We report on phase-shifted holographic interferogram demodulation by singular
value decomposition. Numerical processing of optically-acquired interferograms
over several modulation periods was performed in two steps : 1- rendering of
off-axis complex-valued holograms by Fresnel transformation of the
interferograms; 2- eigenvalue spectrum assessment of the lag-covariance matrix
of hologram pixels. Experimental results in low-light recording conditions were
compared with demodulation by Fourier analysis, in the presence of random phase
drifts.Comment: 4 pages, 3 figure

Atlan, Michael

Lopes, Fernando

English

arXiv

International audienceWe report on phase-shifted holographic interferogram demodulation by singular value decomposition. Numerical processing of optically-acquired interferograms over several modulation periods was performed in two steps : 1- rendering of off-axis complex-valued holograms by Fresnel transformation of the interferograms; 2- eigenvalue spectrum assessment of the lag-covariance matrix of hologram pixels. Experimental results in low-light recording conditions were compared with demodulation by Fourier analysis, in the presence of random phase drifts

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Distributed under a Creative Commons Attribution| 4.0 International License
Singular-value demodulation of phase-shifted holograms
Fernando Lopes, Michael Atlan
To cite this version:
Fernando Lopes, Michael Atlan. Singular-value demodulation of phase-shifted holograms. Optics
Letters, Optical Society of America, 2015, 40, pp.2541 - 2541. ￿10.1364/OL.40.002541￿. ￿hal-01386365￿
Singular value demodulation of phase-shifted holograms
Fernando Lopes1 and Michael Atlan2
1 Institut de Physique du Globe de Paris - Pale´omagne´tisme,
Universite´ Paris Diderot, Sorbonne Paris Cite´, 1 rue Jussieu, Paris, France
2 Centre National de la Recherche Scientifique (CNRS) UMR 7587,
Institut Langevin. Fondation Pierre-Gilles de Gennes,
Institut National de la Sante´ et de la Recherche Me´dicale (INSERM) U 979,
Universite´ Pierre et Marie Curie (UPMC), Universite´ Paris 7. E´cole Supe´rieure de
Physique et de Chimie Industrielles ESPCI ParisTech - 1 rue Jussieu. 75005 Paris. France
(Dated: October 23, 2016)
We report on phase-shifted holographic interferogram demodulation by singular value decomposi-
tion. Numerical processing of optically-acquired interferograms over several modulation periods was
performed in two steps : 1- rendering of off-axis complex-valued holograms by Fresnel transformation
of the interferograms; 2- eigenvalue spectrum assessment of the lag-covariance matrix of hologram
pixels. Experimental results in low-light recording conditions were compared with demodulation by
Fourier analysis, in the presence of random phase drifts.
OCIS codes: 090.1995, 040.2840, 120.2880, 040.3780
Recording optical holograms by time-modulated
interferometry was proposed and demonstrated in early
studies [1–3]. It pioneered the development of phase-
shifting [4] holographic interferometry. Time-modulated
phase-shifting interferometry involves recording of a
sequence of interferograms over at least one modulation
cycle of the intensity. Temporal demodulation consists
in a linear combination of a sequence of recorded
interferograms, whose coefficients are the ones of a
discrete Fourier transform [5–7]. An important feature
of holographic interferometry is to allow spatial signal
modulation in off-axis recording conditions [8], which
permits the discrimination of self-beating and cross-
beating interferometric contributions of the object field
and the reference field. In this configuration, spurious
interferometric contributions can be filtered-off [9]. It
was demonstrated that the combination of spatial and
temporal signal modulation enables low-light imaging
at shot noise levels [10, 11]. Important features and
applications include optical phase imaging [12–15] and
detection tunability in the radiofrequency range for
optical heterodyne detection [16], frequency-resolved
narrowband sensing of scattered optical radiation [17–
19]. Sensitivity is compatible with photon-counting
recording conditions [20–22]. In standard phase-shifting
interferometry, a sequence of interferograms with known
phase shifts between probed and reference optical
waves is recorded. When the modulation is not known,
principal component analysis [23] can be used for non-
parametric, non-iterative demodulation. It identifies
uncorrelated variables in the structure of the data (the
principal components) from the analysis of its correla-
tions. Non-parametric approaches were also applied to
image formation concepts [24], speckle reduction [25–29],
and laser speckle contrast analysis of flows [30].
In this letter, we demonstrate experimentally that
the sensitivity of low-light optical signal retrieval in
time-modulated holographic interferometry can be im-
proved with respect to Fourier transform demodulation
in practical conditions, when random signal fluctuations
are present. For that purpose, a data-driven, non-
parametric signal processing method is used. It consists
of calculating the spectrum of eigenvalues in a singular
value decomposition (SVD) of the lag-covariance matrix
of recorded temporal fluctuations of the hologram’s pix-
els, after spatial Fresnel transformation. The proposed
processing scheme is less affected by random optical
phase drifts occurring during the sampling process than
signal demodulation by discrete Fourier transformation.
In optical heterodyne detection, a field of interest
E(t) = E exp (iωLt), oscillating at the optical frequency
ωL, is non-linearly mixed with a local oscillator field
ELO(t) = ELO exp (i∆ωt) exp (iωLt) that is set at a close-
by intermediate frequency ωL + ∆ω. In this notation,
E and ELO are complex constants and i is the imaginary
unit. The amplitude and phase of the original signal E(t)
can be retrieved in the desired outcome oscillating at the
difference frequency ∆ω, which can be set within the re-
ceiver’s temporal bandwidth. The frequency conversion
process is ensured by non-linear detection of the optical
field E by the array of square-law sensors of a camera,
which respond quadratically with the impinging electric
field’s magnitude. The squared magnitude of the total
field received, I(t) = |E(t) + ELO(t)|
2, has cross-terms
oscillating at the difference frequency ∆ω of the fields E
and ELO, causing modulation of the optical energy flux
I(t) = |E|2+ |ELO|
2+EE∗LOe
−i∆ωt+E∗ELOe
i∆ωt, where ∗
denotes the complex conjugate. This equation describes
the temporal fluctuation of the recorded intensity at a
given pixel. Holographic image rendering, referred to
as spatial demodulation, is then performed onto each
recorded interferogram I by a discrete Fresnel transform.
This transformation involves the free-space propagation
impulse response of the optical field [7, 31]. In off-axis
recording configuration, the Fresnel transform separates
spatially the four interferometric terms of I(t) [8, 9] and
2forms a reconstructed hologram. In the absence of ran-
dom fluctuations, the pixels of the off-axis part of the
hologram carry the signal
X(t) = EE∗LO exp(−i∆ωt) (1)
The purpose of signal demodulation is to retrieve the
magnitude of the oscillating signal at the frequency
∆ω at each pixel. In ideal conditions, Eq. 1 holds and
signal demodulation by discrete Fourier analysis of the
recorded interferograms [6, 7] enables shot-noise limited
detection in dim light [10]; yet in actual experiments,
random amplitude and phase fluctuations limit the
detection sensitivity.
We recorded phase-shifted interferograms over several
modulation periods in low-light and studied signal
retrieval procedures. Optical recording of interferograms
was performed with a frequency-shifted, off-axis Mach-
Zehnder interferometer [32] used for optical heterodyne
detection of the object field E, backscattered from a res-
olution target, beating against a separate local oscillator
field ELO, on a sensor array. The main optical field was
provided by a 150 mW, single-mode laser (wavelength
λ = 532 nm, optical frequency ωL/(2pi) = 5.6× 10
14Hz,
Cobolt Samba-TFB-150). The optical frequency of the
local oscillator beam was shifted by a tunable quantity
∆ω by two acousto-optic modulators (AA-electronics,
MT200-BG9-FIO) driven by phase-locked continuous-
wave radiofrequency signals around 200 MHz. The
observed reflective resolution target was illuminated over
a ∼ 10mm-diameter disk. Interferograms were measured
with a Ximea MQ042MG-CM camera (2048×2048 pixels
CMOSIS CMV4000 sensor array, pixel size d = 5.5µm,
full well charge : 13500 e− (photo-electrons), conversion
gain : 1/G = 0, 075 counts/e−, quantum efficiency at
532 nm η ∼ 0.5), running at an externally-triggered
frame rate of ωS/(2pi) = 20Hz, at 8 bit/pixel quanti-
zation. In the reported experiment, a reflective USAF
target printed on white paper was illuminated in re-
flection, and the recorded intensity of the light in the
object channel accounted for 11.56 counts × G ≃ 154
photo-electrons per pixel per frame, on average. The
average intensity of the light in the LO channel ac-
counted for 63.89 counts × G ≃ 852 photo-electrons
per pixel per recorded frame. In these conditions, a
sequence of 128 consecutive raw interferograms was
recorded for a detuning frequency of the local oscillator
of ∆ω/(2pi) = 5Hz, creating 4-phase (ωS/∆ω = 4)
modulation conditions [32]. Twelve more sequences of
images were acquired after decreasing the optical power
in the object channel by 100.5 from one measurement to
the next, by combining optical densities (O.D., Thorlabs
NE absorptive neutral density filters). The transmitted
power to the target scaled as 10−O.D., from 100 to 10−6;
hence the average number of recorded photo-electrons
per pixel and per frame in the object channel ranged
from 154 to 1.54 × 10−4. The LO-to-object optical
power ratio (heterodyne gain) ranged from GH ≃ 5.5
FFT
demodulation
SVD
demodulation
O
.D
. =
 0
O
.D
. =
 1
O
.D
. =
 2
O
.D
. =
 3
O
.D
. =
 4
O
.D
. =
 5
O
.D
. =
 6
no signal
demodulation
FIG. 1: Amplitude holographic images from 4-phase modu-
lated interferograms acquired over 128 frames. No time de-
modulation (first column). Demodulation by Fourier trans-
form (second column). Demodulation by singular value de-
composition (third column).
to GH ≃ 5.5 × 10
6, because the LO power was kept
constant throughout the experiment. The holographic
images of these two datasets were then demodulated
spatially [31], to yield 661 × 661 complex-valued time
traces X(t) of 128 elements each, cropped from the
2048 × 2048 calculation array, over an off-axis spatial
region of interest where the image was expected. We
then compared two types of signal demodulation proce-
dures to assess the magnitude of the oscillating signal at
the frequency ∆ω in X(t) at each pixel : Fourier anal-
ysis, and singular value analysis of temporal fluctuations.
30 1 2 3 4 5 6
−30
−25
−20
−15
−10
−5
0
5
Optical density (O.D.)
o
p
ti
ca
l 
si
g
n
al
 (
d
B
)
RAW
FFT
SVD
10
log(10
à
O
:D
:)
FIG. 2: Signal magnitude versus optical density (O.D.) in the
object channel from a sequence of N = 128 frames of 4-phase
interferograms. No time demodulation (circles). Demodula-
tion by discrete Fourier transform (squares). Demodulation
by singular value decomposition (triangles).
First, holographic images were rendered without de-
modulation. The squared magnitude of the object field
was averaged over N = 128 frame sequences, and re-
ported in the first column of Fig. 1. The image is not
visible for optical power transmission factors lower or
equal to 10−3 (i.e. O.D. = 3), which corresponds to
19.7 photo-electrons per pixel recorded on average dur-
ing 128 frames. Then, the recorded optical fluctuations
were projected on the expected oscillation to retrieve the
modulated component to improve the detection sensitiv-
ity. The modulation depth of the signal was assessed by
a fast Fourier transform (FFT) X˜ of the sequence xk at
the modulation frequency ωS/4 (5 Hz), calculated on 128
time points. Results are reported in the second column of
Fig. 1. The image is not visible for a power transmission
factor lower or equal to 10−5 (i.e. O.D. = 5), which cor-
responds to 1.97×10−1 photo-electron per pixel recorded
on average during 128 frames. Finally, we performed sin-
gular value analysis of temporal fluctuations. In this ap-
proach, we embedded the time series X = {x1, ..., xN} of
the signal at each pixel in a Hankel matrix of dimension
K × (N − K + 1), where K is the number of principal
components to seek, and N = 128 is the total number
of acquired interferograms. The Hankel matrix D, com-
posed of N −K+1 lag-shifted copies of truncated values
of X(t) has the form
D =


x1 x2 . . . xN−K+1
x2 x3
. . .
...
...
...
. . . xN−1
xK xK+1 . . . xN


(2)
We proceeded by diagonalizing the lag-covariance
matrix of X(t), CX = DD
t∗, where Dt∗ is the adjoint
matrix (Hermitian conjugate) of D. The K eigenvalues
λk, k = 1, ...,K of CX , numbered by decreasing order
of magnitude, were calculated by SVD [33] of CX . The
length of the analysis window K was chosen to be an
integer number of expected periods of the modulated
signal [34] K = p ωS/∆ω. We used a window length
of p = 5 modulation periods (K = 20). Images were
rendered by calculating the amplitude of the second
eigenvalue at each pixel. They are displayed in Fig. 1
in the third column, as a function of the optical density
in the object channel. With this approach, the image is
visible down to an optical power transmission factor of
10−6 (i.e. O.D. = 6), for which the signal wave intensity
accounted for a total of 1.97 × 10−2 recorded photo-
electron per pixel during 128 frames, on average. The
algorithm used for the SVD is based on the orthogonal
triangular decomposition (QR factorization) [35]. SVD
and FFT were performed by the functions svd() and
fft() in Matlab 2012a, run on a CentOS 6 Linux 64-bit
operating system. The response curve of singular value
demodulation was assessed by plotting the quantity
10 log10(〈S〉 / 〈S0〉) (in dB) against the value of the
optical density in the object channel (Fig.2). The
braces 〈 〉 indicate spatial averaging of S, and S0 within
and outside the reconstructed image, respectively.
The expected ideal detection behavior is plotted as
a continuous black line (-10 dB of optical power per
unit of optical density). The demodulation by SVD
is qualitative [36]; yet it enables to retrieve images at
much lower signal levels than FFT demodulation. The
singular value demodulation procedure is illustrated by
the spectrograms of four signals, reported in Fig. 3 : (a)
the complex wave oscillating at ∆ω/(2pi) = 5Hz, which
would be expected in perfect modulation conditions, in
the absence of random phase fluctuation (Eq. 1); (b)
the actual signal X(t) from a pixel at O.D. = 2, which
exhibits a DC contribution and jitter - deviation from
the modulation frequency; and the signal reconstructed
from the first (c) and second (d) eigenvectors and
eigenvalues of the SVD of D. In this example, the DC
term is found in the first component, and the modulated
component at 5 Hz undergoing a phase drift is retrieved
in the second component of the SVD.
In conclusion, we used heuristically a singular value
decomposition procedure to retrieve time-modulated
optical signals in holographic interferometry. In the
reported experiment, over long acquisition times, singu-
lar value demodulation of phase-shifted interferograms
enabled qualitative image visualization at much lower
signal levels than demodulation by discrete Fourier
transformation. This method will be investigated for
signal retrieval in strong phase fluctuations conditions,
phase microscopy, and singular spectrum analysis of
pseudoperiodic modulations.
This work was supported by Agence Nationale de la
Recherche (ANR-09-JCJC-0113, ANR-11-EMMA-046),
Fondation Pierre-Gilles de Gennes (FPGG014), re´gion
4ti
m
e 
(s
)
0
1
2
3
ti
m
e 
(s
)
0
1
2
3
instantaneous beating frequency (Hz)
−10 −5 0 5 10
−10 −5 0 5 10
ti
m
e 
(s
)
0
1
2
3
−10 −5 0 5 10
original
signal
second
eigenvector
first
eigenvector
ti
m
e 
(s
)
0
1
2
3
−10 −5 0 5 10
5 Hz
single tone 
(a)
(b)
(c)
(d)
FIG. 3: Spectrograms of a complex wave oscillating at 5Hz
(a), a hologram’s pixel signal X(t) (b), reconstructed signals
from the first (c) and second (d) elements of the SVD of D.
Ile-de-France (C’Nano, AIMA), the Investments for the
Future program (LabExWIFI: ANR-10-LABX-24, ANR-
10-IDEX-0001-02 PSL*), and European Research Coun-
cil (ERC Synergy HELMHOLTZ).
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Singular-value demodulation of phase-shifted holograms

HAL-INSU

Singular-value demodulation of phase-shifted hologramsFernando Lopes, Michael AtlanTo cite this version:Fernando Lopes, Michael Atlan. Singular-value demodulation of phase-shifted holograms. Op-tics Letters, Optical Society of America, 2015, 40, pp.2541 - 2541. <10.1364/OL.40.002541>.<hal-01386365>HAL Id: hal-01386365https://hal.archives-ouvertes.fr/hal-01386365Submitted on 23 Oct 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Distributed under a Creative Commons Attribution 4.0 International LicenseSingular value demodulation of phase-shifted hologramsFernando Lopes1 and Michael Atlan21 Institut de Physique du Globe de Paris - Pale´omagne´tisme,Universite´ Paris Diderot, Sorbonne Paris Cite´, 1 rue Jussieu, Paris, France2 Centre National de la Recherche Scientifique (CNRS) UMR 7587,Institut Langevin. Fondation Pierre-Gilles de Gennes,Institut National de la Sante´ et de la Recherche Me´dicale (INSERM) U 979,Universite´ Pierre et Marie Curie (UPMC), Universite´ Paris 7. E´cole Supe´rieure dePhysique et de Chimie Industrielles ESPCI ParisTech - 1 rue Jussieu. 75005 Paris. France(Dated: October 23, 2016)We report on phase-shifted holographic interferogram demodulation by singular value decomposi-tion. Numerical processing of optically-acquired interferograms over several modulation periods wasperformed in two steps : 1- rendering of off-axis complex-valued holograms by Fresnel transformationof the interferograms; 2- eigenvalue spectrum assessment of the lag-covariance matrix of hologrampixels. Experimental results in low-light recording conditions were compared with demodulation byFourier analysis, in the presence of random phase drifts.OCIS codes: 090.1995, 040.2840, 120.2880, 040.3780Recording optical holograms by time-modulatedinterferometry was proposed and demonstrated in earlystudies [1–3]. It pioneered the development of phase-shifting [4] holographic interferometry. Time-modulatedphase-shifting interferometry involves recording of asequence of interferograms over at least one modulationcycle of the intensity. Temporal demodulation consistsin a linear combination of a sequence of recordedinterferograms, whose coefficients are the ones of adiscrete Fourier transform [5–7]. An important featureof holographic interferometry is to allow spatial signalmodulation in off-axis recording conditions [8], whichpermits the discrimination of self-beating and cross-beating interferometric contributions of the object fieldand the reference field. In this configuration, spuriousinterferometric contributions can be filtered-off [9]. Itwas demonstrated that the combination of spatial andtemporal signal modulation enables low-light imagingat shot noise levels [10, 11]. Important features andapplications include optical phase imaging [12–15] anddetection tunability in the radiofrequency range foroptical heterodyne detection [16], frequency-resolvednarrowband sensing of scattered optical radiation [17–19]. Sensitivity is compatible with photon-countingrecording conditions [20–22]. In standard phase-shiftinginterferometry, a sequence of interferograms with knownphase shifts between probed and reference opticalwaves is recorded. When the modulation is not known,principal component analysis [23] can be used for non-parametric, non-iterative demodulation. It identifiesuncorrelated variables in the structure of the data (theprincipal components) from the analysis of its correla-tions. Non-parametric approaches were also applied toimage formation concepts [24], speckle reduction [25–29],and laser speckle contrast analysis of flows [30].In this letter, we demonstrate experimentally thatthe sensitivity of low-light optical signal retrieval intime-modulated holographic interferometry can be im-proved with respect to Fourier transform demodulationin practical conditions, when random signal fluctuationsare present. For that purpose, a data-driven, non-parametric signal processing method is used. It consistsof calculating the spectrum of eigenvalues in a singularvalue decomposition (SVD) of the lag-covariance matrixof recorded temporal fluctuations of the hologram’s pix-els, after spatial Fresnel transformation. The proposedprocessing scheme is less affected by random opticalphase drifts occurring during the sampling process thansignal demodulation by discrete Fourier transformation.In optical heterodyne detection, a field of interestE(t) = E exp (iωLt), oscillating at the optical frequencyωL, is non-linearly mixed with a local oscillator fieldELO(t) = ELO exp (i∆ωt) exp (iωLt) that is set at a close-by intermediate frequency ωL + ∆ω. In this notation,E and ELO are complex constants and i is the imaginaryunit. The amplitude and phase of the original signal E(t)can be retrieved in the desired outcome oscillating at thedifference frequency ∆ω, which can be set within the re-ceiver’s temporal bandwidth. The frequency conversionprocess is ensured by non-linear detection of the opticalfield E by the array of square-law sensors of a camera,which respond quadratically with the impinging electricfield’s magnitude. The squared magnitude of the totalfield received, I(t) = |E(t) + ELO(t)|2, has cross-termsoscillating at the difference frequency ∆ω of the fields Eand ELO, causing modulation of the optical energy fluxI(t) = |E|2+ |ELO|2+EE∗LOe−i∆ωt+E∗ELOei∆ωt, where ∗denotes the complex conjugate. This equation describesthe temporal fluctuation of the recorded intensity at agiven pixel. Holographic image rendering, referred toas spatial demodulation, is then performed onto eachrecorded interferogram I by a discrete Fresnel transform.This transformation involves the free-space propagationimpulse response of the optical field [7, 31]. In off-axisrecording configuration, the Fresnel transform separatesspatially the four interferometric terms of I(t) [8, 9] and2forms a reconstructed hologram. In the absence of ran-dom fluctuations, the pixels of the off-axis part of thehologram carry the signalX(t) = EE∗LO exp(−i∆ωt) (1)The purpose of signal demodulation is to retrieve themagnitude of the oscillating signal at the frequency∆ω at each pixel. In ideal conditions, Eq. 1 holds andsignal demodulation by discrete Fourier analysis of therecorded interferograms [6, 7] enables shot-noise limiteddetection in dim light [10]; yet in actual experiments,random amplitude and phase fluctuations limit thedetection sensitivity.We recorded phase-shifted interferograms over severalmodulation periods in low-light and studied signalretrieval procedures. Optical recording of interferogramswas performed with a frequency-shifted, off-axis Mach-Zehnder interferometer [32] used for optical heterodynedetection of the object field E, backscattered from a res-olution target, beating against a separate local oscillatorfield ELO, on a sensor array. The main optical field wasprovided by a 150 mW, single-mode laser (wavelengthλ = 532 nm, optical frequency ωL/(2pi) = 5.6× 1014Hz,Cobolt Samba-TFB-150). The optical frequency of thelocal oscillator beam was shifted by a tunable quantity∆ω by two acousto-optic modulators (AA-electronics,MT200-BG9-FIO) driven by phase-locked continuous-wave radiofrequency signals around 200 MHz. Theobserved reflective resolution target was illuminated overa ∼ 10mm-diameter disk. Interferograms were measuredwith a Ximea MQ042MG-CM camera (2048×2048 pixelsCMOSIS CMV4000 sensor array, pixel size d = 5.5µm,full well charge : 13500 e− (photo-electrons), conversiongain : 1/G = 0, 075 counts/e−, quantum efficiency at532 nm η ∼ 0.5), running at an externally-triggeredframe rate of ωS/(2pi) = 20Hz, at 8 bit/pixel quanti-zation. In the reported experiment, a reflective USAFtarget printed on white paper was illuminated in re-flection, and the recorded intensity of the light in theobject channel accounted for 11.56 counts × G ≃ 154photo-electrons per pixel per frame, on average. Theaverage intensity of the light in the LO channel ac-counted for 63.89 counts × G ≃ 852 photo-electronsper pixel per recorded frame. In these conditions, asequence of 128 consecutive raw interferograms wasrecorded for a detuning frequency of the local oscillatorof ∆ω/(2pi) = 5Hz, creating 4-phase (ωS/∆ω = 4)modulation conditions [32]. Twelve more sequences ofimages were acquired after decreasing the optical powerin the object channel by 100.5 from one measurement tothe next, by combining optical densities (O.D., ThorlabsNE absorptive neutral density filters). The transmittedpower to the target scaled as 10−O.D., from 100 to 10−6;hence the average number of recorded photo-electronsper pixel and per frame in the object channel rangedfrom 154 to 1.54 × 10−4. The LO-to-object opticalpower ratio (heterodyne gain) ranged from GH ≃ 5.5FFTdemodulationSVDdemodulationO.D. = 0O.D. = 1O.D. = 2O.D. = 3O.D. = 4O.D. = 5O.D. = 6no signaldemodulationFIG. 1: Amplitude holographic images from 4-phase modu-lated interferograms acquired over 128 frames. No time de-modulation (first column). Demodulation by Fourier trans-form (second column). Demodulation by singular value de-composition (third column).to GH ≃ 5.5 × 106, because the LO power was keptconstant throughout the experiment. The holographicimages of these two datasets were then demodulatedspatially [31], to yield 661 × 661 complex-valued timetraces X(t) of 128 elements each, cropped from the2048 × 2048 calculation array, over an off-axis spatialregion of interest where the image was expected. Wethen compared two types of signal demodulation proce-dures to assess the magnitude of the oscillating signal atthe frequency ∆ω in X(t) at each pixel : Fourier anal-ysis, and singular value analysis of temporal fluctuations.30 1 2 3 4 5 6−30−25−20−15−10−505Optical density (O.D.)optical signal (dB)RAWFFTSVD10log(10àO:D:)FIG. 2: Signal magnitude versus optical density (O.D.) in theobject channel from a sequence of N = 128 frames of 4-phaseinterferograms. No time demodulation (circles). Demodula-tion by discrete Fourier transform (squares). Demodulationby singular value decomposition (triangles).First, holographic images were rendered without de-modulation. The squared magnitude of the object fieldwas averaged over N = 128 frame sequences, and re-ported in the first column of Fig. 1. The image is notvisible for optical power transmission factors lower orequal to 10−3 (i.e. O.D. = 3), which corresponds to19.7 photo-electrons per pixel recorded on average dur-ing 128 frames. Then, the recorded optical fluctuationswere projected on the expected oscillation to retrieve themodulated component to improve the detection sensitiv-ity. The modulation depth of the signal was assessed bya fast Fourier transform (FFT) X˜ of the sequence xk atthe modulation frequency ωS/4 (5 Hz), calculated on 128time points. Results are reported in the second column ofFig. 1. The image is not visible for a power transmissionfactor lower or equal to 10−5 (i.e. O.D. = 5), which cor-responds to 1.97×10−1 photo-electron per pixel recordedon average during 128 frames. Finally, we performed sin-gular value analysis of temporal fluctuations. In this ap-proach, we embedded the time series X = {x1, ..., xN} ofthe signal at each pixel in a Hankel matrix of dimensionK × (N − K + 1), where K is the number of principalcomponents to seek, and N = 128 is the total numberof acquired interferograms. The Hankel matrix D, com-posed of N −K+1 lag-shifted copies of truncated valuesof X(t) has the formD =x1 x2 . . . xN−K+1x2 x3. . ........... . . xN−1xK xK+1 . . . xN(2)We proceeded by diagonalizing the lag-covariancematrix of X(t), CX = DDt∗, where Dt∗ is the adjointmatrix (Hermitian conjugate) of D. The K eigenvaluesλk, k = 1, ...,K of CX , numbered by decreasing orderof magnitude, were calculated by SVD [33] of CX . Thelength of the analysis window K was chosen to be aninteger number of expected periods of the modulatedsignal [34] K = p ωS/∆ω. We used a window lengthof p = 5 modulation periods (K = 20). Images wererendered by calculating the amplitude of the secondeigenvalue at each pixel. They are displayed in Fig. 1in the third column, as a function of the optical densityin the object channel. With this approach, the image isvisible down to an optical power transmission factor of10−6 (i.e. O.D. = 6), for which the signal wave intensityaccounted for a total of 1.97 × 10−2 recorded photo-electron per pixel during 128 frames, on average. Thealgorithm used for the SVD is based on the orthogonaltriangular decomposition (QR factorization) [35]. SVDand FFT were performed by the functions svd() andfft() in Matlab 2012a, run on a CentOS 6 Linux 64-bitoperating system. The response curve of singular valuedemodulation was assessed by plotting the quantity10 log10(〈S〉 / 〈S0〉) (in dB) against the value of theoptical density in the object channel (Fig.2). Thebraces 〈 〉 indicate spatial averaging of S, and S0 withinand outside the reconstructed image, respectively.The expected ideal detection behavior is plotted asa continuous black line (-10 dB of optical power perunit of optical density). The demodulation by SVDis qualitative [36]; yet it enables to retrieve images atmuch lower signal levels than FFT demodulation. Thesingular value demodulation procedure is illustrated bythe spectrograms of four signals, reported in Fig. 3 : (a)the complex wave oscillating at ∆ω/(2pi) = 5Hz, whichwould be expected in perfect modulation conditions, inthe absence of random phase fluctuation (Eq. 1); (b)the actual signal X(t) from a pixel at O.D. = 2, whichexhibits a DC contribution and jitter - deviation fromthe modulation frequency; and the signal reconstructedfrom the first (c) and second (d) eigenvectors andeigenvalues of the SVD of D. In this example, the DCterm is found in the first component, and the modulatedcomponent at 5 Hz undergoing a phase drift is retrievedin the second component of the SVD.In conclusion, we used heuristically a singular valuedecomposition procedure to retrieve time-modulatedoptical signals in holographic interferometry. In thereported experiment, over long acquisition times, singu-lar value demodulation of phase-shifted interferogramsenabled qualitative image visualization at much lowersignal levels than demodulation by discrete Fouriertransformation. This method will be investigated forsignal retrieval in strong phase fluctuations conditions,phase microscopy, and singular spectrum analysis ofpseudoperiodic modulations.This work was supported by Agence Nationale de laRecherche (ANR-09-JCJC-0113, ANR-11-EMMA-046),Fondation Pierre-Gilles de Gennes (FPGG014), re´gion4time (s)0123time (s)0123instantaneous beating frequency (Hz)−10 −5 0 5 10−10 −5 0 5 10time (s)0123−10 −5 0 5 10originalsignalsecondeigenvectorfirsteigenvectortime (s)0123−10 −5 0 5 105 Hzsingle tone (a)(b)(c)(d)FIG. 3: Spectrograms of a complex wave oscillating at 5Hz(a), a hologram’s pixel signal X(t) (b), reconstructed signalsfrom the first (c) and second (d) elements of the SVD of D.Ile-de-France (C’Nano, AIMA), the Investments for theFuture program (LabExWIFI: ANR-10-LABX-24, ANR-10-IDEX-0001-02 PSL*), and European Research Coun-cil (ERC Synergy HELMHOLTZ).[1] A. Macovski, Applied Physics Letters 14, 166 (1969).[2] C. C. Aleksoff, Appl. Phys. Lett 14, 23 (1969).[3] R. Da¨ndliker, in In: Progress in optics. Volume 17.(A81-13109 03-74) Amsterdam, North-Holland Publishing Co.,1980, p. 1-84., vol. 17 (1980), vol. 17, pp. 1–84.[4] I. Yamaguchi and T. Zhang, Optics letters 22, 1268(1997).[5] K. Creath, Applied Optics 24, 3053 (1985).[6] K. Freischlad and C. Koliopoulos, Journal of the OpticalSociety of America A: Optics and Image Science, andVision 7, 542 (1990).[7] U. Schnars and W. P. O. Juptner, Meas. Sci. Technol.13, R85 (2002).[8] E. Leith and J. Upatnieks, JOSA 52, 1123 (1962).[9] E. Cuche, P. Marquet, and C. Depeursinge, Applied Op-tics 39, 4070 (2000).[10] M. Gross and M. Atlan, Optics Letters 32, 909 (2007).[11] M. Lesaffre, N. Verrier, and M. Gross, Applied optics 52,A81 (2013).[12] E. Cuche, P. Marquet, and C. Depeursinge, Applied Op-tics 38, 6994 (1999).[13] F. Charrie`re, T. Colomb, F. Montfort, E. Cuche, P. Mar-quet, and C. Depeursinge, Applied optics 45, 7667(2006).[14] N. Pandey and B. Hennelly, 3D Research 2, 1 (2011).[15] Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss,P. Magistretti, P. Marquet, and C. Depeursinge, NaturePhotonics 7, 113 (2013).[16] F. Bruno, J. Laurent, D. Royer, and M. Atlan, AppliedPhysics Letters 104, 083504 (2014).[17] M. Gross, P. Goy, B. Forget, M. Atlan, F. Ramaz, A. Boc-cara, and A. Dunn, Opt. Lett. 30, 1357 (2005).[18] M. Atlan, P. Desbiolles, M. Gross, and M. Coppey-Moisan, Opt. Lett. 35, 787 (2010).[19] H. Ruan, M. L. Mather, and S. P. Morgan, JOSA A 30,1409 (2013).[20] M. Yamamoto, H. Yamamoto, and Y. Hayasaki, Opticsletters 34, 1081 (2009).[21] P. Latorre Carmona, B. Javidi, F. Pla Ban˜o´n, and E. Ta-jahuerce (2013).[22] N. Demoli, H. Skenderovic´, and M. Stipcˇevic´, Optics Let-ters 39, 5010 (2014).[23] J. Vargas, J. A. Quiroga, and T. Belenguer, Optics letters36, 1326 (2011).[24] M. Bertero and E. Pike, Opt. Acta. 29, 727 (1982).[25] F. A. M. Rodriguez, A. Federico, and G. H. Kaufmann,Optics Communications 275, 38 (2007).[26] M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kauf-mann, Optics and lasers in engineering 45, 723 (2007).[27] S. Equis and P. Jacquot, Opt. Express 17, 611 (2009).[28] W.-H. Su, C.-K. Lee, and C.-W. Lee, Optics and Lasersin Engineering 48, 212 (2010). Fringe Projection Tech-niques.[29] M. Leo, R. Piccolo, C. Distante, P. Memmolo,M. Paturzo, and P. Ferraro, Optical Engineering 53,112314 (2014).[30] A. Humeau-Heurtier, P. Abraham, and G. Mahe, MedicalImaging, IEEE Transactions on PP, 1 (2014).[31] M. K. Kim, L. Yu, and C. J. Mann, Journal of Optics A:Pure and Applied Optics 8, S518 (2006).[32] M. Atlan, M. Gross, and E. Absil, Optics Letters 32,1456 (2007).[33] G. H. Golub and C. Reinsch, Numerische Mathematik14, 403 (1970).[34] R. Vautard and M. Ghil, Physica D: Nonlinear Phenom-ena 35, 395 (1989).[35] G. H. Golub and C. F. Van Loan, Matrix computations,vol. 3 (JHU Press, 2012).[36] D. Broomhead and G. King, Physica D 20, 217 (1986).

Hal-Diderot

Singular-value demodulation of phase-shifted holograms
Fernando Lopes, Michael Atlan
To cite this version:
Fernando Lopes, Michael Atlan. Singular-value demodulation of phase-shifted holograms. Op-
tics Letters, Optical Society of America, 2015, 40, pp.2541 - 2541. <10.1364/OL.40.002541>.
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Singular value demodulation of phase-shifted holograms
Fernando Lopes1 and Michael Atlan2
1 Institut de Physique du Globe de Paris - Pale´omagne´tisme,
Universite´ Paris Diderot, Sorbonne Paris Cite´, 1 rue Jussieu, Paris, France
2 Centre National de la Recherche Scientifique (CNRS) UMR 7587,
Institut Langevin. Fondation Pierre-Gilles de Gennes,
Institut National de la Sante´ et de la Recherche Me´dicale (INSERM) U 979,
Universite´ Pierre et Marie Curie (UPMC), Universite´ Paris 7. E´cole Supe´rieure de
Physique et de Chimie Industrielles ESPCI ParisTech - 1 rue Jussieu. 75005 Paris. France
(Dated: October 23, 2016)
We report on phase-shifted holographic interferogram demodulation by singular value decomposi-
tion. Numerical processing of optically-acquired interferograms over several modulation periods was
performed in two steps : 1- rendering of off-axis complex-valued holograms by Fresnel transformation
of the interferograms; 2- eigenvalue spectrum assessment of the lag-covariance matrix of hologram
pixels. Experimental results in low-light recording conditions were compared with demodulation by
Fourier analysis, in the presence of random phase drifts.
OCIS codes: 090.1995, 040.2840, 120.2880, 040.3780
Recording optical holograms by time-modulated
interferometry was proposed and demonstrated in early
studies [1–3]. It pioneered the development of phase-
shifting [4] holographic interferometry. Time-modulated
phase-shifting interferometry involves recording of a
sequence of interferograms over at least one modulation
cycle of the intensity. Temporal demodulation consists
in a linear combination of a sequence of recorded
interferograms, whose coefficients are the ones of a
discrete Fourier transform [5–7]. An important feature
of holographic interferometry is to allow spatial signal
modulation in off-axis recording conditions [8], which
permits the discrimination of self-beating and cross-
beating interferometric contributions of the object field
and the reference field. In this configuration, spurious
interferometric contributions can be filtered-off [9]. It
was demonstrated that the combination of spatial and
temporal signal modulation enables low-light imaging
at shot noise levels [10, 11]. Important features and
applications include optical phase imaging [12–15] and
detection tunability in the radiofrequency range for
optical heterodyne detection [16], frequency-resolved
narrowband sensing of scattered optical radiation [17–
19]. Sensitivity is compatible with photon-counting
recording conditions [20–22]. In standard phase-shifting
interferometry, a sequence of interferograms with known
phase shifts between probed and reference optical
waves is recorded. When the modulation is not known,
principal component analysis [23] can be used for non-
parametric, non-iterative demodulation. It identifies
uncorrelated variables in the structure of the data (the
principal components) from the analysis of its correla-
tions. Non-parametric approaches were also applied to
image formation concepts [24], speckle reduction [25–29],
and laser speckle contrast analysis of flows [30].
In this letter, we demonstrate experimentally that
the sensitivity of low-light optical signal retrieval in
time-modulated holographic interferometry can be im-
proved with respect to Fourier transform demodulation
in practical conditions, when random signal fluctuations
are present. For that purpose, a data-driven, non-
parametric signal processing method is used. It consists
of calculating the spectrum of eigenvalues in a singular
value decomposition (SVD) of the lag-covariance matrix
of recorded temporal fluctuations of the hologram’s pix-
els, after spatial Fresnel transformation. The proposed
processing scheme is less affected by random optical
phase drifts occurring during the sampling process than
signal demodulation by discrete Fourier transformation.
In optical heterodyne detection, a field of interest
E(t) = E exp (iωLt), oscillating at the optical frequency
ωL, is non-linearly mixed with a local oscillator field
ELO(t) = ELO exp (i∆ωt) exp (iωLt) that is set at a close-
by intermediate frequency ωL + ∆ω. In this notation,
E and ELO are complex constants and i is the imaginary
unit. The amplitude and phase of the original signal E(t)
can be retrieved in the desired outcome oscillating at the
difference frequency ∆ω, which can be set within the re-
ceiver’s temporal bandwidth. The frequency conversion
process is ensured by non-linear detection of the optical
field E by the array of square-law sensors of a camera,
which respond quadratically with the impinging electric
field’s magnitude. The squared magnitude of the total
field received, I(t) = |E(t) + ELO(t)|
2, has cross-terms
oscillating at the difference frequency ∆ω of the fields E
and ELO, causing modulation of the optical energy flux
I(t) = |E|2+ |ELO|
2+EE∗LOe
−i∆ωt+E∗ELOe
i∆ωt, where ∗
denotes the complex conjugate. This equation describes
the temporal fluctuation of the recorded intensity at a
given pixel. Holographic image rendering, referred to
as spatial demodulation, is then performed onto each
recorded interferogram I by a discrete Fresnel transform.
This transformation involves the free-space propagation
impulse response of the optical field [7, 31]. In off-axis
recording configuration, the Fresnel transform separates
spatially the four interferometric terms of I(t) [8, 9] and
2forms a reconstructed hologram. In the absence of ran-
dom fluctuations, the pixels of the off-axis part of the
hologram carry the signal
X(t) = EE∗LO exp(−i∆ωt) (1)
The purpose of signal demodulation is to retrieve the
magnitude of the oscillating signal at the frequency
∆ω at each pixel. In ideal conditions, Eq. 1 holds and
signal demodulation by discrete Fourier analysis of the
recorded interferograms [6, 7] enables shot-noise limited
detection in dim light [10]; yet in actual experiments,
random amplitude and phase fluctuations limit the
detection sensitivity.
We recorded phase-shifted interferograms over several
modulation periods in low-light and studied signal
retrieval procedures. Optical recording of interferograms
was performed with a frequency-shifted, off-axis Mach-
Zehnder interferometer [32] used for optical heterodyne
detection of the object field E, backscattered from a res-
olution target, beating against a separate local oscillator
field ELO, on a sensor array. The main optical field was
provided by a 150 mW, single-mode laser (wavelength
λ = 532 nm, optical frequency ωL/(2pi) = 5.6× 10
14Hz,
Cobolt Samba-TFB-150). The optical frequency of the
local oscillator beam was shifted by a tunable quantity
∆ω by two acousto-optic modulators (AA-electronics,
MT200-BG9-FIO) driven by phase-locked continuous-
wave radiofrequency signals around 200 MHz. The
observed reflective resolution target was illuminated over
a ∼ 10mm-diameter disk. Interferograms were measured
with a Ximea MQ042MG-CM camera (2048×2048 pixels
CMOSIS CMV4000 sensor array, pixel size d = 5.5µm,
full well charge : 13500 e− (photo-electrons), conversion
gain : 1/G = 0, 075 counts/e−, quantum efficiency at
532 nm η ∼ 0.5), running at an externally-triggered
frame rate of ωS/(2pi) = 20Hz, at 8 bit/pixel quanti-
zation. In the reported experiment, a reflective USAF
target printed on white paper was illuminated in re-
flection, and the recorded intensity of the light in the
object channel accounted for 11.56 counts × G ≃ 154
photo-electrons per pixel per frame, on average. The
average intensity of the light in the LO channel ac-
counted for 63.89 counts × G ≃ 852 photo-electrons
per pixel per recorded frame. In these conditions, a
sequence of 128 consecutive raw interferograms was
recorded for a detuning frequency of the local oscillator
of ∆ω/(2pi) = 5Hz, creating 4-phase (ωS/∆ω = 4)
modulation conditions [32]. Twelve more sequences of
images were acquired after decreasing the optical power
in the object channel by 100.5 from one measurement to
the next, by combining optical densities (O.D., Thorlabs
NE absorptive neutral density filters). The transmitted
power to the target scaled as 10−O.D., from 100 to 10−6;
hence the average number of recorded photo-electrons
per pixel and per frame in the object channel ranged
from 154 to 1.54 × 10−4. The LO-to-object optical
power ratio (heterodyne gain) ranged from GH ≃ 5.5
FFT
demodulation
SVD
demodulation
O
.D
. =
 0
O
.D
. =
 1
O
.D
. =
 2
O
.D
. =
 3
O
.D
. =
 4
O
.D
. =
 5
O
.D
. =
 6
no signal
demodulation
FIG. 1: Amplitude holographic images from 4-phase modu-
lated interferograms acquired over 128 frames. No time de-
modulation (first column). Demodulation by Fourier trans-
form (second column). Demodulation by singular value de-
composition (third column).
to GH ≃ 5.5 × 10
6, because the LO power was kept
constant throughout the experiment. The holographic
images of these two datasets were then demodulated
spatially [31], to yield 661 × 661 complex-valued time
traces X(t) of 128 elements each, cropped from the
2048 × 2048 calculation array, over an off-axis spatial
region of interest where the image was expected. We
then compared two types of signal demodulation proce-
dures to assess the magnitude of the oscillating signal at
the frequency ∆ω in X(t) at each pixel : Fourier anal-
ysis, and singular value analysis of temporal fluctuations.
30 1 2 3 4 5 6
−30
−25
−20
−15
−10
−5
0
5
Optical density (O.D.)
o
p
ti
ca
l 
si
g
n
al
 (
d
B
)
RAW
FFT
SVD
10
log(10
à
O
:D
:)
FIG. 2: Signal magnitude versus optical density (O.D.) in the
object channel from a sequence of N = 128 frames of 4-phase
interferograms. No time demodulation (circles). Demodula-
tion by discrete Fourier transform (squares). Demodulation
by singular value decomposition (triangles).
First, holographic images were rendered without de-
modulation. The squared magnitude of the object field
was averaged over N = 128 frame sequences, and re-
ported in the first column of Fig. 1. The image is not
visible for optical power transmission factors lower or
equal to 10−3 (i.e. O.D. = 3), which corresponds to
19.7 photo-electrons per pixel recorded on average dur-
ing 128 frames. Then, the recorded optical fluctuations
were projected on the expected oscillation to retrieve the
modulated component to improve the detection sensitiv-
ity. The modulation depth of the signal was assessed by
a fast Fourier transform (FFT) X˜ of the sequence xk at
the modulation frequency ωS/4 (5 Hz), calculated on 128
time points. Results are reported in the second column of
Fig. 1. The image is not visible for a power transmission
factor lower or equal to 10−5 (i.e. O.D. = 5), which cor-
responds to 1.97×10−1 photo-electron per pixel recorded
on average during 128 frames. Finally, we performed sin-
gular value analysis of temporal fluctuations. In this ap-
proach, we embedded the time series X = {x1, ..., xN} of
the signal at each pixel in a Hankel matrix of dimension
K × (N − K + 1), where K is the number of principal
components to seek, and N = 128 is the total number
of acquired interferograms. The Hankel matrix D, com-
posed of N −K+1 lag-shifted copies of truncated values
of X(t) has the form
D =


x1 x2 . . . xN−K+1
x2 x3
. . .
...
...
...
. . . xN−1
xK xK+1 . . . xN


(2)
We proceeded by diagonalizing the lag-covariance
matrix of X(t), CX = DD
t∗, where Dt∗ is the adjoint
matrix (Hermitian conjugate) of D. The K eigenvalues
λk, k = 1, ...,K of CX , numbered by decreasing order
of magnitude, were calculated by SVD [33] of CX . The
length of the analysis window K was chosen to be an
integer number of expected periods of the modulated
signal [34] K = p ωS/∆ω. We used a window length
of p = 5 modulation periods (K = 20). Images were
rendered by calculating the amplitude of the second
eigenvalue at each pixel. They are displayed in Fig. 1
in the third column, as a function of the optical density
in the object channel. With this approach, the image is
visible down to an optical power transmission factor of
10−6 (i.e. O.D. = 6), for which the signal wave intensity
accounted for a total of 1.97 × 10−2 recorded photo-
electron per pixel during 128 frames, on average. The
algorithm used for the SVD is based on the orthogonal
triangular decomposition (QR factorization) [35]. SVD
and FFT were performed by the functions svd() and
fft() in Matlab 2012a, run on a CentOS 6 Linux 64-bit
operating system. The response curve of singular value
demodulation was assessed by plotting the quantity
10 log10(〈S〉 / 〈S0〉) (in dB) against the value of the
optical density in the object channel (Fig.2). The
braces 〈 〉 indicate spatial averaging of S, and S0 within
and outside the reconstructed image, respectively.
The expected ideal detection behavior is plotted as
a continuous black line (-10 dB of optical power per
unit of optical density). The demodulation by SVD
is qualitative [36]; yet it enables to retrieve images at
much lower signal levels than FFT demodulation. The
singular value demodulation procedure is illustrated by
the spectrograms of four signals, reported in Fig. 3 : (a)
the complex wave oscillating at ∆ω/(2pi) = 5Hz, which
would be expected in perfect modulation conditions, in
the absence of random phase fluctuation (Eq. 1); (b)
the actual signal X(t) from a pixel at O.D. = 2, which
exhibits a DC contribution and jitter - deviation from
the modulation frequency; and the signal reconstructed
from the first (c) and second (d) eigenvectors and
eigenvalues of the SVD of D. In this example, the DC
term is found in the first component, and the modulated
component at 5 Hz undergoing a phase drift is retrieved
in the second component of the SVD.
In conclusion, we used heuristically a singular value
decomposition procedure to retrieve time-modulated
optical signals in holographic interferometry. In the
reported experiment, over long acquisition times, singu-
lar value demodulation of phase-shifted interferograms
enabled qualitative image visualization at much lower
signal levels than demodulation by discrete Fourier
transformation. This method will be investigated for
signal retrieval in strong phase fluctuations conditions,
phase microscopy, and singular spectrum analysis of
pseudoperiodic modulations.
This work was supported by Agence Nationale de la
Recherche (ANR-09-JCJC-0113, ANR-11-EMMA-046),
Fondation Pierre-Gilles de Gennes (FPGG014), re´gion
4ti
m
e 
(s
)
0
1
2
3
ti
m
e 
(s
)
0
1
2
3
instantaneous beating frequency (Hz)
−10 −5 0 5 10
−10 −5 0 5 10
ti
m
e 
(s
)
0
1
2
3
−10 −5 0 5 10
original
signal
second
eigenvector
first
eigenvector
ti
m
e 
(s
)
0
1
2
3
−10 −5 0 5 10
5 Hz
single tone 
(a)
(b)
(c)
(d)
FIG. 3: Spectrograms of a complex wave oscillating at 5Hz
(a), a hologram’s pixel signal X(t) (b), reconstructed signals
from the first (c) and second (d) elements of the SVD of D.
Ile-de-France (C’Nano, AIMA), the Investments for the
Future program (LabExWIFI: ANR-10-LABX-24, ANR-
10-IDEX-0001-02 PSL*), and European Research Coun-
cil (ERC Synergy HELMHOLTZ).
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Singular value demodulation of phase-shifted holograms

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