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Abstract
An achromatic snapshot full-Stokes imaging polarimeter (ASSIP) that enables the acquisition of 2D-spatial full Stokes parameters from a single exposure is presented. It is based on the division-of-aperture polarimetry using an array of four-quadrant achromatic elliptical analyzers as polarization state analyzer (PSA). The optimization of PSA is addressed for achieving immunity of Gaussian and Poisson noises. An extended eigenvalue calibration method (ECM) is proposed to calibrate the system, which considers the imperfectness of retarder and polarizer samples and the intensity attenuation of polarizer sample. A compact prototype of ASSIP operating over the waveband of 450-650 nm and an optimized calibration setup are developed. The achromatic performance is evaluated at three bandwidths of 10, 25, and 200 nm, respectively. The results show that the prototype with an uncooled CMOS camera works well at each bandwidth. The instrument matrix determined at the narrower bandwidth is more applicable to the wider one. The uncertainties of the calibrated instrument matrices and reconstructed Stokes parameters are improved by using the extended EMC at each bandwidth. To speed up the acquisition of high-contrast images, wide bandwidth along with short exposure time is preferable. The snapshot capability was verified via capturing dynamic scenes.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Full-Stokes imaging polarimetry is a powerful tool to capture the states of polarization (SOPs) across a 2D scene and thus to characterize targets’ surface features, shape, shading, and roughness with enhanced contrast [1,2]. According to temporal resolution, the full-Stokes imaging architectures are divided into two types: time-sequential and single-shot. The time-sequential system usually employs polarization switching elements to implement a set of measurements in sequence. They may not adapt to the dynamic objects or the static objects that located at rapidly changing environments. Even more, the working platform of the system should be stationary during measurements. In contrast, the single-shot system performs all measurements in parallel from a single exposure [3,4]. It can avoid mechanical disturbance and temporal registration error that may appear in time-sequential systems. Currently, there are four single-shot techniques: division of focal plane (DoFP) [5–10], division of amplitude (DoAM) [11], division of aperture (DoA) [12,13], and channeled imaging polarimetry (CIP) [14]. In this paper, we focus on the DoA technique with a single focal plane array (FPA). No matter what kind of technique one takes, polarization state analyzer (PSA) is a key component for polarization modulation and analysis. The number of analysis states of the PSA must be no less than four to reconstruct the full Stokes parameters (S0, S1, S2, S3) with inverse or pseudoinverse estimators. However, to maximize the spatial resolution of each sub-aperture projected on the FPA, the PSA with four-quadrant sub-apertures must be the optimal choice.
The question is what configuration of four analysis states is optimal [15]. As we all know, there are kinds of noise: Gaussian noise, Poisson noises or compound noises, frequently encountered in passive optical imaging systems. The noises influence the precision of the measured four Stokes parameters [16,17]. Especially, for the single-shot system that works at video rate in real time, multiple frames cannot be acquired within a single exposure time. It is not applicable to average multiple frames at post processing for the reduction of noise. The consequences of Poisson or Gaussian noises are that the sum of the noise variances on the four estimated Stokes parameters is not minimized, and the noise variances on each of the last three Stokes parameters are not equalized. Another consequence of Poisson noise is that the noise variances on each estimated Stokes parameter is not independent of input SOPs. Therefore, it is challenged to acquire high signal-to-noise ratio Stokes polarization images and fairly compare the practical proportion of the last three Stokes parameters to the first one. The high signal-to-noise ratio is particularly significant in applications requiring precise polarization measurements and is also important in the acquisition of high-quality digital images. Fortunately, optimized PSA can make the single-shot system robust to diverse scenes or environments. Although several single-shot systems based on DoA technique have been developed, but some systems have one or more limitations [18]: only linear Stokes parameters are analyzed [19], noise immunity is not considered [20], and the noise perturbation is not considered during polarimetric calibration [19,20].
Polarimetric calibration is the most vital step to vitalize the rough system in which the instrument matrix of the system must be determined unambiguously before applications. However, the calibration measurements are also inevitably perturbed by the noise, and the estimation precision of instrument matrix would be degraded by the noisy data. The calibration methods can be classified into component-wise parameterization method [21] and modular accumulation method [22,23]. While the former needs high-quality parametrizable and characterized components, the latter considers the system as a black box and automatically accounts the systematic errors and the effects of imperfect components. The modular accumulation method includes two methods: Pseudo-inverse Calibration Method (PCM) [22] and Eigenvalue Calibration Method (ECM) [23]. The PCM usually needs a standard polarization state generator (PSG) to produce a set of reference polarization states. In our opinion, this method only adapts to narrowband full-Stokes non-imaging systems or broadband linear-Stokes imaging systems. For the calibration of broadband full-Stokes imaging system, it is hard to get a retarder with well-known retardances over a wide waveband across a large field of view (FOV). Thus, the generation of accurate elliptical polarization states is impossible. In contrast, the ECM is robust to calibrate such system. Although the ECM needs both the PSG and reference samples, the actual values of them can be determined accurately by calibration itself. No precise positioning systems or specific optical elements (such as exact quarter- or half-wave plates) are needed for the samples. Furthermore, if the PSG is optimized, the estimation precision of the instrument matrix will be enhanced. However, the use of ideal models of retarder and polarizer and the neglected intensity attenuation of polarizer may degrade the calibration accuracy and precision.
In this paper, an achromatic snapshot full-Stokes imaging polarimeter (ASSIP) based on the DoA technique is presented. We aim to optimize the four analysis states of the four-quadrant PSA and let them has the immunity to both Gaussian and Poisson noises. An extended ECM is proposed to calibrate the instrument matrix. The extended ECM uses the general Muller matrix models to characterize the imperfect performance of the polarizer and retarder samples, such as the residual diattenuation, residual phase delay, and depolarization. The intensity attenuation induced by the polarizer sample is also considered. The designed achromatic snapshot system is calibrated and evaluated at different bandwidths of 10, 25, and 200 nm, respectively. As far as we know, ASSIP is the first achromatic snapshot polarimeter which was optimized to both Gaussian and Poisson noises. In Section 2, the general principle of the optimized system is presented. Section 3 provides the theory of extended ECM. Section 4 describes the imaging and calibration setups followed by calibration results in Section 5. The imaging results are presented in Section 6, and the conclusion in Section 7.
2. General principle
2.1. Snapshot full-Stokes imaging polarimeter
The general principle of ASSIP system is depicted in Fig. 1. It is straightforward. Target is first imaged by an objective lens (OL) onto an intermediate image plane. A field stop (FS) is placed at the plane to limit the FOV of system. The rays from the FS is then collimated by a collimating lens (CL). The collimated light rays are analyzed simultaneously by a four-quadrant full-Stokes PSA located at the pupil plane. The PSA includes a retarder array (RA) followed a horizontal linear polarizer (P). The rays from the four sub-apertures are filtered by a bandpass filter and imaged by a four-quadrant lens array (LA) onto the FPA. The separately four-quadrant polarization sub-images are obtained directly in a single exposure time. The 2D-spatial distribution of the four Stokes parameters are then reconstructed with data-reduction matrix. The spatial resolution obtained in the system depends on the size of the FPA.
Fig. 1 Optical layout of a snapshot full-Stokes imaging polarimeter based on division-of-aperture technique. Abbreviations: OL objective lens, FS field stop, CL collimating lens, RA retarder array, P polarizer, BF bandpass filter, LA lens array, FPA focal plane array.
2.2. Optimization of the four-quadrant PSA
The number of retarders in each sub-aperture of the RA can be diverse as shown in Fig. 2, such as each sub-aperture consists of a single retarder, or two retarders, or hybrid. We ever found that, for the DoA technique, each sub-aperture consists of a single retarder in Fig. 2(a) would be less sensitive to the systematic error such as the misalignment of fast-axis azimuths and the deviation of retardance [24]. Therefore, we employ the RA architecture in Fig. 2(a) as the four-quadrant PSA. The four analysis states of the PSA, determined by the fast-axis azimuths θi and retardances δi of retarders, influence the estimation precision of the noisy four Stokes parameters. The optimal set of fast-axis azimuths and retardances should be determined first.
Fig. 2 The architecture of the four-quadrant RA. (a) Each sub-aperture consists of a single retarder; (b) each of two sub-apertures consists of a single retarder, and each of the other two sub-apertures consists of two retarders; (c) each sub-aperture consists of two retarders.
The linear measurement model of a general Stokes polarimeter is
where S denotes the 4 × 1 incident Stokes vector with four parameters Sk (k = 0, 1, 2, 3), I represents the 4 × 1 measured intensity vector with elements Ii (i = 1, 2, 3, 4), A indicates the 4 × 4 instrument matrix with elements Ai,k. The incident Stokes vector can be reconstructed with estimatorwhere B is the inverse matrix of A, also termed the data-reduction matrix. If the measured intensities are perturbed by signal-independent Gaussian noise with variance σ2, the noise variance on each estimated Stokes parameter will beThe Gaussian variances of the four Stokes parameters also are independent of the incident signal. The method to minimize and equalize noise variances is to optimized the instrument matrix A and thus the data-reduction matrix B. The frequently-used factors of merit are the L2-norm condition number (CN2) [25], the equally weighted variance (EWV) [26] and the polarization modulation efficiency (PME) [27]. However, with these factors of merit the optimized instrument matrices are diverse [28], because there are multiple selections for the fast-axis azimuths θi and retardances δi. Table 1 lists several optimal sets of (θi, δi). As seen, the four retarders can have the same retardance in the sets (i) or (ii), or the two different retardances in the sets (iii) or (iv), or the three different retardances in the set (v), or the four different retardances in the set (vi). Correspondingly, the tetrahedrons formed by the last three columns of the normalized A within a unit Poincaré sphere are, respectively, depicted in Fig. 3. Each trajectory on the sphere represents a retardance δi with the rotated fast-axis azimuth. The vertices of tetrahedrons distribute on the trajectories [26]. The orientations of tetrahedrons are diverse, meaning there are many different solutions. Although one of solutions also has immunity to Poisson noise, it is hard to find it unambiguously just using the above factors of merit.
Table 1. Optimal sets of the fast-axis azimuths θi and retardances δi, determined with the CN2, and the corresponding values of CN2 and BCPN of the instrument matrix A.
Fig. 3 The tetrahedrons formed by the last three columns of the normalized A within a unit Poincaré sphere and the A is calculated from the optimal set in Table 1. Each trajectory on the sphere represents a retardance δi with the continuously rotated fast-axis azimuth. The vertexes of tetrahedrons correspond to the optimal retardances and optimal fast-axis azimuths.
If the measurements are affected by the signal-dependent Poisson noise, the noise variance on each estimated Stokes parameter will be [29,30]
where and , P indicates the degree of polarization (DoP), and s denotes the normalized Stokes vector formed by the last three Stokes parameters. Contrary to the Gaussian variances, the Poisson variances depend on the incident signal P. To let the variances be independent of P, and to minimize and equalize the variances, we ever proposed the optimally balanced condition for Poisson noise (BCPN) [31],where N is the number of the analysis states. The first term roughly determines the optimal solution for Gaussian noise, analogous to the CN2, EWV or PME. The second term extracts the specific solution for Poisson noise. Using the BCPN [31], the optimal set (iv) in Table 1 can be unambiguously selected, and the corresponding instrument matrix A is [30–32]This matrix has the immunity to both noises or their combination. As a result, the Gaussian variances become where νk = 1 if k = 0 and νk = 3 if k = otherwise; the Poisson variances become Note that has the same expressions as , only with variance σ2 replaced by S0/2. The total variance is minimized, and the last three variances are equalized. Therefore, the optimal set (iv) in Table 1 is employed to design the four-quadrant RA. Since the four vertices in Fig. 3(d) represent four elliptical states, the four-quadrant polarization modulator is equivalent to an array of elliptical analyzers.3. Polarimetric calibration model
3.1. ECM model
The measurement configurations involved in the EMC method are usually classified into two types: reflection and transmission [23]. The transmission-limited configuration is suitable for the calibrations of Stokes polarimeters and transmission-type Mueller matrix polarimeters as shown in Fig. 4. It consists of a light source, a complete PSG, samples, a complete PSA, and detector. Theoretically, a set of samples, including air, retarder, and polarizer, should be employed. In the absence of noise during calibration, only the choice of samples affects the accuracy of calibration for any PSG and PSA. However, in the presence of noise, not only the optimal samples but also the well-balanced PSG and PSA are needed to ensure the accuracy and precision of calibration. With the well-balanced 4 × 4 matrices W and A, De Martino et al [33] numerically determined a suitable set of reference samples that includes air (M0), a linear polarizer (M1, M2) set at two orientations (θ1 = 0° and θ2 ≈90°), and a retarder (M3) with any retardance Δ within the region of 110° ± 30° and a fast-axis azimuth of θ3 ≈30°. Wherein, except θ1 should be equal to zero accurately, no precise positioning systems or specific optical elements are needed for other parameters (θ2, θ3, Δ).
The first three necessary steps of ECM are as follows. (i) A set of measurements are implemented as
with i = 1, 2, 3, η0 and ηi indicate the intensity of source for each sample. (ii) Then the auxiliary matrix is calculated as(iii) Since the matrix Ti and Mi have the same eigenvalues λj (j = 1, 2, 3, 4), the feature parameters (τp, τR, ϕ) of Mi are derived and added to the ideal Mueller matrix models of an unrotated polarizer and retarder, respectively, aswhere τp and τR are transmissions, and ϕ is retardance. The successive steps are to use these models to find W from Eq. (8) and then A from Eq. (7). As seen, the key step is to get the accurate feature parameters of the samples, since their precisions determine that of W and A.3.2. The general model of the sample’s Muller matrix
The samples’ Muller matrices in Eq. (9) are ideal models. Practically, the polarizer sample may have residual diattenuation and induced retardance, the retarder sample may have the dichroism on its fast and slow axes, and the retardance of achromatic retarder is not a constant over the waveband and FOV. The ideal models in Eq. (9) cannot be directly adapted to these kinds of imperfect components. To improve calibration precision, a generalized model that accounts for imperfect effects is proposed
where G = P or R. For the ideal polarizer: τP = 0.5, aP = 1, and bP = cP = 0. For the practical polarizer, the appearance of b and c means induced retardance, and a ≠ 1 means residual dichroism. For the ideal retarder: τR = 1, aR = 0, bR = cosϕ, and cR = sinϕ. For the practical retarder, the appearance of a means potential dichroism, b2 + c2 ≠ 1 means the retardance is compound. Correspondingly, these parameters are derived from the eigenvalues of Ti as where imag denotes imaginary term, λr1, λr2, λc1, and λc2 are the real and complex eigenvalues, with λr1 > λr2 and imag (λc1) > imag(λc2). The generalized model just for the polarizer was ever theoretically proposed for calibrating Muller polarimeter [34], however the performance enhancement based on this model has not been validated. In Sections 5.2 and 6.1, the improvement based on the general models of both polarizer and retarder will be presented.3.3. Illumination intensity
The calibration measurements in Eq. (7) will also be affected by noise as
where n represents noise matrix. As a result, the auxiliary matrix becomesThe derived eigenvalues will deviate from that of Mi, and thus influence the precision of the recovered W and A. Ideally, the air and the retarder samples have a unit transmission, and the polarizer sample has a transmission of τP = 0.5. So, the measured intensities of the air and retarder samples are 2 times larger than that of the polarizer sample at the same integration time. Further, if the dichroic polarizer sample is used, the measured intensity will be weaker (such as τP = 0.3) due to its additional absorption. Correspondingly, the noise influence on the measurement of polarizer sample would become obvious. The theoretical analysis of noise errors is beyond the scope of this paper. We propose two ways to suppress the noise effect on the polarizer sample: increasing integration time or intensifying illumination. Relatively, the latter is preferable, since it will not change the dark noise and read noise of the detector. In Section 5.2, we will let η1,2 be larger than 2η0 and thus let the . The performance improvement also will be presented in Sections 5.2 and 6.1. For the convenience, the ideal model, the general model, and the general model with intensified intensity are termed as model #1, model #2, and model #3, respectively.3.4. Optimized PSG
Since the instrument matrix A is optimized, it is also necessary to build an optimal PSG W for suppressing calibration noise. We have proposed a PSG that consists of a horizontal linear polarizer P followed by the QWP1 and QWP2 in tandem [28]. The fast-axis azimuths of the two QWPs are also optimized using the BCPN in Eq. (5). An optimal set of fast axis azimuths (θ1, θ2) are ( ± 87.84°, ± 70.15°) and ( ± 19.14°, ± 42.82°). The resulted CN2 and BCPN for the PSG are similar to that of the set (iv) in Table 1, meaning that the PSG also has the immunity to both Gaussian and Poisson noises. We ever used this PSG configuration to achieve PCM calibration, and the noise perturbations were suppressed in the reconstructed Stokes parameters [22]. Therefore, this configuration is employed to implement extended ECM calibration.
4. Optical design
4.1. The ASSIP system setup
A photograph of the ASSIP prototype for general imaging application is shown in Fig. 5(a). The OL is a medium telephoto lens (Cannon, EF 100mm, f/2). This lens can be replaced with other objective and coupled to multiple imaging modalities, such as telescope, microscopes, ophthalmoscope, endoscopes, and others. The FS made of black anodized aluminum plate is placed at the intermediate image plane of the objective for spatial sampling. The CL is a standard photographic lens (Cannon, EF 50mm, f/1.4). The PSA consists of a custom four-quadrant achromatic RA followed by a dichroism polarizer P (Edmund, #85922). The nominal retardances and fast-axis azimuths of the RA are δ = (102°, 142°) and θ = ( ± 72°, ± 35°) over the waveband of 450-650 nm. The re-imaging LA is a 2 × 2 array in which each sub-aperture comprises of two VIS-NIR coated achromatic lenses. The effective focal length of the compound lens is 50mm, then the magnification between the CL and the LA is one. The FPA is a large format mono CMOS camera (JAI, #SP-20000-PMCL). The volume of ASSIP is 300 × 70 × 70 mm, and the weight is about 1 kg. The most space and weight are from the fore relay lenses which can be removed for infinite imaging applications.
Fig. 5 (a) A photograph of the ASSIP prototype which is assembled on a 60mm cage system. Abbreviations: OL objective lens, FS field stop, CL collimating lens, RA retarder array, P polarizer, LA lens array, AA aperture array. (b) The calibration setup for the ASSIP system.
4.2. The achromatic four-quadrant PSA
The used CMOS camera just has a well-depth of 104 which is an order of magnitude less than that of a typical astronomical CCD with a well-depth of 105 or more. Thus, the measurable polarization degree is first limited by the number of collected photons in the case of Poisson statistics [35]. However, the noise distributions may not only obey Poisson statistics but also Gaussian statistics, which will degrade measurable precision further. To let the PSA immune to both noises over a wide waveband, each retarder is achromatized using true zero-order quartz and MgF2 waveplates with orthogonal azimuths on a fusion quartz substrate. The resulted achromatic retarders have a retardance variation of ± 2.5° for the 102° retarder and ± 3.6° for the 142° retarder over the waveband. The larger deviation appears for 142° retarder due to the accumulation of thickness. Further, the retardance also varies with the ASSIP’s FOV. The accumulated retardance variations would make the practical instrument matrix A deviating from the ideal one in Eq. (6). As a result, the noise immunity would be degraded. To evaluate the effects of such non-ideal achromatic retarders, the ASSIP will be analyzed with different bandwidths of 10, 25, and 200 nm at the same central wavelength of 550 nm. The linear polarizer (Edmund, #85922) is made of PMMA plastic makes it lightweight. It features an extinction ratio ≥ 9 × 103 and transmission ≥ 40% of unpolarized light over the waveband of 400 ~700 nm and across a FOV of ± 10°. Its working waveband and FOV are larger than that of the ASSIP system. The high extinction ratio is important for the achievement of noise immunity [12]. In addition, since the polarizer is fixed, the polarization diverse response of the COMS is avoided in contrast to the traditional solution using multiple polarizer orientations [19,20].
4.3. Lens array
The 2 × 2 re-imaging LA is assembled using the off-the-shelf VIS-NIR coated achromatic doublet (Edmund, #49333, 100mm F.L.). Each re-imaging lens is combined with the two achromatic doublets in tandem to form the effective focal length of 50mm. Comparing to a single achromatic doublet with the 50mm focal length, the compound lens can suppress aberrations across the FOV. The ray tracing of the compound lenslet is shown in Fig. 6(a). The lenslet is diffraction limited as the RMS spot sizes approach to the size of Airy disk in Fig. 6(b). The LA is supported using a custom black anodized aluminum plate with a hole array of 13 mm pitch. The imaging diameter of the reimaging lens is about 17 mm which overlaps with that of adjacent lens to maximize the usable COMS area. However, the current ASSIP prototype does not fully use the whole CMOS pixels. The square FOV of the system is about ± 3.4° and the instantaneous FOV (IFOV) is 3.6 × 10−3 deg. The optimal diameter of each sub-aperture is about 5mm and the resulted f-number of the reimaging lens is F/10. There is a trade-off between the FOV and f-number.
Fig. 6 The ray tracing of a compound re-imaging lens modeled using Zemax optics software. (a) The layout and (b) the spot diagrams at different FOVs and wavelengths.
4.4. Calibration setup
Figure 5(b) shows the calibration setup. An integrating sphere is illuminated by a stabilized tungsten lamp to produce unpolarized uniformly extended source. The PSG consists of a dichroic sheet polarizer (CVI, FPG-50.8-5.3) followed by two custom achromatic retarders AQWP1 and AQWP2. The polarizer features an extinction ratio ≥ 104 and about 30% transmission of unpolarized light over the waveband of 380 ~780nm and across the FOV of 90°.The retardances of both AQWPs also vary with the wavelength and FOV. The PSG must deviate from the nominal states. The polarizer sample is also a dichroic sheet polarizer (CVI, FPG-50.8-5.3). Its 30% transmission is far smaller than that of the retarder sample. As we stated in Sec. 3.3, the illumination intensity should be enhanced for the polarizer sample. The retarder sample is a standard zero-order precision QWP @ 633 nm (Meadowlark, NQ-200-633). Its retardance locates among the optimal values of 80 - 140 deg over the waveband of 450 - 650 nm. All components are mounted on manually rotatable mounts. These mounts are integrated into a compact cage system. Since the clear aperture of the cage system adapts to the system’s FOV, all pixels of the system can be calibrated in parallel.
5. Calibration
5.1. Image preprocess
Flat field and image registration should be implemented before polarimetric calibration. For flat field, the PSG and sample in Fig. 5(b) are temporally removed from the calibration system. First, a reference blank flat image is acquired when the system directly images the exit port of the integrating sphere. The four sub-images present diverse intensities, and there was critical vignetting at the marginal FOV due to the imperfect transmittance and mismatched aperture. In addition, there were parallaxes among the four sub-images. We cropped the common FOV (1200 × 1200 pixels) on each sub-image. Then, twenty dark images were acquired at the same exposure time. Subtracting the average dark frame and dividing the flat image by all subsequent measurements enabled normalization of the intensity.
Images registration is important and challenged for multiple-channel imaging polarimeters. The misregistration of sub-images are mainly contributed by the four-quadrant PSA and the LA. The fore relay optic OL and CL do not lead to mis-registration because they affect the four sub-images in the same way. To implement image registration, an unpolarized target with high frequency information was directly imaged by the system. One of sub-images is treated as the truth, and the other three images were adjusted to overlay the truth. A sub-pixel image registration algorithm, speeded up robust features (SURF) [36], was employed to register the images. Figure 7 shows the registration results for a resolution test target which was placed at the minimum working distance of 0.8 m. As seen, the original three sub-images (see Figs. 7(a)–7(c)) had different mis-registrations (see Figs. 7(e)–7(g)) relative to the fourth sub-image (see Fig. 7(d)) before registration, especially obvious at the edges. The translation misregistration was quantified via a single-step discrete Fourier transform algorithm that computes the cross-correlation of up-sampled images [37]. The maximum row and column translations were about 5.864 and 1.682 pixels. After registration, the row and column translations were reduced to 0.002 pixels as shown in Figs. 7(i)–7(k). The edge effects were almost disappeared. The amplitude of residual intensity difference (see Fig. 7(l)) was almost equal to that of noise perturbation, which only was a small proportion to the maximum intensity in Figs. 7(a)–7(d). Further, the image quality was acceptable across the FOV owe to the use of compound re-imaging lens. As shown in Fig. 7(h), the smallest resolvable line pair was the group 3 element 3 that corresponds to 10-line pairs per millimeter or 7.2 × 10−3 deg IFOV at the object plane. According to the Nyquist sampling theorem, the tested IFOV was equal to the two times of the designed IFOV.
Fig. 7 The image registration results for a 1951 USAF negative resolution target. The original four sub-images (a), (b), (c), and (d) acquired by the four sub-apertures respectively. The absolute differences of the former three images (a), (b), and (c) from the fourth image (d) are (e), (f), and (g) before registration, and (i), (j), and (k) after registration, respectively. The central part on the fourth image (d) encompassed by yellow rectangle is magnified (h) for clarification. The histogram (l) of (k).
5.2. Polarimetric calibration
To implement the EMC polarimetric calibration, the PSG and sample were added in front of the ASSIP system as shown in Fig. 5(b). The 4 polarization states were generated by the PSG with the rotation of the AQWP1 and AQWP2 to the optimal pairs of fast-axis azimuths. The 4 samples: air, the polarizer at 0 deg, the polarizer at 90 deg, the QWP at 30 deg were measured successively at each PSG state. Totally, 16 measurements were implemented, and each measurement consists of 4 sub-images. The instrument matrix across the FOV (1200 × 1200 pixels) was calculated with the ECM from these measured sub-images. The polarimetric calibrations were repeated at the bandwidths of 10, 25, and 200 nm respectively. Further, the measurements were repeated after intensifying illumination level that corresponds to the polarizer sample.
The full well charge of the CMOS is 15000 electrons/pixel and the RMS noise is 8 electrons/pixel, then its dynamic range is 1875. To fully utilize the dynamic range, we let the CMOS work at 10-bit digital number (DN) readout although 12-bit is selectable. Correspondingly, each grayscale will occupy about 1.8 electron/pixel. All measurements were performed at the same integration time of 100 ms, then the readout noises were relatively uniform. As a result, the mean DN output of dark frames was around 33. To ensure there is no saturated pixel for all noisy calibration measurements, we let the mean DN output is no more than 950 for the brightest sub-image. As a result, the mean DN was around 40 for the darkest sub-image at the normal illumination level for the polarizer sample. The darkest sub-image almost approached to the dark background at the normal illumination. Thus, we intensified the illumination for the polarizer sample, and the resulted mean DN of the darkest sub-image approached to 55 which was brighter than that of the dark background. Correspondingly, the useful signal was increased trebly after the subtraction of background. However, further increase is subject to the saturation of the brightest sub-image. It should be noted that these DNs were raw values and they would be scaled by the flat-field coefficients. As an example, the statistical distributions of a set of 16 flat-field sub-images for 0 deg polarizer sample with intensified illumination are depicted in Fig. 8. As seen, most sub-images obeyed Gaussian distribution, two darker sub-images complied Poisson fitting (mean μ equals variance σ2), and other sub-images have complex statistical distributions. In addition, we found that other 48 sub-images for other 3 samples also have similar complex statistical distributions. That is the calibration measurements included compound statistical distributions. It is better to let the calibration method have immunity to these noise distributions. Our proposed calibration process has this capability.
Fig. 8 The histograms of the measured 16 sub-images with intensified illuminations at 25nm bandwidth for the 0 deg polarizer sample. Each PSG state corresponds to 4 sub-images. While the red solid line represents the Gauss fitting using the statistical mean μ and variance σ2, the blue solid line indicates the Poisson fitting with the mean μ.
Table 2 lists the calibrated instrument matrices A at each bandwidth corresponding to the models #1, #2, and #3, respectively. The standard deviation (STD) denotes the uncertainty around the mean value across the FOV. It is noted that the mean values were slightly different at each bandwidth. Per bandwidth, the STDs for the model #2 were smaller than that for the model #1, and the STDs for the model #3 were the smallest. That means, the calibration precision was improved with the general model and intensified illumination. However, the mean values slightly deviated from the optimally nominal matrix in Eq. (6), mainly due to the imperfection of components. Figure 9 shows the statistical distributions of the CN2 corresponds to the instrument matrices A. All the mean values μ of the CN2 were slightly larger than the optimally nominal value of 1.732. That meant the noise immunity of the ASSIP system was degraded a little bit relative to the nominal matrix. The mean CN2 corresponding to the models #2 and #3 were smaller than that to the model #1, meaning they have better immunity to noise. The STDs of CN2 for the model #3 were the smallest, meaning the intensified illumination was effective to reduce the uncertainty.
Table 2. The mean value and standard deviation of the calibrated instrument matrices at each bandwidth (10, 25, and 200 nm) corresponding to the models #1, #2, and #3 respectively.
Fig. 9 The histogram of the L2-norm CN2 corresponds to the instrument matrices (A) at each bandwidth. The blue area, the mean μ1, and the STD σ1 for the model #1; the green area, μ2, and σ2 for the model #2; the red area, μ3, and σ3 for the model #3.
6. Image results
6.1. The performance on measuring SOP
A set of known SOPs can be generated to verify the performance of the above calibrated instrument matrices for different models. Usually, it is hard to produce accurate elliptical polarization states, since the function of a retarder varies with the FOV and bandwidth. In contrast, the polarizer has the stable performance over relatively larger FOV and wider waveband. Herein, we just kept the polarizer sample in front of the ASSIP (see Fig. 5(b)), then performed M = 19 measurements from 0° to 180° azimuth angles in 10° steps at each bandwidth. The exposure time and illumination were similar at each bandwidth. The corresponding theoretical Stokes parameters were calculated for comparison.
Per bandwidth, the upper rows in Figs. 10–12 depict the measured Stokes parameter using the instrument matrices of models #1, #2, and #3, respectively. Wherein, the dot markers with error bars indicated the average Stokes parameter across the effective FOV (1200 × 1200 pixels) and the solid lines denoted the theoretical values. As seen, the measured values were close to the theoretical values. The middle rows in Figs. 10–12 presented the absolute errors between the theoretical values and experimental values. The maximum absolute error was smaller than 0.05 for each model and each bandwidth, which were mainly induced by the minor residual mis-registrations. The lower rows in Figs. 10–12 presented the STDs for each Stokes parameter. All STDs approached to the order of 10−2. The maximum STD was smaller than 0.06 for each model and each bandwidth. Per bandwidth, the STDs of S1, S2, and S3 approached to each other, which meant the ASSIP system had the noise immunity. The accuracies and precisions of the reconstructions were almost independent of bandwidths, that meant the instrument matrices estimated at each bandwidth were impartial, mainly because the extended ECM was a modular method which already took all imperfect factors and high-order effects into account. Both the mean errors and STDs of the model #2 was smaller than that of model #1, and the model #3 was smallest.
Fig. 10 Upper row: the Stokes parameters of a polarizer’s azimuth angle rotated from 0° to 180° in 10° steps. Middle row: the absolute differences between the measured and theoretical values. Lower row: the STD of each Stokes parameter. Dot markers indicate ASSIP’s data correspond to the instrument matrix of model #1, and solid lines denote theoretical values. The error bar indicates the standard deviation across the FOV.
To compare the reconstruction uncertainty among the models #1, #2, and #3, we defined the root-mean-square errors (RMSEs) between the measured and theoretical output as
where the superscript meas and theo denote the experimental value and theoretical value respectively, k indicates the order of Stokes parameter, M indicates the total number of measurements related to the rotation angle of the polarizer, X and Y indicate the total row pixels and column pixels related to the FOV respectively. The RMSEs corresponding to each model and each bandwidth were listed in Table 3. Per bandwidth, the RMSE of each Stokes parameter corresponding to the model #3 was smallest and were approximately 1.2 and 1.1 times smaller than that of the models #1 and #2, respectively. The use of model #3 improved the precisions of the calibration and thus reconstruction. Therefore, the instrument matrices calibrated using the model #3 were employed in the following discussion.Table 3. The RMSE between the experimental mean value and theoretical output for all Stokes parameters across the FOV of the ASSIP and the azimuthal angles of the polarizer.
In the above analysis, we just used the instrument matrix estimated at individual bandwidth to reconstruct the Stokes parameters from the data measured at the same bandwidth. Thus, the determined instrument matrix worked well at each bandwidth. Usually the system with a wider bandwidth is welcomed due to its high flux. However, it should be noted that the residual retardances of achromatic retarders in the ASSIP system vary with the bandwidth. An interesting question is thus to make sure whether the instrument matrix estimated at the wider bandwidth can be used to process the data measured at the narrower bandwidth, since the broadband instrument may encounter the target with narrow bandwidth. To answer this question, we made the cross-validation among the three bandwidths. The estimated instrument matrix at one bandwidth was used to reconstruct the Stokes parameters from the data measured at the other two bandwidths.
Table 4 lists the total RMSEs of the cross-validations, where the diagonal elements in the table correspond to the self-validations. Obviously, the RMSEs of the cross-validations were larger than that of the self-validations, mainly because the exchange of filters introduced inevitable systematic errors. However, it is interesting to find that the upper triangular elements were approximately 1.1 times larger than the lower triangular elements, respectively, meaning that the instrument matrices estimated at the narrower bands were more applicable to the data obtained at the wider band. As a summary, it is better to let the bandwidth of the system be close to, or at least be narrower than, that of target. In fact, the self-validations were equivalent to the results of using the narrowband matrices to measure wideband targets, because the illumination source is broadband. A priori information for target is useful for the improvement of estimation precision.
Table 4. The RMSEs of the cross-validations at different bandwidthsa, and the diagonal elements correspond to the self-validations results in Table 3 at the same bandwidthb.
6.2. The performance on enhancing image contrast
The estimations of the SOPs were demonstrated in the former section. However, the Stokes imaging polarimeters not only can be used to measure the SOPs but also can be employed to improve the contrast of a target in a complex scene. Generally, there are two ways to improve contrast with the polarimeter: directly polarimetric imaging from each analysis state and recovering the Stokes images. It has been theoretically and experimentally demonstrated that the optimal analysis states, for maximizing the contrast of the direct images, are also that in the nominal instrument matrix in Eq. (6) [38–40]. However, the comparison between the direct images and Stokes images has not been paid enough attention. To verify the contrast power of Stokes images, we made a transparent scene sample and illustrated it with the integrating sphere. The background of the scene consists of a superposition of a polarizer sheet, a layer of plastic film and a translucent adhesive tape having birefringent properties. The target is a smaller square piece of translucent adhesive tape placed on the background. The four aperture sub-images were captured simultaneously by the ASSIP at each bandwidth with two different exposure times, and the Stokes images were reconstructed from the measured direct images.
Figure 13 presents the measured data at the bandwidth of 25 nm with the two exposure times of 50 ms and 150 ms, respectively. As can be seen, although the target was visible in the original sub-images, the contrasts were diverse and inhomogeneous. It is interesting to note that the contrast became higher and homogeneous in the reconstructed S1, S2, and S3 images, although the contrast was almost zero in the S0 image. Further, the contrast increased with the exposure time. Figure 14 presents the analogous data measured at the bandwidth of 200 nm. Compared to the results in Fig. 13, it is noted that the contrast increased obviously with the bandwidth, mainly due to the increase of integral intensity over a wider band [41]. As a summary, to speedily get the high contrast images, the ASSIP should work at wider bandwidth with the shorter exposure time in the case of without considering the precision of SOP.
Fig. 13 The measured data at the bandwidth of 10 nm with the exposure times of 50 and 150 ms, respectively. (a) and (c) The original sub-image for each sub-aperture, (b) and (d) the reconstructed image for each Stokes parameter. The curves in the third row correspond to the central cross sections of the images along the pink horizontal lines, respectively. The black, red, green, and blue lines correspond to the images of apertures A1-A4 respectively, and the images of S0 - S3 respectively.
Fig. 14 The measured data at the bandwidth of 200 nm with the two exposure times of 50 and 150 ms, respectively.
6.3. The performance on capturing dynamic scenes
To demonstrate the snapshot performance, a dynamic scene was imaged at a video rate of 20 fps by the system with the bandwidth of 25nm. The scene consists of a rotated glass window with the five linear polarizers and two circular polarizers, and it was illuminated by the integrating sphere. The polarizers were cemented on the window with double-stick tape on the edges. The outer five linear polarizers have the tangentially oriented principal transmission directions, and the right- and left-hand circular polarizers stand in the center. Figure 15 depicts a frame of the reconstructed image video (see Visualization 1). From the S1, S2, and S3 images in the left two columns, the linear and the circular polarizers were recognized unambiguously. All of them complied with the theoretical predictions. To enhance the human perception of the polarization information on a single image, the measured data can be combined with the color fusion techniques [42–45]. One of polarization-color mapping strategies is to directly map polarization into HSV (hue, saturation, and value) space [42]. For just displaying the linear polarization components on a single image, we mapped the angle of polarization (AoP) into H, the degree of linear polarization (DoLP) into S, and the intensity (S0) into V. For only displaying the circular polarization components on a single image, we mapped the handedness (the sign on S3) into H, the degree of circular polarization (DoCP) into S, and the intensity (S0) into V. The AoP, DoLP, and DoCP are calculated with
Fig. 15 The recovered images of S0, S1, S2, and S3 (left and middle columns) for a glass window sample with the outer five linear polarizers and the inner left- and right-hand circular polarizers; and the polarization-HSV normal color fusion images (right column) (see Visualization 1).
The right column in Fig. 15 depicts the two polarization-HSV color mapping images (see Visualization 1). For the linear polarization components, the dark red, dark cyan, dark violet, dark green, and dark blue colors, respectively, represent the AoPs of 12°, 85°, 159°, 47°, and 122°. The dark color is induced by the low irradiance of polarizer relative to the void area. Maximum saturation represents a DoLP = 1.0, while gray indicates no linear components. For the circular polarization components, dark red and dark green indicate the left and right circular polarization, respectively. Maximum saturation represents a DoCP = 1.0, while gray indicates no circular components. Obviously, normally mapping polarization parameters into HSV color space could perceptually enhance the recognition of each polarization information.
In the above laboratory scene, since the irradiance and the DoP of targets were all high, the normal color fusion of polarization-HSV was encouraging. However, if the irradiance is low and the DoP is high, the chromatic properties of mapped polarization will be faint. This case usually happens at the outdoor natural scene. As an example, we captured moving automobiles on an urban road at a cloudy day. Figure 16 shows a frame of reconstructed data video (see Visualization 2). Although the DoLP (maximum 0.73, average 0.55) on the rear windows of two cars were high, the fused chromatic properties were perceptually weak due to very low irradiance. As a result, it is hard to recognize the polarization information on the fused images. Tyo et al [43] recently proposed an adaptive color fusion method to resolve this issue, as
and the parameters in the brackets are user-defined. Herein, we defined , , and . The polarization-HSV adaptive color fusion images are depicted in Fig. 17 (see Visualization 3). the linear polarization components on the rear windows of two cars perceptually became distinct. The weak circular polarization component is still invisible. The ASSIP system has the capability to capture dynamic targets in real time.Fig. 16 The recovered images of S0, S1, S2, and S3 (left and middle columns) for the automobiles on an urban road; and the polarization-HSV normal color fusion images (right column) (see Visualization 2).
7. Conclusion
The optimization, design, calibration, and validation of an ASSIP prototype has been presented. The ASSIP and calibration setup are theoretically optimized to both Gaussian and Poisson noises using the BCPN condition. With the extended EMC that accounts for imperfectness of samples, the instrument matrix of the ASSIP is calibrated across the FOV at three bandwidths of 10, 25, and 200 nm, respectively. It is found that the instrument matrix calibrated with the proposed general model of samples is better than the ideal model at each bandwidth. After intensifying the illumination intensity for the polarizer sample, the calibrated instrument matrix has the best precision. The L-2 norm CN2 of the calibrated instrument matrix is close to the ideal value of 1.732, meaning the ASSIP prototype has the noise immunity. The accuracy and precision of the system are validated using the calibrated instrument matrices to measure the well-known polarization states. The instrument matrix corresponding to the general model with the intensified illumination obtains the most accurate and precise Stokes parameters. The precision is approximate to the order of 10−2. However, applying the instrument matrix calibrated at a specified bandwidth to the data measured at another bandwidth will increase uncertainty, mainly because there are the residual retardances in the achromatic retarders of the system. The instrument matrix determined at the narrower bandwidth is more applicable to the data obtained at wider bandwidth. The contrast ability of polarimetric imaging is validated using a transparent scene at different bandwidths and exposure times. The results show that although the target is visible in the original sub-images, the contrasts are enhanced in the reconstructed S1, S2, and S3 images. Further, the contrast increases with both the exposure time and bandwidth. One can speedily get high contrast image at wider bandwidth along with short exposure time. The snapshot capability of the ASSIP system is verified by capturing dynamic scenes. The polarization features of moving vehicles are traced in real time. Using the polarization-HSV color fusion method, it is convenient to perceptually recognize the polarization targets on a single image. The compact ASSIP has the potential applications on micro-unmanned air vehicles for surveillance, on ground vehicles for target identification, or on microscopy platforms for in-vivo imaging.
Funding
National Natural Science Foundation of China (NSFC) (61775176, 61405153, 41530422); Fundamental Research Funds for the Central Universities of China (xjj2017105).
References
1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef] [PubMed]
2. Q. Liu, C. Bai, J. Liu, J. He, and J. Li, “Fourier transform imaging spectropolarimeter using ferroelectric liquid crystals and Wollaston interferometer,” Opt. Express 25(17), 19904–19922 (2017). [CrossRef] [PubMed]
3. N. Hagen, R. T. Kester, L. Gao, and T. S. Tkaczyk, “Snapshot advantage: a review of the light collection improvement for parallel high-dimensional measurement systems,” Opt. Eng. 51(11), 111702 (2012). [CrossRef] [PubMed]
4. T. Mu, F. Han, D. Bao, C. Zhang, and R. Liang, “Compact snapshot optically replicating and remapping imaging spectrometer (ORRIS) using a focal plane continuous variable filter,” Opt. Lett. 44(5), 1281–1284 (2019). [CrossRef] [PubMed]
5. W. L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014). [CrossRef] [PubMed]
6. T. Mu, S. Pacheco, Z. Chen, C. Zhang, and R. Liang, “Snapshot linear-Stokes imaging spectropolarimeter using division-of-focal-plane polarimetry and integral field spectroscopy,” Sci. Rep. 7(1), 42115 (2017). [CrossRef] [PubMed]
7. J. S. Tyo, “Hybrid division of aperture/division of a focal-plane polarimeter for real-time polarization imagery without an instantaneous field-of-view error,” Opt. Lett. 31(20), 2984–2986 (2006). [CrossRef] [PubMed]
8. A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Optimal bandwidth micropolarizer arrays,” Opt. Lett. 42(3), 458–461 (2017). [CrossRef] [PubMed]
9. I. J. Vaughn, A. S. Alenin, and J. S. Tyo, “Channeled spatio-temporal Stokes polarimeters,” Opt. Lett. 43(12), 2768–2771 (2018). [CrossRef] [PubMed]
10. M. Garcia, T. Davis, S. Blair, N. Cui, and V. Gruev, “Bioinspired polarization imager with high dynamic range,” Optica 5(10), 1240–1246 (2018). [CrossRef]
11. J. Mudge, M. Virgena, and P. Dean, “Near-infrared simultaneous Stokes imaging polarimeter,” Proc. SPIE 7461, 74610L (2009). [CrossRef]
12. T. Mu, C. Zhang, and R. Liang, “Demonstration of a snapshot full-Stokes division-of-aperture imaging polarimeter using Wollaston prism array,” J. Opt. 17(12), 125708 (2015). [CrossRef]
13. N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365, eaax1839 (2019).
14. M. W. Kudenov, M. J. Escuti, E. L. Dereniak, and K. Oka, “White-light channeled imaging polarimeter using broadband polarization gratings,” Appl. Opt. 50(15), 2283–2293 (2011). [CrossRef] [PubMed]
15. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimization and performance criteria of a Stokes polarimeter based on two variable retarders,” Opt. Express 18(10), 9815–9830 (2010). [CrossRef] [PubMed]
16. F. Goudail and J. S. Tyo, “When is polarimetric imaging preferable to intensity imaging for target detection?” J. Opt. Soc. Am. A 28(1), 46–53 (2011). [CrossRef] [PubMed]
17. D. Lara and C. Paterson, “Stokes polarimeter optimization in the presence of shot and Gaussian noise,” Opt. Express 17(23), 21240–21249 (2009). [CrossRef] [PubMed]
18. I. J. Vaughn and B. G. Hoover, “Noise reduction in a laser polarimeter based on discrete waveplate rotations,” Opt. Express 16(3), 2091–2108 (2008). [CrossRef] [PubMed]
19. J. L. Pezzaniti and D. B. Chenault, “A division of aperture MWIR imaging polarimeter,” Proc. SPIE 5888, 58880V (2005). [CrossRef]
20. W. Zhang, J. Liang, L. Ren, H. Ju, E. Qu, Z. Bai, Y. Tang, and Z. Wu, “Real-time image haze removal using an aperture-division polarimetric camera,” Appl. Opt. 56(4), 942–947 (2017). [CrossRef] [PubMed]
21. H. Gu, X. Chen, H. Jiang, C. Zhang, and S. Liu, “Optimal broadband Mueller matrix ellipsometer using multi-waveplates with flexibly oriented axes,” J. Opt. 18(2), 025702 (2016). [CrossRef]
22. T. Mu, D. Bao, C. Zhang, Z. Chen, and J. Song, “Optimal reference polarization states for the calibration of general Stokes polarimeters in the presence of noise,” Opt. Commun. 418, 120–128 (2018). [CrossRef]
23. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999). [CrossRef] [PubMed]
24. T. Mu, C. Zhang, Q. Li, and R. Liang, “Error analysis of single-snapshot full-Stokes division-of-aperture imaging polarimeters,” Opt. Express 23(8), 10822–10835 (2015). [CrossRef] [PubMed]
25. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25(16), 1198–1200 (2000). [CrossRef] [PubMed]
26. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000). [CrossRef] [PubMed]
27. J. C. del Toro Iniesta and M. Collados, “Optimum modulation and demodulation matrices for solar polarimetry,” Appl. Opt. 39(10), 1637–1642 (2000). [CrossRef] [PubMed]
28. T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal configurations of full-Stokes polarimeter with immunity to both Poisson and Gaussian noise,” J. Opt. 18(5), 055702 (2016). [CrossRef]
29. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009). [CrossRef] [PubMed]
30. F. Goudail, “Equalized estimation of Stokes parameters in the presence of Poisson noise for any number of polarization analysis states,” Opt. Lett. 41(24), 5772–5775 (2016). [CrossRef] [PubMed]
31. T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal design and performance metric of broadband full-Stokes polarimeters with immunity to Poisson and Gaussian noise,” Opt. Express 24(26), 29691–29704 (2016). [CrossRef] [PubMed]
32. C. F. LaCasse, T. Ririe, R. A. Chipman, and J. S. Tyo, “Spatio-temporal modulated polarimetry,” Proc. SPIE 8160, 81600K (2011). [CrossRef]
33. A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). [CrossRef] [PubMed]
34. H. Hu, E. Garcia-Caurel, G. Anna, and F. Goudail, “Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration,” Appl. Opt. 52(25), 6350–6358 (2013). [CrossRef] [PubMed]
35. W. Sparks, T. A. Germer, J. W. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt. 51(22), 5495–5511 (2012). [CrossRef] [PubMed]
36. H. Bay, T. Tuytelaars, and L. Van Gool, “SURF: speeded up robust features,” in Computer Vision-ECCV 2006 (Springer, 2006), pp. 404–417.
37. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33(2), 156–158 (2008). [CrossRef] [PubMed]
38. F. Goudail and M. Boffety, “Optimal configuration of static polarization imagers for target detection,” J. Opt. Soc. Am. A 33(1), 9–16 (2016). [CrossRef] [PubMed]
39. F. Goudail and M. Boffety, “Performance comparison of fully adaptive and static passive polarimetric imagers in the presence of intensity and polarization contrast,” J. Opt. Soc. Am. A 33(9), 1880–1886 (2016). [CrossRef] [PubMed]
40. F. Goudail and M. Boffety, “Fundamental limits of target detection performance in passive polarization imaging,” J. Opt. Soc. Am. A 34(4), 506–512 (2017). [CrossRef] [PubMed]
41. M. Boffety, H. Hu, and F. Goudail, “Contrast optimization in broadband passive polarimetric imaging,” Opt. Lett. 39(23), 6759–6762 (2014). [CrossRef] [PubMed]
42. J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15(2), 367–374 (1998). [CrossRef]
43. J. S. Tyo, B. M. Ratliff, and A. S. Alenin, “Adapting the HSV polarization-color mapping for regions with low irradiance and high polarization,” Opt. Lett. 41(20), 4759–4762 (2016). [CrossRef] [PubMed]
44. A. W. Kruse, A. S. Alenin, I. J. Vaughn, and J. S. Tyo, “Perceptually uniform color space for visualizing trivariate linear polarization imaging data,” Opt. Lett. 43(11), 2426–2429 (2018). [CrossRef] [PubMed]
45. A. W. Kruse, A. S. Alenin, and J. S. Tyo, “Review of visualization methods for passive polarization imaging,” Opt. Eng. 58(08), 082414 (2019). [CrossRef]
