Single-shot Sub-Rayleigh Imaging with Sparse Detection

0.15 MB·English

by  Wenlin Gong

Single-shot Sub-Rayleigh Imaging with Sparse Detection
∗
Wenlin Gong and Shensheng Han
Key Laboratory for Quantum Optics and Center for Cold Atom Physics of CAS,
Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
(Dated: January 18, 2013)
For conventional imaging, the imaging resolution limit is given by the Rayleigh criterion. Ex-
ploiting theprior knowledge of imaging object’s sparsity and fixed optical system, imaging beyond
the conventional Rayleigh limit, which is backed up by numerical simulation and experiments, is
achievedbyilluminatingtheobjectwithsingle-shotthermallightanddetectingtheobject’sinforma-
3 tionattheimagingplanewithsomesparse-arraysingle-pixeldetectors. Thequalityofsub-Rayleigh
1
imaging with sparse detection is also shown to be related to the effective number of single-pixel
0
detectors and thedetection signal-to-noise ratio at theimaging plane.
2

n PACSnumbers: 42.25.Kb,42.30.Va,42.30.Lr
a
J
7 For standard conventional imaging of directly record- remotesensing[22,23],compressiveimaging[24–28],and
1 ing the object’s intensity distribution with a chargecou- magnetic resonance imaging with success [29]. For con-
pled device (CCD) camera, both the imaging system’s ventional imaging, the optical system is usually fixed

] Rayleigh limit and the camera’s pixel-resolution restrict or known, so the optical system’s point spread function
s
c the opticalsystem’s imagingresolution[1]. Forexample, (PSF) can be also taken as a priori. Furthermore, based
i the imaging resolution is mainly determined by the op- on the object’s sparse priori property, measurements be-
t
p tical system’s Rayleigh limit in remote sensing because low Nyquist rate are required to exactly restore the ob-
o the numerical aperture (N.A.) of imaging lens is usually ject. Therefore, exploiting the object’s sparsity assump-
.
s small relative to the detection distance. While in long- tion in a known basis and the prior knowledge of PSF,
c wavelengthradiationbandsuchasinfraredandterahertz single-shotsub-Rayleighimagingis possibleevenif some
i
s imaging, because the camera with large planar arrays sparse-arraysingle-pixeldetectors are used to recordthe
y
is very hard to manufacture, the imaging resolution is object’s information at the imaging plane.
h
mainly limited by the camera’s pixel-resolution.
p
(c )
[ Compared with overcoming the camera’s pixel- 1

resolution to imaging resolution, many methods are in-
1
vented to overcome the imaging system’s Rayleigh limit
v
7 at present[2–14]. Exploiting the evanescentcomponents
0 at the object’s immediate proximity, sub-Rayleighimag-
1 ing can be achieved, but this method is only applied
4 in the near-field range [2–4]. Several microscopy tech-
.
1 niques based on fluorescence are also introduced to im-
0 prove imaging resolution. However, it requires scanning FIG. 1: Schematic of thermal-light single-shot imaging with
3 orrepetitiveexperiments,thuswhichlimitsreal-timeap- sparse detection.
1
plications [5–9]. In addition, using additional a priori
:
v informationofopticalsystem,theimagingresolutionbe- Fig. 1 presentsexperimentaldemonstrationschematic
i yond Rayleigh diffraction limit can be obtained. How- of thermal-light single-shot sub-Rayleigh imaging with
X
ever, the improvement degree is limited in practice be- sparsedetection. Theuniformlightemittingfromahalo-
r
a cause of the influence of detection noise [10–14]. gen lamp is filtered by an optical filter (with the center
wavelength λ=650 nm and the bandwidth ∆λ=10 nm)
Recently,theimage’ssparsityhasbeentakenaspopu-
and then collimated by a lens with the focal length f .
lar a priori, which is a quite generalassumption because 0
The object is illuminated by the collimation light and
most of the natural objects are sparse in a known basis
then imaged onto the imaging plane x by a standard
(or under a suitable basis transform), an image can be t
conventionalimagingsetup. Differentfromrecordingthe
stably reconstructed by sparse reconstruction technique
object’s transmission information by a CCD camera at
evenifthemeasurementnumberislessthanNyquistrate
the imaging plane x , the detection system D we pro-
andthistechnique,tosomeextent,isrobusttonoise[15– t t
pose is characterized by some sparse-array single-pixel
19]. Moreover, this technique has already been applied
detectors.
to de-noising [18, 19], super-resolution imaging [20, 21],
The distances z , z and the focal length of the lens f
1 2
obey Gaussian thin-lens equation: 1 + 1 = 1. Based
z1 z2 f
onRayleighcriterion[1],theresolutionlimit∆x ofcon-
s
∗Electronicaddress: [email protected] ventional imaging shown in Fig. 1 is determined by the
2
wavelength λ and the N.A. of the lens f, namely algorithm[30],theobjectT canbereconstructedbysolv-
ing the following convex optimization program:
λ λz
∆xs =0.61N.A. ≃1.22 L1. (1) T =|t′|; which minimizes: 1(cid:13)(cid:13)I (x )−h(x)⊗|t′(x)|2(cid:13)(cid:13)2
2(cid:13) t i (cid:13)2
where L is the effective transmission aperture of the +τ(cid:13)(cid:13)Ψ{|t′(x)|2}(cid:13)(cid:13) ,∀ =1···M.(4)
imaging lens f and N.A. is approximate to L . (cid:13) (cid:13) s
2z1 1
When the light is fully spatially incoherent and uni-
where τ is a nonnegative parameter which is used to ad-
form, the intensity at the imaging plane x is the con-
t just the sparse weight in the image restoration process,
volution of the intensity at the object plane with the
M is the number ofsingle-pixeldetectors at the imaging
incoherent point spread function (PSF) [10]
plane x and Ψ denotes the transform operator to the
t
sparse basis. kVk and kVk represent the Euclidean
I (x)=I (x)⊗h(x)+I (x). (2) 2 1
t obj noise norm and the ℓ -norm of V, respectively.
1
where h(x) is the optical system’s PSF, I (x) is the
obj
object’s intensity distribution, Inoise(x) is the intensity (a) (b1) (c1)
distribution of the noise and I (x) is the intensity distri-
t
bution at the imaging plane x . For the optical system
t
shown in Fig. 1, the PSF is
(b ) (b ) (b ) (b ) (b )
2 3 4 5 6
L
h(x)=sinc2[ x]. (3)
λz
1
where sinc(x)= sin(πx).
πx (c2) (c3) (c4) (c5) (c6)
From Eqs. (2) and (3), compared with the original
object, it is a blurred image with low spatial resolution
at the imaging plane x when the transmission aperture
t
of the imaging lens f is small. In order to restore a
high-resolution image, the relation shown in Eq. (2)
usually yields F{I (x)} = F{I (x)}H +F{I (x)} in FIG. 2: The simulated demonstration of noiseless sub-
t obj obj
Rayleigh imaging with sparse detection. (a) The original
the spatial-frequency domain, where F denotes Fourier
object; (b1) the intensity distribution recorded by M=4096
transformand H is the optical transfer function. There-
sparse-arraysingle-pixeldetectorsattheimagingplanextand
fore, the problem is that we wish to reconstruct the ob-
the distance between two singe-pixel detectors in lateral di-
ject but the measured image is smeared by a low-pass rection ∆x=6.45 µm; (b2) M=1024 and ∆x=12.90 µm; (b3)
filter. Althoughmanyiterativemethodswereusedtore- M=256 and ∆x=25.80 µm; (b4) M=64 and ∆x=51.60 µm;
store the object, the reconstruction quality strongly de- (b5) M=36 and ∆x=64.50 µm; (b6) M=72 and ∆x=64.50
pendsonthesignal-to-noiseratio(SNR)inthemeasured µm; (c1-c6) are corresponding sparse reconstruction results
data [10–14]. Furthermore, for the optical system shown with respect to (b1-b6).
in Fig. 1, because the detection system D is charac-
t
terized by some sparse-array single-pixel detectors and In order to verify the idea, Fig. 2 and Fig. 3 have
many low-frequency information will be also missed, it given the simulated and experimental demonstration of
becomesmuchmoredifficulttorestoretheobjectbytra- sub-Rayleigh imaging with sparse detection, using the
ditional iterative methods, compared with recording all optical system depicted in Fig. 1. The original object
thesmearedimage’slow-frequencyinformationbyaCCD (64×64 pixels, and the pixel size is 6.45 µm×6.45 µm),
camera. as shown in Fig. 2(a) and Fig. 3(a), is a double-slit
Different from the method described above, we try to (slit width a=30 µm, silt height h=240 µm and center-
directly restore the object borne on the optical system to-center separation d=60 µm) and it comprises of 370
showninFig. 1,exploitingthesparsityassumptionofthe nonzero values in real-space domain. The parameters
object in a known basis and the prior knowledge of the listed in Fig. 1 are set as follows: z =z =800 mm, the
1 2
fixed optical system. According to sparse reconstruction focallengthofthelensf=400mmanditseffectivetrans-
theory,evenifthemeasurementprocessisnoiseless,there mission aperture L=8.0 mm. In addition, the pixel size
areaninfinitenumberofimages,which−afterbeingcon- of the single-pixel detector is 6.45 µm×6.45 µm. There-
volutedbythePSF−willresultinthesmearedimage,our fore, based on Eq. (1), the imaging system’s resolution
convex optimization program is how to find the sparsest limitis∆x ≃80µmandtheobject’simageattheimag-
s
one. Thesparsereconstructiontechniquehasmathemat- ing plane x can not be resolved for conventional imag-
t
ically demonstrated that if the object is sparse enough, ing [Fig. 2(b ), Fig. 3(b )]. However, both the simu-
1 1
then any sparsity-basedreconstruction method is bound lated and experimental results illustrated in Fig. 2 and
to find the sparsest solution [15–19]. Here, we have em- Fig. 3 clearly demonstrate that sub-Rayleigh imaging
ployed the gradient projection for sparse reconstruction with sparse detection can be achieved by exploiting the
3
(a) (b ) (c ) are depicted in Fig. 2(c1-c6) and Fig. 3(c1-c6), respec-
1 1
tively. As the distance betweentwo singe-pixeldetectors
is increased, the restoration quality will be reduced be-
cause of the decrease of the sampling number [Fig. 2(c -
1
c ) and Fig. 3(c -c )]. However, if the sampling number
(b ) (b ) (b ) (b ) (b ) 5 1 5
2 3 4 5 6 at the imaging plane x is increased, sub-Rayleighimag-
t
ing with sparse detection can be stably restored, even if
thedistancebetweentwosinge-pixeldetectors(∆x=64.5
µm) is large than the double-slit’s center-to-center sepa-
(c2) (c3) (c4) (c5) (c6) ration [Fig. 2(c5-c6) and Fig. 3(c5-c6)]. In addition, as
showninFig. 2(c )andFig. 3(c ),onlyM=72measure-
6 6
mentsareusedtorestorethe object’simage,whichisfar
fewer than suggested by Shannon’s sampling theorem.
As shown in Fig. 4, we also perform the dependance
FIG. 3: The experimental demonstration of sub-Rayleigh of sub-Rayleigh imaging with sparse detection on the
imaging with sparse detection. (a) The original object im- detection SNR at the imaging plane x . Fig. 4(a-f)
t
agedbyaconventionalopticalimagingsetupwithlargeN.A.; present the intensity distributions recorded by the de-
(b1-b6) and (c1-c6) are the intensity distributions record by tectors at the imaging plane x in different SNR. Based
t
thedetectorsattheimagingplanext andtheircorresponding on the sparse reconstruction technique described by Eq.
sparse reconstruction results, the same as Fig. 2.
(4), the corresponding restoration results are displayed
in Fig. 4(a -f ). From Fig. 4(a -d ), even if the de-
1 1 1 1
(a) (b) (c) (d) (e) (f) tection SNR is 2 dB, we can approximately reconstruct
sub-Rayleighimaging using M=1024measurements and
the reconstruction quality will be improved as the in-
crease of the detection SNR. While in the case of the
(a ) (b ) (c ) (d ) (e ) (f )
1 1 1 1 1 1 detection SNR=15 dB, high-quality sub-Rayleigh imag-
ing with sparse detection, as shown in Fig. 4(d -f ), can
1 1
be still restored by using the measurements far below
Nyquist rate.
FIG. 4: Sub-Rayleigh imaging with sparse detection in dif- In conclusion, we have realized single-shot sub-
ferent detection SNR and (M, ∆x). (a) The intensity Rayleigh imaging with sparse detection, exploiting the
distributions recorded by M=1024 (∆x=12.90 µm) sparse- imagingobject’ssparsityandthepriorknowledgeoffixed
array single-pixel detectors and the detection SNR=2 dB
optical system. Both the simulated and experimental
at the imaging plane xt; (b) M=1024 (∆x=12.90 µm) and results have demonstrated that using the measurement
SNR=5 dB; (c) M=1024 (∆x=12.90 µm) and SNR=10 dB;
below Nyquist rate, sub-Rayleigh imaging with sparse
(d) M=1024 (∆x=12.90 µm) and SNR=15 dB; (e) M=120
detection can break through the limitation of both the
(∆x=38.70µm)andSNR=15dB;(f)M=72(∆x=64.50µm)
optical system’s Rayleigh limit and the camera’s pixel-
andSNR=15dB;(a1-f1)arecorrespondingsparserestoration
results with respect to (a-f). resolution to imaging resolution. Furthermore, the ex-
perimenthasalsoshownthatsub-Rayleighimagingwith
sparsedetectioncanstillbeapproximatelyreconstructed
prior knowledge of the object’s sparsity and the optical inthecaseofmeasuredSNR=2dB.Thistechniqueisvery
system’s PSF.When the number of singe-pixeldetectors usefultomicroscopyoflivingcellsorbacteria,andimag-
(namely the total effective measurement number M) at ing ofatomscapturedby iontrap,etc, where the images
the imaging plane x is M=4096, 1024, 256, 64, 36, and are enough sparse.
t
72,Fig. 2(b -b ) and Fig. 3(b -b ) presentthe intensity The workwassupported by the Hi-Tech Researchand
1 6 1 6
distributions recorded by the detectors, and their corre- DevelopmentProgramofChinaunderGrantProjectNo.
sponding sparse reconstruction results borne on Eq. (4) 2011AA120101and No. 2011AA120102.
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