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The Study of Spectral Camera based on Ghost Imaging via Sparsity Constraints with Sunlight Illumination Zhentao Liu, Shiyu Tan, Jianrong Wu, Enrong Li, 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.
[email protected]
Abstract: Spectral camera based on ghost imaging via sparsity constraints can acquire information at a rate significantly below Nyquist. We compare the results reconstructed by ghost imaging and ghost imaging via sparsity constraints with sunlight illumination. OCIS codes: (110.0110) Imaging systems; (110.6150) Speckle imaging; (110.4234) Multispectral and hyperspectral imaging;
1. Introduction Applying sparse representation of images[1] and combining with compressive sensing theory[2, 3], ghost imaging via sparsity constraints (GISC) [4] has become an interesting research area. By combining spectral imaging with ghost imaging, spectral camera based on ghost imaging via sparsity constraints is proposed[5]. GISC spectral camera can acquire the information at a rate significantly below the Nyquist rate, and opens a way of approaching the Shannon Limit determined by Information Theory in optical imaging instruments. Under the framework of ghost imaging (GI), the intensity distribution detected by CCD can be expressed as (1) I is rt I i0 ri , l hI rt ; ri , l dri dl ; i r , t where hI rt ; ri , l is the incoherent intensity impulse response function, is the coordinate and plane
rt is the coordinate in detection plane, ri
l is the wavelength in object plane. I ri , l is the spectral intensity distribution in the 0 i
ri , l . GISC spectral camera modulates the true thermal light into a spatial fluctuating pseudo-thermal light
by using the diffraction of a spatial random phase modulator[6], which converts ensemble statistics for time t into ensemble statistics for space rt because of the ergodic property of the pseudo-thermal light. The light field of different wavelength modulated by the spatial random phase modulator generates the uncorrelated speckles where the spatial random phase modulator is acting as a random grating. The spatial intensity fluctuation of the pseudothermal light is pre-determined in GISC spectral camera. Assuming the height autocorrelation function of the spatial random phase modulator
R r0 , r0 = r0 r0
where r0 and r0
2 r0 r0 exp =R r0 , r0 =r0 r0 2
are the height of the spatial random phase modulator at
(2)
r0 and r0 , and are
respectively the height standard deviation and transverse correlation length of the spatial random phase modulator. The correlation function of intensity fluctuations between the spatial intensity fluctuation in the pre-determined reference arm and the test arm is G 2 ri , l
1 z12 z22 z1 z2
2
Ti ri , kl
ri
2 2 2 z2 ri 2 exp 2 2 n 1 kl exp 1 exp 2 n 1 kl z z 1 2 ri k
l
(3) where kl 1
l , kl 1 l ,
denotes the operation of convolution. Equation (3) shows that we can calculate
the correlation function of intensity fluctuations G 2 ri , l to separate the spectral images Ti ri , l . Under the framework of GISC, GISC spectral camera is formulated as follows. Assuming that the number of spectral channels is L, the size of spatial image pixels is N. A spatial pixel with a narrowband wavelength as light source generates the speckle, denote the i-th speckle intensity recorded by the CCD detector as Ii, and reshape it into a column vector. After L × N observations, the measurement matrix A (which satisfies the restricted isometry property of CS theory) is totally completed. The speckle intensity through the imaging broadband filter before the
kl
ATu1J.4.pdf
CLEO:2016 © OSA 2016
objective lens recorded by the CCD detector is denoted as Ii and reshape it into a column vector Y. Therefore, X can be reconstructed from Y = AX by using non-linear optimization algorithm[7]. 3. Experiments, Data, Discussion
Fig. 1. Experimental setup of GISC spectral camera.
With sunlight illumination, the experimental setup of GISC spectral camera is shown in Fig. 1. The imaging system with focal length of f=180mm projects the object image onto the first image plane, a beam splitter (BS) with split ratio 50:50 splits the light field, CCD1 detector with the pixel size 3.45um×3.45um is placed on the first image plane to obtain the conventional image of the object for comparison, a spatial random phase modulator disperses the image with different wavelengths as a random grating and modulates the image to generate the speckles, a microscope objective with magnification β=10 and the numerical aperture N.A.=0.25 which magnifies the speckles, the magnified speckles is recorded by CCD2 detector with the pixel size 13um×13um. The first image plane is divided into Nx×Ny=140×140 pixels. The number of spectrum bands for single exposure is 7, and we get two wavelength ranges of 520~580nm and 620~680nm with two exposures. The spectral images can be respectively reconstructed by ghost imaging and ghost imaging via sparsity constraints.[7-9] The object consists of the green aloe & red pot and its image acquired by a color camera is shown in Fig. 2. The 520~580nm and the 620~680nm band pass filter respectively placed in front of the objective lens ensures that only the subset of the data cube corresponding to this band of wavelengths is detected. The spectral images reconstructed by GI and GISC with 1:3 sampling rate is respectively shown in Fig. 3(a) and Fig. 3(b).
Fig. 2. The green aloe & red pot
(a)
(b)
Fig. 3. The spectral images reconstructed by (a) GI and (b) GISC with 1:3 sampling rate.
The comparison between Fig.3 and Fig.4 shows that the spectral images quality reconstructed by GISC is better than reconstructed by GI. One explanation is that GI reconstruction uses the arithmetic average to approximate the expect value that theoretically can be used to achieve the ideal image, which may be vulnerable to statistical fluctuation especially when the number of measurements is small. GISC reconstruction, on the other hand, simply extracts the spectral images from all the measurements with no such assumption. 5. References 1. 2. 3. 4. 5. 6. 7. 8. 9.
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