A compressive imaging spectropolarimeter is proposed in this paper, capable of simultaneously acquiring full polarization, spatial and spectral information of the object scene. The spectral and polarization information is modulated through a combination of high-order retarders, a dispersion prism and a polarizer filter wheel. Using a random coded aperture, compressive sensing is applied to eliminate the channel crosstalk and resolution limitation of traditional channeled spectropolarimeters. The forward sensing model and inverse problem are developed. Computer simulation results are reported, followed by experimental demonstrations.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Intensity, wavelength, coherence, and polarization are the main physical quantities characterizing an optical field. The intensity of optical radiation over some wavelengths of interest can be taken by panchromatic cameras. The spatial intensity over a number of wavelengths can be captured by imaging spectrometers. While the spectrum provides information on the distribution of material components, the vector nature of the optical field, i. e., polarization, identifies the object shape, shading, surface features and roughness in a scene. The information provided by polarization is largely uncorrelated with spectral and intensity images. Thereby, polarization is often used to enhance the contrast of the optical field [1–3].
An imaging spectropolarimeter detects three-dimensional spatio-spectral datacube for each of the Stokes parameters, simultaneously . Typically, such a detection requires scanning specific domains, such as the spatial domain in channeled spectropolarimetry , the optical path difference domain in Fourier transform imaging spectropolarimetry , and polarization domain in Stokes imaging spectropolarimetry . Recently, some systems have been developed to eliminate the scanning requirements. Some of them are implemented by direct measurement such as integrating integral field spectrometry with division-of-aperture imaging polarimetry . Others are implemented by computational strategies including combining channeled spectropolarimetry with computed tomography imaging spectrometry , combining channeled spectropolarimetry with image mapping spectrometer , or combining imaging spectrometry with compressive sensing imaging . However, the channeled spectropolarimetry based systems have limitations such as noise sensitivity, channel cross-talk, and spectral resolution limitations which are mainly introduced by the processing based on Fourier tansform [11, 12]. The spatial resolution is highly decreased in the systems based on the division of aperture or focal plane array .
Compressive sensing is an efficient signal measurement strategy that enables accurate signal reconstruction from coded projections at rates far lower than those required by the Nyquist-sampling theorem . In the field of imaging, the sub-Nyquist condition is achieved by exploiting the sparsity property of natural images. Compressive sensing indicates that it is feasible to reconstruct the underlying images from a relatively small collection of well-chosen measurements, generally by an iterative algorithm. According to the sparsity assumption, when images are expressed in a proper function basis, most coefficients are negligible or zero-valued. Recently, Ma et al. first proposed and proved an adaptive data-driven compressive sensing framework, to effectively improve the compression efficiency compared to the traditional compressive sensing approaches with random projections , and applied the compressive sensing methods to solve the computational lithography problems [16, 17]. Based on the compressive sensing theory, a coded aperture snapshot spectral imager (CASSI) was developed, where real-time spectral imaging was achieved through the use of a block-unblock coded aperture and a dispersive element . To overcome the drawbacks of channeled spectropolarimetry based systems, a compressive sensing based reconstruction method was developed for a channeled linear imaging polarimeter and a linear channeled spectropolarimeter [19, 20]. However, neither system can acquire spatial, spectral and full polarization information simultaneously.
Inspired by compressive sensing, a single-pixel polarimetric imaging spectrometer was designed by combining the single-pixel camera and a linear polarizer . A sequence of measurements must be taken for the successive configurations of the analyzer. A compressive snapshot color polarization imager that encodes spatial, spectral, and polarization information using a liquid crystal modulator was developed to capture the spatial distribution of four polarization states . Only three color channels were acquired by using a Bayer filter. Other imaging spectropolarimeters were developed based on CASSI [10, 22–25]. A coded aperture snapshot spectral polarization imaging was developed through replacing the dispersion prism in CASSI by a birefringent crystal pair to generate polarization-selective displacement and wavelength-dependent dispersion to all of the propagating images . In this approach, only the first two linear polarization states, S0 and S1, could be reconstructed. A compressive spectral polarization imager for spectral linear polarization imaging was developed by rotating the dispersion prism in CASSI and adding a micropolarizer array to match the colored detector [23, 24]. Combining the channeled spectropolarimetry and CASSI, a compressed channeled imaging spectropolarimeter was developed to acquire the spatial, spectral, and complete Stokes parameters information of the targets . The polarization modulation is achieved by encoding each band with a constant which limits the reconstruction quality.
In this work, we present a compressive imaging system, referred to channeled compressive imaging spectropolarimeter (CCISP), which is capable of providing the multidimensional information (two-dimensional spatial (x,y), one-dimensional spectral (λ), and four-dimensional polarization (S0, S1, S2, and S3) information) of the light reflected by an object. The channeled spectropolarimetry and CASSI schemes are both combined into this system. The integration of polarizer filters wheel and channeled spectropolarimetry implemented efficient encoding in the polarization domain. The feasibility is verified via the simulation and laboratory experiment. The system model is established according to the sensing procedure. The polarization and spectral modulation introduced by channeled spectropolarimetry are integrated into the linear system model. Thereby, the reconstruction is readily implemented by compressive sensing reconstruction methods. Due to the absence of Fourier transform and spatial filtering, the limitations in the traditional channeled spectropolarimetry based imaging spectropolarimeters are eliminated.
2. Channeled compressive imaging spectropolarimeter
A schematic of CCISP is shown in Fig. 1. The light reflected from the object is first directed to the collimating lens L1. The parallel wavefront passes through a polarizer filters wheel (PFW) with 5 filter positions. In the first position, there is no polarizer. Thereby, the light remains unchanged while passing through this position. Different polarizers are used in the other four positions sequentially. The angles between the axis of the four polarizers and the x axis of system are 0°, 45°,90°, and 135°, respectively. R1 and R2 are two high-order retarders with thicknesses t1 and t2 followed by an analyzer P1. The orientation of the retarder’s fast axes relative to the transmission axis of the analyzer are 0° and 45° for R1 and R2, respectively. The combination of R1, R2, and P1 is referred to a channeled spectral modulator . The modulated light passes through the relay lens L2 and is then modulated by the coded aperture (CA). The spatial encoded light is re-imaged by the imaging lens L3 and dispersed by the double Amici prism (DAP). The two-dimensional coded projection image is then captured by the camera.
As an example, the data modulation procedure while PFW rotates to position 1 is illustrated in Fig. 2. The datacubes S0, S1, S2, and S3 denote the desired spatial, spectral, and full Stokes polarization information. The four datacubes are modulated by the wavelength-dependent constants introduced by the channeled spectral modulator. That is, each band of the four datacubes is multiplied by a constant. The spectral modulated datacubes are denoted as S0M, S1M, S2M, and S3M, respectively. When the wavefront hits the CA, the sum of these four modulated datacubes SM is then modulated by a binary mask in the spatial domain. The spatial encoded datacube is denoted as SCA. When the coded wavefront passes through the DAP, a spatial wavelength-dependent shift occurred due to the dispersion. The dispersed datacube is denoted as SD. Sum up SD along the wavelength, a two-dimensional coded projection image I is obtained.
3. System model
As the schematic diagram of CCISP is shown in Fig. 1, the objective is denoted as a four-dimensional datacube , where x and y represent the two spatial dimensions, λ is the wavelength, and represents the index of different Stokes parameters. The intensities of the wavefront passing through the PFW and channeled spectral modulator is given by Eq. (2), the polarization modulation parameter with respect to Sp while PFW is rotated to the position is denoted as .
Similar to CASSI, the measured image on detector while PFW rotates to the position can be expressed as Eq. (3) is given by Eq. (4) is expressed as Eq. (5) couldbe rewritten as
Thereby, the system could be modeled as
Concisely, Eq. (9) can be rewritten as
To visualize the measurement matrix H, a datacube with spatial resolution of 4 × 4 and 4 spectral bands is considered. The transmittance is given as 1. The coded aperture is generated as random binary 2-D matrix. The polarization modulation parameters depicted in Eq. (2) can be determined by Fig. 3. The figures of the measurement matrices are shown as a table with 5 lines and 4 columns. The figures shown in columns 1 to 4 represent the measurement matrices of S0, S1, S2, and S3, respectively. Correspondingly, the measurement matrices shown in lines 1 to 5 display the measurement matrices while the PFW is rotated to different positions. Only at position 1, all four Stokes parameters are modulated. While the PFW is rotated to the other four positions, there are only two of the four Stokes parameters being measured. The first Stokes parameter S0, which represents the total intensity of light, is detected in every measurement. The second Stokes parameter S1, which denotes the difference between horizontal and vertical linear polarization components, is detected while PFW is rotated to positions 2 and 4. The third Stokes parameter S2 , which denotes the diagonal linear polarization components, is detected while PFW is rotated to the positions 3 and 5. The Stokes parameter S3, which represents the circular polarization component, is encoded while the PFW is rotated to the position 1.
4. Reconstruction algorithm
The traditional linear inversion methods, such as pseudo-inverse or least-squares, are ineffective in reconstructing the object since the system equations in the measurements are ill-conditioned or underdetermined. One must apply additional regularizing constraints in order to obtain useful solutions. Compressive sampling, which can be used with the sparse approximation conception, allows one to estimate the object according to given fewer number of measurements than the traditional Nyquist required. To obtain the sparse solutions, many methods such as two-step iterative shrinkage/thresholding (TwIST) and gradient projection for sparse reconstruction (GPSR) algorithms were proposed over the past decade [27, 28]. TwIST usually uses regulation functions such as L1-norm or total variation to constrain the estimation. TwIST provides the visually pleasing reconstructions while being an effective algorithm that solves the Lagrangian unconstrained formulation of constrained optimization problems. In this paper, we use the TwIST algorithm to reconstruct the Stokes parameter images. According to TwIST, the estimation of S can be represented as10].
In this section, we use a polarization dataset obtained by direct measurement to verify the feasibility of CCISP. Four datacubes are used in the simulation with size of corresponding to the polarization components I0, I45, I90, and I135, respectively. The Stokes parameters S0, S1, S2, and S3 are obtained by 30]. In the simulation, the circular polarization Stokes parameter is assumed as
The false color images shown in Figs. 4(a)–4(e) correspond to the measurements while the PFW are located in positions 1 to 5. The reconstruction process relies on the TwIST algorithm .This method is widely used for reconstruction of hyperspectral datacubes because of its efficiency, but other methods such as SALSA  and approximate message passing algorithms  can be utilized. The reconstructed results of S0, S1, S2, and S3 are shown in Figs. 4(f)–4(i), respectively. To quantitatively evaluate the reconstructed results, the peak signal to noise ratio (PSNR) of each band with respect to different Stokes parameters is calculated . It is found that most of the reconstructed images are accurate as shown in Table 1. According to the simulations, CCISP is an efficient approach to attain the spatial, spectral, and full Stokes polarization information, simultaneously.
6.1. Hardware implementation
Figure 5 depicts the experimental prototype of CCISP developed at the computational imaging and spectroscopy laboratory at the University of Delaware. L1 is an achromatic lens (Thorlabs, AC254-050-A-ML). PFW is a setup modified from the filters wheel (Thorlabs, FW102CWNEB). R1 and R2 are the custom-made calcite retarders. The thicknesses of R1 and R2 are 8 mm and 4 mm, respectively. P is a polarizer (Thorlabs, LPVISE100-A). The relay lens L2 is composed by two lenses (Thorlabs, AC254-100-A-ML). The coded aperture is implemented by a DMD (Digital Mirror Device, Discovery 1100) which includes 1024 × 768 elements of random binary pattern with mirror pitch. The imaging lens L3 is composed by two lenses (Thorlabs, AC254-100-A-ML). Double Amici prism is a custom-made component with a central wavelength of 550 nm. The measured image is taken by a CMOS camera (Ximea, MQ042RG-CM) with pixel size of 5.5 and resolution of 2048 × 2048.
In CCISP, the reconstruction performance is related to the modelling of the measurement matrix H. Thus, H is calibrated experimentally, which considers errors such as the aberrations caused by the optical components and the sub-pixelmisalignment. The calibrations include the spectral and polarization mapping as detailed next.
The spectral calibration is implemented in two steps. In the first step, the wavelength and shifting positions are determined. The target is replaced by a standard whiteboard which is illuminated by a ring illuminator. The source is a monochromator whose wavelength range is from 450 to 700 nm. The wavelength interval is 5 nm. Then, we identify the central column indexes of the coded aperture images with different wavelengths. A fitted equation between the position index c and wavelength λ is achieved by least square method. According to the fitted equation, we can get the position of each wavelength from 450 to 700 nm with intervals of 1 nm. The central wavelength is selected as the wavelength corresponding to the position index. The variation of the position with wavelength represents the dispersion effects of prism. The dispersion curves are shown in Fig. 6. It can be seen that 12 wavelengths are selected out to be detected. Also, the response curve also is measured in this procedure.
Accordingly, the central wavelengths corresponding to different position indexes could be determined. The coded aperture images at different wavelengths can be obtained. In order to eliminate the random noise, 10 coded aperture images were measured and averaged for each wavelength. In this procedure, the exposure time is constant. A coded aperture image of a white noise mask at wavelength 636 nm is shown in Fig. 7.
The polarization calibration is used to get the wavelength-dependent parameters in Eq. (2). Traditionally, a Fourier transform based method is used to determine these parameters . In the calibration, the key problem is to determine the values of ϕ1 and ϕ2. However, the accuracy of a channeled spectropolarimeter would be sensitive to the error introduced by the angles of the high-order retarders [35–37]. In order to overcome this problem, we propose an approach without the need of a Fourier transform to get these parameters. Considering the non-ideal condition and Eq. (1), it can be obtained that
Figure 8 shows the polarization calibration system to measure the polarization modulation parameters introduced by the channeled spectral modulator. PBS is a polarized beamsplitter (Thorlabs, CCM1-PBS251); P1 and P2 are two polarizers; R1 and R2 are retarders; L1 is a collimating lens; FS is a fiber spectrometer (Ocean Optics USB2000+) which is located on the focal plane of L1. R1, R2, and P2 compose the channeled spectral modulator. The incident light is the collimated white light. PBS and P1 are installed on kinematic rotation mounts.
The calibration procedure is is accomplished by the following steps. Through rotating operation of PBS, various linear polarization light could be generated. By rotating P1 to generate different polarized light with polarization angles of 0, 45, 90 and 135 degrees, the linear components of Stokes vectors, , are measured. Six linear polarized lights with different states are generated by rotating the PBS six times. For each state, the Stokes vectors are obtained by rotating P1 while the channeled spectral modulator is removed. The modulated intensity is obtained while P1 is moved away. An equation corresponding to the linear polarized light is then obtained asEq. (16). Theoretically, ϕ2 can be obtained according to . In the calibration, we obtain the Hilbert transform of . Via unwrapping algorithm, we obtain the retardation phase . Also, the ideal values of ϕ1 and ϕ2 can be calculated by Eq. (11). In this system, the retarders are custom-made by quartz. The birefringence is 0.009 approximately. The ideal and retrieved phases are shown in Fig. 9. It can be seen that the two curves are not identical to each other. The difference is due to the birefringence dispersion and machining error.
6.3. Experiment results
As shown in Fig. 10(a), a target made up by a red plastic rod and six pieces of coloured chalk labelled as 1 to 6 is used. The plastic rod is labeled as 7. A polarizer film was placed next to the plastic rod. Figures 10(b)-10(f) are the measured images taken by CCISP, while the PFW was located in positions 1 to 5, respectively. Correspondingly, the polarization states of the polarizers are 0°, 45°, 90°, 135° and no polarizer (NP).
Using TwIST algorithm, the spectral and polarization information can be reconstructed. Figure 11 shows the reconstructed S0 with 12 bands between 450 and 700 nm. The intensities of S1, S2, and S3 are too weak to display using the same scale parameter as S0. In order to display them more visible, all the Stokes parameters are Gamma corrected and displayed as 3-D datacubes which are usually applied in hyperspectral remote sensing. Figures 11(c)-11(f) present the datacubes of S0, S1, S2, and S3, respectively. The x-axis and y-axis of the datacube are spatial domain. The z-axis is spectral domain. From the color bars of the datacubes, it can be found that the intensities of S1, S2, and S3 are much less than that of S0. It can be illustrated by the properties of the Stokes parameters. S0 represents the totally intensity of the light and includes both the polarized and unpolarized components. S1 is the difference of the horizontal and vertical polarized lights. S2 is the difference of the polarized lights at 45° and 135°. Both S1 and S2 represent the linear polarized components. S3 is the difference of the left and right circular polarized lights and represents the circular polarized component. The main component of the light reflected by the natural scene is unpolarized light. The measured target is a natural scene. Thereby, the unpolarized component is dominant. It should be noticed that the polarized film in the target is not readily seen in the measured imaged due to its darker color. Figures 11(b) shows the spectral profiles of the red chalk. The blue and red curves denoted as SFS and represent the spectral profiles sensed by FS and CCISP, respectively. It is found that the two spectral profiles are very similar to each other. The difference between them is introduced by the reconstruction and calibration.
In order to quantitatively evaluate the performance of the reconstructed spectral profiles, we computed the correlation coefficient R between the spectral profiles sensed by FS and CCISP. For each chalk, we evaluated the spectral profiles of 100 points for each Stokes parameter. For the plastic rod, the spectral profiles of 25 points were evaluated. The evaluation result is shown in Table 2. It can be noticed that most of the reconstructed spectral profiles of S0
are characterized by a better quality. For the reconstructed S1, S2, and S3, the spectral profiles of most points have good quality. Overall, the correlation coefficients of S1, S2, and S3 are lower than those of S0. This can be the result that S0 is considerably larger than the other three Stokes parameters.
The degree of polarization (DoP) and angle of polarization (AoP) are defined as
The DoP and AoP images with 12 bands obtained based on Eq. (17) are shown in Figs. 12(a) and 12(b), respectively. The datacubes of AoP and DoP are shown in Figs. 12(c) and 12(d), respectively. The regions covered by the polarizer film, without polarizer film, and edges among different parts are distinguished from images shown in Fig. 12. The region covered by the polarizer film is characterized as negative AoP and larger DoP. The region without polarizer film is characterized as small positive AoP and DoP. The region corresponding to the edge is characterized as bigger AoP and medium DoP. The polarizer film which is invisible in the spectral images can be identified in the AoP and DoP images. It illustrates that polarization information provides us a more powerful tool to discriminate surface information such as edges and roughness than simple spectral imagery.
A channeled compressive imaging spectropolarimeter system based on CASSI and channeled spectropolarimetry is developed in this work. CCISP presents an integration of multi-dimensional information (spatial, spectral, and polarization). The polarization encoding is implemented by the channeled spectral modulator, which encodes the full Stokes parameters into one output. The polarization encoded light is sensed by CASSI, which compressively measures the spectral images. A polarizer filters wheel is introduced to change the polarization states of the incident light fields. Compared with traditional imaging spectropolarimeters based on channeled spectropolarimetry, a compressive sensing based linear model is established. The reconstruction is implemented by reconstruction algorithms such as TwIST. Fourier transform and spatial filter are not required. Thereby, the limitations existing in the traditional imaging spectropolarimeters based on channeled spectropolarimetry such as noise sensitivity, spectral resolution reduction, and channel crosstalk are eliminated in CCISP. In addition, a polarization calibration to determine the parameters used in the measurement matrix has been developed without a Fourier transform. Acquisition of the full Stokes parameter images with 12 bands in the range of 450 to 700 nm using CCISP is demonstrated. The feasibility has been verified by the simulation and experiment.
China Scholarship Council (201706305022); National Natural Science Foundation of China (11504297, 61673314, 11664004, 41605118, 41605118); Ministry of Science and Technology of the People’s Republic of China (2017YFC0403203); Northwest A&F University (2452015225, 2452015226, 2452017168,2452017166, Z109021504, Z109021508); Natural Science Foundation of Shaanxi Province (2016KTZDGY05-02); Guangxi Teachers Education University (2015GXESPKF03).
All the authors thank Xiao Ma, Tianyi Mao, Angela Cuadros, Carlos Mendoza, Edgar Salazar, Juan F. Florez, and Karelia Pena who provided constructive suggestions on our work.
2. J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013). [CrossRef]
4. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Optics Letters 24, 1475–1477 (1999). [CrossRef]
6. V. C. Chan, M. Kudenov, C. Liang, P. Zhou, and E. Dereniak, “Design and application of the snapshot hyperspectral imaging fourier transform (shift) spectropolarimeter for fluorescence imaging,” Proc. SPIE 8949, 894903 (2014). [CrossRef]
7. T. Mu, C. Zhang, Q. Li, Y. Wei, Q. Chen, and C. Jia, “Snapshot full-stokes imaging spectropolarimetry based on division-of-aperture polarimetry and integral-field spectroscopy,” Proc. SPIE 9298, 92980D (2014).
8. D. S. Sabatke, A. M. Locke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. K. Hamilton, and R. W. McMillan, “Snapshot imaging spectropolarimeter,” Optical Engineering 41, 1048–1055 (2002). [CrossRef]
9. R. T. Kester and T. S. Tkaczyk, “Image mapped spectropolarimetry,” US Patent 9,239,26319 January (2016).
11. A. M. Locke, D. Salyer, D. S. Sabatke, and E. L. Dereniak, “Design of a swir computed tomographic imaging channeled spectropolarimeter,” Proc. SPIE 5158, 507136 (2003).
12. E. L. D. Julia, M. Craven, and Michael W. Kudenov, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 7419 (2009).
13. N. A. Hagen, L. S. Gao, T. S. Tkaczyk, and R. T. Kester, “Snapshot advantage: a review of the light collection improvement for parallel high-dimensional measurement systems,” Optical Engineering 51, 111702 (2012). [CrossRef] [PubMed]
14. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Magazine 25, 21–30 (2008). [CrossRef]
16. X. Ma, Z. Wang, H. Lin, Y. Li, G. R. Arce, and L. Zhang, “Optimization of lithography source illumination arrays using diffraction subspaces,” Optics Express 26, 3738–3755 (2018). [CrossRef] [PubMed]
17. X. Ma, Z. Wang, Y. Li, G. R. Arce, L. Dong, and J. Garcia-Frias, “Fast optical proximity correction method based on nonlinear compressive sensing,” Optics Express 26, 14479–14498 (2018). [CrossRef] [PubMed]
19. D. J. Lee, C. F. LaCasse, and J. M. Craven, “Compressed channeled linear imaging polarimetry,” Proc. SPIE 10407, 104070D (2017).
21. F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, “Single-pixel polarimetric imaging spectrometer by compressive sensing,” Applied Physics B 113, 551–558 (2013). [CrossRef]
23. C. Fu, H. Arguello, G. R. Arce, and V. O. Lorenz, “Compressive spectral polarization imaging,” Proc. SPIE 9109, 91090D (2014).
24. C. Fu, H. Arguello, B. M. Sadler, and G. R. Arce, “Compressive spectral polarization imaging by a pixelized polarizer and colored patterned detector,” Journal of the Optical Society of America A 32, 2178–2188 (2015). [CrossRef]
25. G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Processing Magazine 31, 105–115 (2014). [CrossRef]
26. W. Ren, C. Fu, and G. R. Arce, “The first result of compressed channeled imaging spectropolarimeter,” in Applied Industrial Optics: Spectroscopy, Imaging and Metrology, (Optical Society of America, 2018), pp. JTu4A.
27. M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE Journal of Selected Topics in Signal Processing 1, 586–597 (2007). [CrossRef]
28. J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing 16, 2992–3004 (2007). [CrossRef] [PubMed]
29. D. H. Goldstein, Polarized Light, 3rd Edition (CRC, 2003). [CrossRef]
30. G. W. Kattawar, “A search for circular polarization in nature,” Optics and Photonics News 5, 42–43 (1994). [CrossRef]
31. M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Transactions on Image Processing 19, 2345–2356 (2010). [CrossRef] [PubMed]
32. J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive hyperspectral imaging via approximate message passing,” IEEE Journal of Selected Topics in Signal Processing 10, 389–401 (2016). [CrossRef]
33. Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electronics Letters 44, 800–801 (2008). [CrossRef]
34. J. Craven-Jones, M. W. Kudenov, M. G. Stapelbroek, and E. L. Dereniak, “Infrared hyperspectral imaging polarimeter using birefringent prisms,” Applied Optics 50, 1170–1185 (2011). [CrossRef] [PubMed]
38. F. W. King, Hilbert transforms, vol. 1 (Cambridge University, 2009).
39. J. Novak, “Computer analysis of interference fields using matlab,” in MATLAB conference, (2002), pp. 406–410.