates a source composed of a small num-
ber of mutually incoherent effective sources, i.e.
herent modes, to describe measurements.
sentially equivalent to the low entropy assumption ,
i.e. a nearly pure quantum state in the quantum ana-
logue. This assumption is valid for lasers, synchrotron
and table-top X-ray sources , and K¨ ohler illumination
in optical microscopes . An additional requirement for
LRMR to succeed is that measurements are “incoherent”
with respect to the eigenvectors of the matrix, i.e. the
measured energy is approximately evenly spread out be-
tween modes [15, 22]. Diffraction certainly mixes the co-
herent modes of the source rapidly, so we expect LRMR
to perform well for classical PST. The same expectation
for QST has already been established .
This is es-
The two–point correlation function of a classical sta-
tionary quasi–monochromatic partially spatially coher-
ent field is the mutual intensity function 
J(x1,x2) = ?g∗(x1)g(x2)?, (1)
where ?·? denotes ensemble average.
The measurable quantity of the classical field, i.e. the
arXiv:1109.1322v2 [physics.optics] 4 Nov 2011
intensity, after propagation by distance z is ,
which can be expressed in operator form as
I = tr(PxoJ), (3)
where P denotes the operator that combines both the
quadratic phase and Fourier transform operations in Eq.
2, tr(·) computes the trace, and xodenotes the lateral co-
ordinate at the observation plane. (For the coherent case,
see [22, 23].) By changing variables x = (x1+ x2)/2,
x?= x1−x2and Fourier transforming the mutual inten-
sity with respect to x we obtain the Ambiguity Function
(AF) [24, 25]
Eq. 2 can be written as [6–8, 24],
˜I(u?;z) = A(u?,λzu?),(5)
where˜I is the Fourier transform of the vector of mea-
sured intensities with respect to xo. Thus, radial slices
of the AF may be obtained from Fourier transforming the
vectors of intensities measured at corresponding propaga-
tion distances, and from the AF the mutual intensity can
be recovered by an additional inverse Fourier transform,
subject to sufficient sampling.
To formulate a linear model for compressive PST, the
measured intensity data is first arranged in Ambiguity
space. The mutual intensity is defined as the “sparse”
unknown to solve for. To relate the unknowns (mutual in-
tensity) to the measurements (AF), the center–difference
coordinate–transform is first applied, expressed as a lin-
ear transformation T upon the mutual intensity J, fol-
lowed by Fourier transform F, and adding measurement
noise e as
A = F · T · J + e.(6)
The mutual intensity propagation operator is unitary
and Hermitian, since it preserves energy. We use eigen-
value decomposition to determine the basis where the
measurement is sparse. The resulting basis, i.e. the set
of eigenvectors, is also known as coherent modes in classi-
cal optical coherence theory, whereas the whole process is
known as coherent mode decomposition . The goal of
the LRMR method is to minimize the number of coherent
modes to describe measurements. By doing LRMR, we
impose two physically meaningful priors: (1) existence of
the coherent modes , and (2) sparse representation of
the partially coherent field in terms of coherent modes.
FIG. 1: Experimental arrangement
19 26 34 41 50 58 67 77 87 99 111 125 141 160 181 207 239 280 334 409
FIG. 2: Intensity measurements at several unequally
spaced propagation distances.
Mathematically, if we define all the eigenvalues λiand
the estimated mutual intensity asˆJ, the method can be
subject to A = F · T ·ˆJ,
λi= 1. (7)
Direct rank minimization is NP–hard; however, it can be
accomplished by solving instead a proxy problem: con-
vex minimization of the “nuclear norm” (?1norm) of the
matrix J [19, 27]. In our implementation, we used the
singular value thresholding (SVT) method . Numer-
ical simulations for Gaussian-Schell model sources with
varying degree of coherence showed that SVT yields re-
constructions with mean-square-error less than 3% given
the same number of measurements as in our experiment
in the noiseless case. The reconstruction degrades grace-
fully as the signal-to-noise ratio decreases (see Supple-
The experimental arrangement is illustrated in Fig-
ure 1. The illumination is generated by an LED with
620nm central wavelength and 20nm bandwidth. To gen-
erate partially coherent illumination, a single slit of width
355.6µm (0.014??) is placed immediately after the LED
and one focal length (75 mm) to the left of a cylindrical
lens. One focal length to the right of the lens, we place
FIG. 3: Real part of the reconstructed mutual intensity
from (a) FBP; (b) SVT. For imaginary parts, see
FIG. 4: SVT Estimated eigenvalues.
the second single slit of width 457.2µm (0.018??), which
is used as a one–dimensional (1D) object.
The goal is to retrieve the mutual intensity immedi-
ately to the right of the object from a sequence of in-
tensity measurements at varying z-distances downstream
from the object, as described in the theory. We measured
the intensities at 20 z-distances, ranging from 18.2mm to
467.2mm, to the right of the object. The data are given in
Figure 2. Each 1D intensity measurement consists of 512
samples, captured by a CMOS sensor with 12µm pixel
size. The dimension of the unknown mutual intensity
matrix to be recovered is 512 × 512. Since only intensi-
ties at positive z, i.e. downstream from the object, are
accessible, we can only fill up the top right and bottom
left quadrants of Ambiguity space. The other two quad-
rants are filled symmetrically, i.e. assuming that if the
field propagating to the right of the object were phase
conjugated with respect to the axial variable z, it would
yield the correct field to the left of the object, i.e. nega-
tive z [8, 13]. Under this assumption, a total of 40 radial
slices are sampled in Ambiguity space. This is only 7.8%
of the total number of entries in the unknown mutual
The reconstructions from the filtered–backprojection
(FBP) and SVT methods are compared in Figure 3. The
FBP reconstruction suffers from three types of artifacts.
First, it yields an unphysical result with lower values
along the diagonal of the matrix. However, a correla-
tion function should always have maximum value at zero
separation. Second, the estimated degree of coherence is
lower than the theoretical prediction. The third is the
high frequency noise due to undersampling between the
radial slices. All these artifacts are greatly suppressed or
completely removed in the SVT reconstruction. In the
real part of the reconstruction, the width of the square
at the center is approximately 456µm (38 pixels), which
agrees with the actual width of the slit. The imaginary
part is orders of magnitude smaller than the real part (see
Supplemental Material). The SVT estimated eigenvalues
are shown in Figure 4.
We further validated our compressive estimates by
measuring the field intensity immediately to the right
of the illumination slit (Figure 5a). Assuming that the
illumination is spatially incoherent (a good assumption
in the LED case), the mutual intensity of the field imme-
diately to the left of the object is the Fourier transform
of the measured intensity, according to the Van Cittert–
Zernike theorem . This calculated mutual intensity,
based on the measurement of Figure 5(a) and screened by
the object slit, is shown in Figure 5(b). The eigenvalues
computed by coherent mode decomposition are shown in
Figure 5(c) and are in good agreement with the SVT es-
timates, as compared in Figure 5(d). It is seen that 99%
of the energy is contained in the first 13 modes, which
confirms our low-rank assumption. The compressive re-
construction may also be compared quantitatively to the
FBP reconstruction in terms of the global degree of co-
imentally found as 0.46 and 0.12, respectively; whereas
the estimate yielded by the Van Cittert–Zernike theo-
rem is 0.49. The eigenvectors of each individual mode
are shown and compared in the Supplemental Material.
Small errors in the compressive estimate are because the
missing cone is still not perfectly compensated by the
compressive approach, and other experimental imperfec-
In this classical experiment, we have the benefit that
direct observation of the one–dimensional object is avail-
able; thus, we were able to carry out quantitative analysis
of the accuracy of the compressive estimate. In the quan-
tum analogue of measuring a complete quantum state,
direct observation would have of course not been possi-
ble, but the accuracy attained through the compressive
estimate should be comparable, provided the low entropy
assumption holds .
Here we followed a much simplified version of the ap-
proach described in  which showed that the com-
plex operators describing the measurements should be
herence parameters ¯ µ =
i|λi|, which were exper-
?1500 ?1000?5000500 10001500
05 10 1520 25 30
0510 1520 2530
FIG. 5: (a) Intensity measured immediately to the right
of the illumination slit; (b) the real part of the Van
Cittert–Zernike theorem estimated mutual intensity
immediately to the right of the object slit (for imaginary
part, see Supplemental Material); (c) eigenvalues of the
mutual intensity in (b); (d) the absolute error between
the eigenvalues in Figure 4 and 5(c) versus mode index.
uniformly distributed in the n–dimensional unit sphere,
whereas we simply utilized free space propagation, the
classical analogue of rotated quadrature phase opera-
tors [10, 13]. The phase masks described in  to im-
plement optimal sampling are outside the scope of the
The authors thank Baile Zhang, Jonathan Petruccelli,
and Yi Liu for helpful discussions and one anonymous
reviewer for constructive criticism.
was provided by Singapore’s National Research Founda-
tion through the Centre for Environmental Sensing and
Modeling (CENSAM) and BioSystems and bioMechanics
(BioSyM) independent research groups of the Singapore-
MIT Alliance for Research and Technology (SMART),
the US National Institutes of Health and by the Chevron–
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