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Abstract
This article intends to provide all the experimental insights and analyze the best polarimetric calibration method for a division of aperture polarimetric imager considering the different implications it has on the experimental set-up and its performance. Polarimetric cameras require careful calibration for the correct measurement of polarization information. The calibration methods are introduced, intermediate results are presented, and the ability of the set-up to estimate Stokes vectors and Mueller matrices of the samples in passive and active imaging modes is evaluated. Polarization information recovery achieves accuracy errors below the 10% for all polarization modes when the Data Reduction Matrix or the Eigenvalue Calibration Method are used. Such performance, however, degrades significantly when using the Polarizer Calibration Method. To the best of our knowledge, this is the first time such a detailed comparison of calibration methods is presented in the literature, and it is also the first time the Polarizer Calibration Method is applied to a division of aperture polarimeter.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light polarization can be retrieved across a scene using imaging polarimeters. There exist several optical architectures like division of time, division of amplitude, division of focal plane and division of aperture (DoAP) [1–4] to acquire the polarization information based on different physical approaches. The last architecture has the advantage of acquiring in a single shot the intensity sub-images, projected into the active camera sensor’s surface, needed for the recovery of the polarization information. In a DoAP, each sub-image corresponds to a sub-aperture where different polarization components act as an analyzer with a specific state of polarization. The total set of elements (composed of the polarization components, optical elements, and the sensor) is thus called the polarization state analyzer (PSA) of the system and constitutes the system matrix ${\boldsymbol A}$ of the polarimeter [5].
Each time any sensor is used for quantitative measurements, it needs precise calibration. Generally in imaging, a radiometric calibration of the bare sensor is required to account for the offsets and inhomogeneities present in the pixels of the sensor, as well as a geometric calibration of the whole device is required to account for the existing optical aberrations and set a clear pixel to pixel correspondence between images. In a polarimetric imaging system, an additional calibration regarding polarization is essential to consider the real polarimetric characteristics of the components for a good retrieval of the polarization information. This system’s behaviour is summarized in the experimental retrieval of the instrument matrix ${\boldsymbol A}$ of the system. Several calibration methods have been developed in the last years for this purpose [1,6–9]. The majority of them were designed according to the system to be calibrated and very few developed a general calibration independent of it.
The purpose of the present article is to set a reference by addressing, for the first time as far as we know, a detailed evaluation of different methods for polarimetric calibration using an optimised DoAP imaging polarimeter. In concrete, three methods, which claimed to be general, are presented and implemented: the Data Reduction Matrix method (DRM) [10], the Eigenvalue Calibration Method (ECM) [11], and the recently proposed Polarizer Calibration Method (PCM) [12]. Since each of them relies on a different approach, they are tested on the same DoAP imaging polarimeter to determine which is the most adequate for this type of system under different experimental configurations, essentially under passive or active polarization imaging. For the first time, the DRM is applied using the minimal optimum states against Gaussian and Poisson noise, the possible sample combinations needed in ECM are analyzed, and the PCM is implemented on a broadband, optimised DoAP imaging polarimeter using its associate polarimetric model. The intermediate and final results of the calibrations are presented for passive and active polarization imaging modes. The performances of the methods are compared and discussed for each mode. Finally, a compilation of the advantages and disadvantages of each method is given together with the most suitable calibration method for the system.
2. Polarization calibration methods
In this section, we briefly present the set-up used and the three state-of-the-art methods chosen to be evaluated for the calibration of a broadband DoAP polarimeter. Previous to polarimetric calibration, radiometric calibration is mandatory to get rid of the inhomogeneities in the sensitivity of the pixels in the sensor, including offset signal, flat field, and differences in quantum efficiency of single pixels. In addition, since the polarimeter uses different images, a geometrical calibration for a correct pixel-pixel correspondence is required. Thereafter, the imaging polarimeter can be calibrated for recovering the polarization information. The three methods proposed differ in the mathematical approach for recovering the system matrix ${\boldsymbol A}$ of the DoAP camera. This matrix contains the response of the system transforming the incident polarization of the scene (described by Stokes vector $\vec S $) into intensity images following:
The performance of the system is determined by the optical design and the calibration of the system. As optimal performance is sought, some prior considerations about the device should be addressed. The polarization states that compose the PSA are recommended to have been optimized to estimate the polarization information in the shortest possible time. The minimal number of states required for Stokes vector recovery is N = 4. Moreover, it is worth stating that the system can be immunized against Gaussian and Poisson noise by choosing wisely the states used in the PSA [13–15]. This previous work will help the calibration algorithms to be more accurate in the final results.
2.1 DoAP imaging polarimeter
The DoAP system calibrated for measuring polarization information is based on [14–17]. Its PSA consists of four apertures corresponding to the optimum theoretical reference polarization states (RPS) to constitute an optimized instrument matrix ${\boldsymbol A}$. Its components consist of custom retarders followed by a linear polarizer, similar to the system described in [17]. All the components have their working bandwidth in the visible range (400-700nm) allowing using white light illumination for the measurement of polarization information.
The dynamic range of the sensor is 12 bits and the exposure time used in the measurements is 120ms. The polarimetric camera based on DoAP is presented in Fig. 1 together with its intensity image when being illuminated by linearly polarized light at 70°. It can be appreciated that each aperture shows a different intensity value due to the response of the corresponding RPS of the PSA to the input polarization.
Fig. 1. (Left) DoAP imaging polarimeter working in the visible band. (Right) The intensity detected shows the four apertures when the DoAP camera detects a 70° linear polarization.
2.2 Data reduction matrix method
The DRM is the first calibration method considered, and it consists in directly solving the Eq. (1) pixel-wisely [10]. It can be rewritten in a matrix form considering that the incident polarization states $({\mathbf S})$ generated by the polarization state generator (PSG) are known and measured by the PSA giving the flux matrix ${\mathbf I}$:
Assuming the PSA is formed by four linearly independent analyzers, it is demonstrated that by using 4-input polarization states the ${{\mathbf W}_{_{\mathbf{DRM}}}}$ calibration matrix can be derived. ${{\mathbf W}_{_{\mathbf{DRM}}}}$ has rank four and is non-singular. Therefore, its inverse exists and is unique.
To obtain ${{\mathbf W}_{_{\mathbf{DRM}}}}$, the inverse of ${\mathbf I}$ must be calculated. Since it may be ill-conditioned, Ref. [18] recommends calculating its pseudoinverse by means of singular value decomposition. This allows truncating the singular values up to the range of solutions of the system, so the full Stokes vector is composed of four parameters. This truncation provides a more stable calibration matrix.
2.3 Eigenvalue calibration method
ECM is based on the eigenvalue’s recovery from the intensity matrix instead of using its pseudoinverse. It is quite an extended method used for Stokes and Mueller polarimeters [11,19]. This method requires illuminating reference samples with the polarization states of the PSG to recover not only the instrument matrix of the system $({{\boldsymbol A}_{_{\mathbf{ECM}}}})$ but the one of the PSG $({\boldsymbol G})$ as well. They are calculated through the polarization parameters of each reference sample coming from the eigenvalues of an intensity-related matrix $({{\boldsymbol C}_{\boldsymbol i}})$. These parameters are introduced in the corresponding Mueller matrix model and the cost function is optimized to get the calibration matrices.
Normally, a linear polarizer (LP) and a quarter waveplate (QWP), at specific angles, are assumed as convenient samples fulfilling the conditions established in [11]. It is usually assumed that at least one sample of each type, a dichroic sample and a retarder, are needed for a unique calibration. The fundamental operations of the calibration method are enumerated in the following, for a complete description see Ref. [11]:
- 1. Measure the flux matrix ${{\boldsymbol I}_{\boldsymbol{air}}}$ of the PSG states through the air. Eq. 1 can be written as: ${{\boldsymbol I}_{\boldsymbol{air}}} = {{\boldsymbol A}_{\boldsymbol{ECM}}}{\boldsymbol G}$.
- 2. Measure the intensity matrix ${{\boldsymbol I}_{\boldsymbol i}}$ for the ith sample, knowing the model of the Mueller matrix. The final equation is: ${{\boldsymbol I}_{\boldsymbol i}} = {{\boldsymbol A}_{\boldsymbol{ECM}}}{{\boldsymbol M}_{\boldsymbol i}}{\boldsymbol G}$.
- 3. Calculate for each sample the product:${{\boldsymbol C}_{\boldsymbol i}} = {\boldsymbol I}_{\boldsymbol{air}}^{ - 1}{{\boldsymbol I}_{\boldsymbol i}} = {{\boldsymbol G}^{ - 1}}{{\boldsymbol M}_{\boldsymbol i}}{\boldsymbol G}$. ${{\boldsymbol C}_{\boldsymbol i}}$ and ${{\boldsymbol M}_{\boldsymbol i}}$ share the same eigenvalues. From the eigenvalues of ${{\boldsymbol C}_{\boldsymbol i}}$, the polarization properties of the samples (like the transmission coefficient, the retardance or the angle of the optical axis) can be recovered and introduced in the model of the Mueller matrix ${\boldsymbol (}{{\boldsymbol M}_{\boldsymbol i}})$.
- 4. Defining the error function as ${{\boldsymbol M}_{\boldsymbol i}}{\boldsymbol G} - {\boldsymbol G}{{\boldsymbol C}_{\boldsymbol i}} = 0$, ${\boldsymbol G}$ can be derived from the null space of the equations system and it is unique.
- 5. The instrument matrix of the system ${{\boldsymbol A}_{\boldsymbol{ECM}}}$ can be obtained through the relation: ${{\boldsymbol A}_{\boldsymbol{ECM}}} = {{\boldsymbol I}_{\boldsymbol{air}}}{{\boldsymbol G}^{ - 1}}$.
For the implementation of the method, square matrices are essential for the existence of the matrix inverse. Therefore, during this article and for a rigorous comparison among calibration methods, ECM is done using the same polarization states generated by the PSG in the DRM method proposed in Section 2.2, and the DoAP device is already optimized with a PSA of N = 4. It should be reminded that the ECM must be applied to each pixel of the sensor.
2.4 Polarizer calibration method
This third calibration method is a variation of the already explained ECM where only one sample, in particular an LP, is employed. PCM, thus, does not need any retarder or air samples to calculate the two instrument matrices of the system simultaneously, ${{\boldsymbol A}_{\boldsymbol{PCM}}}$ and ${\boldsymbol G}$. Removing the retarder lets the algorithms be free of the principal errors which appear when they are used: chromaticity effects and misalignment errors in the fast axis.
PCM relies on treating the LP Mueller matrix as a projector's product. Hence, the basis of all the mathematical derivations is to find the projectors that are the eigenvectors of certain intensity-related matrices. Since all the measurements are performed with an LP, PCM only initially recovers the components of the instrument matrices that are associated with linear polarization. For a complete calculation of the matrices, the polarization model of the system under calibration is required. Through the introduction of this model, which englobes all the polarization behaviour of the system, the circular-related components of the matrices can be derived.
The main steps for PCM are summarized in the following and should be followed for all the pixels in the image, for further details see Ref. [12]:
- 1. Measure the intensity matrix ${{\boldsymbol I}_{\boldsymbol i}}$ of the different polarization angles ${\alpha _i}$. It can be written in projectors’ notation as ${{\boldsymbol I}_{\boldsymbol i}} = \frac{1}{2}\; \vec {{b_i}} \cdot{\vec{c}_i}$.
- 2. Calculate the symmetric matrices ${{\boldsymbol I}_{\boldsymbol i}}{\boldsymbol I}_{\boldsymbol i}^{\boldsymbol T}$ and ${\boldsymbol I}_{\boldsymbol i}^{\boldsymbol T}{{\boldsymbol I}_{\boldsymbol i}}$ to identify $\vec{{b_i}} $ and ${\vec{c}_i}$, respectively. They are the unique eigenvectors of each symmetric matrix.
- 3. Calculate ${{\boldsymbol P}_{{\boldsymbol n\vec{b}}}}$ and ${{\boldsymbol P}_{{\boldsymbol n\vec{c}}}}$ by stacking the n matrices coming from the relation ${P_{\vec{b}}}$:$${P_{\vec{b}}} = \left[ {\begin{array}{{c}} {b_i^{(2 )}{{\vec{p}}_i}^T - b_i^{(1 )}{{\vec {{p_i}} }^T}{{\vec{0}}^T}{{\vec{0}}^T}}\\ {{{\vec{0}}^T}b_i^{(3 )}{{\vec {{p_i}} }^T} - b_i^{(2 )}{{\vec {{p_i}} }^T}{{\vec{0}}^T}}\\ {{{\vec{0}}^T}{{\vec{0}}^T}b_i^{(4 )}{{\vec {{p_i}} }^T} - b_i^{(3 )}{{\vec {{p_i}} }^T}} \end{array}} \right]$$
- 4. The instrument matrices are derived by solving the matrix systems: ${{\boldsymbol P}_{{\boldsymbol n\vec{b}}}} \cdot \vec a = \vec{0}$ and ${{\boldsymbol P}_{{\boldsymbol n\vec{c}}}}\vec {\cdot g} = \vec{0}$. The vectors $\vec{a}$ and $\vec{g}$ are the vectorised instrument matrices and are the associated eigenvectors of the smallest eigenvalue of ${{\boldsymbol P}_{{\boldsymbol n\vec{b}}}}$ and ${{\boldsymbol P}_{{\boldsymbol n\vec{c}}}}$ matrices removing the 4jth columns.
- 5. The circular components of $\vec{a}$ and $\vec{g}$ are calculated by the relation between fully-polarized Stokes components. The signs of the components are adjusted accordingly to the polarimetric model of the system.
- 6. ${{\boldsymbol A}_{\boldsymbol{PCM}}}$ and ${\boldsymbol G}$ come from converting the complete $\vec{a}$ and $\vec{g}$ into 4 x4 matrices.
Once more, for a proper comparison between the calibration methods, during the PCM the RPS used in the PSG are the same as in the previous methods and the PSA corresponds to the already explained DoAP polarimeter.
2.5 Evaluation of the calibration
In this subsection, the quantitative analysis of the performance of the calibration procedures is described. There are three main groups of analysis: the one related to the instrument matrices, the one evaluating the accuracy of the Stokes vector recovery, and the final one studying the performance in the estimation of the Mueller matrix.
In order to quantify the correctness of the calibration matrices before polarization recovery, two main values are of interest: the condition number and its associated equally weighted value (EWV). The condition number quantifies how much well-conditioned is the matrix. In this case, the condition number is calculated using the 2-norm (${{\boldsymbol \kappa }_2}$). A low condition number in the matrix yields low error propagation when recovering the polarization information. Several studies have used the condition number as a figure of merit for optimizing the systems [20–23] and they conclude that the optimum value for polarization calibration matrices of N = 4 systems is $\sqrt 3 $ [23]. On the other hand, the EWV stands for the analysis of the variance introduced by the matrix when recovering the polarimetric information [24].
Once the calibration is performed, the DoAP imaging polarimeter can be employed to recover two different polarization information: the Stokes vector or the Mueller matrix. The estimation of the Stokes vector is straightforward by solving pixel-wisely Eq. (2). When measuring this type of polarimetric information, the DoAP system is said to be a passive imaging polarimeter. In this case, the system only measures the light coming from the scene, being totally or partially polarized.
Beyond Stokes parameters $({S_0},{S_1},{S_2},{S_3})$, some other advanced parameters can be calculated from them. The degree of polarization (DOP) measures the portion of input light that is polarized. The angle of polarization (AOP) indicates the angle of the principal axis of the polarization. Equations (3),(4) show the mathematical expressions for these parameters.
To quantify the performance of the system when recovering the Stokes vector and related parameters, the maximum absolute errors $({\varepsilon _i})$ and the spatial standard deviation ${\boldsymbol (}{{\boldsymbol \delta }_i})$ of the system are derived. The errors show the deviation from the ideal value meanwhile the deviations state the spatial variations in the surface of the sensor.
In the second application, the system is said to be an active imaging polarimeter. To estimate the Mueller matrix of a scene, it must be illuminated by the polarization states compounding the PSG matrix ${\boldsymbol G}$ used in the illumination subsystem during calibration. This involves that the Mueller matrix is measured performing 4 shots instead of 16, thanks to the use of the DoAP system, which is one of the main advantages of this architecture. For the recovery, the two resulting matrices of calibration need to be used in Eq. (5) at each pixel:
${\boldsymbol M}$ is a 4 × 4 matrix that contains complete information about the polarization behaviour of the sample such as depolarization, diattenuation and retardance variation.
Following a similar study, four parameters are defined for measuring the accuracy of the experimental Mueller matrix. The Mueller deviation matrix $\textrm{(}{\boldsymbol \varDelta M}\textrm{)}$ gathers all the non-ideal effects of the system in a single matrix [7]:
The mean absolute error $({\varepsilon _{M}})$ corresponds to the mean difference between the ideal and the recovered Mueller matrices divided by the number of components of the Mueller matrix. The percentage average error per coefficient $({{\Delta }_{{{M}}}})$ [19] differs from the previous one since it applies the norm instead of the mean to the absolute error matrix. It is divided by 16 and multiplied by 100 to have a percentage value. Besides, one parameter related to variance is extracted: the mean-variance $({\delta _M})$. It is calculated in the same manner as the ${\varepsilon _{M}}$ but using the matrix formed by the variance of each component.
3. Calibration results and discussion
This section presents and analyzes the results of the three different calibration methods tested for the recovery of polarization both in the passive and the active mode for the DoAP system.
3.1 Passive imaging mode
The different calibrations are done using the DoAP system presented in subsection 2.1, with the N = 4 optimal analyzers in the PSA, and the same PSG in the illumination, whose RPS for calibration were selected to be the complementary optimal states with N = 4. This set is derived theoretically in literature when optimizing polarimetric systems and its analyzers draw a tetrahedron with maximum volume in the Poincaré sphere [25]. In addition, ECM and PCM require a previous study of the adequacy of the sample sets for system calibration.
In passive imaging, the performance of the three calibration methods (DRM, ECM and PCM) is analyzed by measuring the polarization states of a fixed LP at 0° followed by an achromatic QWP whose fast axis is rotated from 0° to 180° in steps of 10°. For the sake of brevity, the images of the best configuration are displayed and the results of the remaining configurations are gathered in tables for each method.
3.1.1 Data reduction matrix
The set-up of the DRM comprises a white light source together with an integrating sphere and the DoAP system. The PSG is composed of one LP (EO #47-316) followed by two achromatic QWPs (Thorlabs AQWP10M-580). The PSG generates the states that are measured by the PSA of the DoAP system giving the intensity matrix, see Fig. 2.
Table 1 gathers the parameters of the components in PSG for generating the RPS set for calculating the instrument matrix ${{\boldsymbol A}_{DRM}}$ together with the associated ${{\boldsymbol \kappa }_2}({{{\boldsymbol A}_{{\boldsymbol DRM}}}} )$ and EWV of the DRM. The theoretical and the recovered matrices in DRM for the DoAP system are shown in Appendix A. The experimental condition number and EWV are close to the ideal optimum values of 1.73 and 10.0, respectively, for a DoAP system with 4 states in the PSA [23,24].
Table 1. RPS used for DRM and the figures of merit of the resulting matrix: ${{\boldsymbol \kappa }_2}({{{\boldsymbol A}_{{\boldsymbol DRM}}}} )$ and ${\boldsymbol EVW}$
The experimental matrix ${{\boldsymbol A}_{DRM}}$ is used for calculating the Stokes parameters of the test sample: a fixed LP at 0° followed by an achromatic QWP whose fast axis is rotated from 0° to 180°. The DoAP creates a map for each of the input Stokes parameters (Fig. 3) from the measured intensity matrix, the experimental matrix ${{\boldsymbol A}_{DRM}}$ and Eq. (2), following the set-up sketched in Fig. 2. From these maps, the advanced parameters DOP and AOLP are retrieved as well. The statistical distributions of the recovered values are displayed as histograms. As an example of Stokes vector measurement, Fig. 3 shows the Stokes parameters maps and distributions when the fast axis of the rotating QWP is at 70°. They present a Gaussian distribution and their mean values and variances are saved for later analysis. Figure 4 displays the mean Stokes parameters and their associated errors, from maps like in Fig. 3, retrieved by the camera at each azimuth angle of the QWP. It can be appreciated that the errors are below the 8% error. Table 2 comprises the maximum absolute error for each Stokes parameter as well as for the advanced ones, and their associated spatial variance in the recovered maps.
Fig. 3. (Top) Recovered maps of the Stokes vector parameters corresponding to a fixed LP at 0° followed by a QWP with the fast axis at 70° using the ${A_{DRM}}$ matrix. (Bottom) Related histograms over the region of interest for each Stokes map.
Fig. 4. (Top) Average recovered Stokes parameters corresponding to a fixed LP at 0° followed by a QWP rotated between [0,180]° using DRM with N = 4. (Bottom) Related errors of the previous parameters.
Table 2. Maximum absolute errors for the Stokes parameters (S1, S2 and S3), the DOP and AOLP together with their associated spatial variances using the DRM
The DRM results show good performance regarding accuracy since the Stokes parameters are seen to be lower than 0.10 upper bound, together with the DOP. The AOLP has an estimation error below 8° and the spatial variance along the whole sensor surface is below 1%. The errors can be conditioned for the use of a QWP in the test sample since errors are below the 0.05 limit when employing a unique rotating LP.
These results can be compared with other systems in the literature that employed this method. The first one uses a division of time based on ferroelectric crystal cells [26]. They employed different RPS in the PSG and calibrated it in the green band (520-550 nm). Their errors are lower in DOP and AOLP but they only calibrate in the 30 nm band meanwhile the optimum DoAP system covers 300 nm. They state that for blue (395-480 nm) and red (>600 nm) bands the condition numbers of the instrument matrix vary and the behaviour to noise changes with respect to the green one. However, they do not show intermediate results of calibration like the calibration matrix or the Stokes parameters curves. Another division of time polarimeter based on a CMOS sensor reported that using the DRM method with 6 optimum RPS states in the PSG and the PSA made the method reach an error of around 10% using off-the-shelf components for linear polarization recovery [27]. A division of amplitude system using two colour micro-polarizers arrays was also calibrated using DRM [28]. They employed 18 non-optimised RPS to calibrate the system at the peak bands of red, green and blue. The condition number of the matrices was worse than the one presented here, the DOP errors were similar, and the variance was higher. They do not show the Stokes curves and errors nor the AOLP values. A third system based on DoAP uses 6 states in the PSA and used 6 RPS [17]. The DOP errors were shown and 0.06 was their maximum, being similar to the here discussed. They do not provide the ${{\boldsymbol \kappa }_2}$, the EWV, the Stokes parameters, the AOLP and their errors.
3.1.2 Eigenvalue calibration method
The set-up for a calibration based on the ECM is the same as used for the DRM, but introduces samples in the optical path before the DoAP system (Fig. 5). The calibration samples here used are an LP and an achromatic QWP.
Before testing the calibration, the best sample combination is sought for the DoAP system. The ECM is carried out using the four sample combinations, extracted from Ref. [11]. Table 3 gathers the different combinations of the samples. The LP should be measured at two different axis angles $({\mathrm{\theta }_{\textrm{L}{\textrm{P}_1}}},{\mathrm{\theta }_{\textrm{L}{\textrm{P}_2}}})$ and the QWP only once at ${\mathrm{\theta }_{\textrm{QWP}}}$. It should be stated that two types of Mueller matrix models were tested in parallel: the one which considers the samples as ideal, and the general model where both samples are assumed to be dichroic retarders to account for the experimental non-idealities of the components. The general model shows more accurate matrices and it is employed during the ECM. The figures of merit of the calibration matrices are presented in Table 3 to see the influence of the sample set choice.
Table 3. Figures of merit ${{\boldsymbol \kappa }_2}({{{\boldsymbol A}_{\boldsymbol{ECM}}}} )$, ${{\boldsymbol \kappa }_2}({{{\boldsymbol G}}} )$ and EWV for the calibration matrix of the ECM for the different combinations of samples using the general model of the Mueller matrix. ${\boldsymbol (}{{\boldsymbol \theta }_{{\boldsymbol L}{{\boldsymbol P}_{\boldsymbol 1}}}}{\boldsymbol ,}{{\boldsymbol \theta }_{{\boldsymbol L}{{\boldsymbol P}_{\boldsymbol 2}}}}{\boldsymbol ,}{{\boldsymbol \theta }_{{\boldsymbol QWP}}}{\boldsymbol )}$ stands for the two angles of the polarizer and the QWP angle
In this method, the instrument matrix ${\boldsymbol G}$ is obtained and its associated condition number is closer to the theoretical value than the ${{\boldsymbol A}_{\boldsymbol{ECM}}}$ values. Looking at the figures of merit for each set-up, different behaviours may be appreciated from the combinations of the samples. Although the best condition number may be found for Set II, it shows an EWV lower than the theoretical one (EWV = 10.0), making it doubtful for a good estimation. This is checked through the recovery of the polarization information from the fixed LP followed by a rotating QWP for all the cases.
To be as concise and synoptic as possible, only the test for recovering the Stokes vector using Set I is shown since it is the best combination as discussed in the following. The evolution of the mean Stokes parameters of the fixed LP at 0° followed by the rotating QWP and their related errors for the best combination are shown in Fig. 6. The error and standard variances of the Stokes and advanced parameters relative to this method for all the combinations of the calibration samples are gathered in Table 4.
Fig. 6. (Top) Average recovered Stokes parameters of a fixed LP at 0° followed by a QWP rotated between [0,180]° using ECM with Set I. (Bottom) Errors of the previous parameters.
Table 4. Maximum absolute errors for the Stokes parameters (S1, S2 and S3), the DOP and AOLP together with their associated spatial variances using the ECM.
Data in Table 4 confirm the ECM shows the best polarization measuring performance with Set I (LP at 0°, LP at 90° and QWP at 30°). It yields the lowest errors in all parameters being below 0.15. The mean accuracy of this method can be comparable to that obtained in the DRM. The presence of the QWP in the test sample makes the error accuracy higher in this method since when using only a rotating LP the maximum error is 5%. The generated input polarization states from the fixed LP and the rotating QWP can differ from the expected theoretical ones due to two causes: misalignment between the optical axis of the LP and the fast axis of the QWP and variations in the effective retardance of the QWP. Although the QWP is achromatic, the illumination is in the visible broadband and the nominal retardance may appear to change due to wavelength-dependent variations.
The conclusion that ECM demonstrates equivalent performance to DRM is supported by some division of time polarimetric systems reported in the bibliography that employ this type of combination but do not show the intermediate [29] or global results of the calibration [30]. Both systems use the ECM method with Set I and claim to have the best results although they do not study the other combinations.
Set II, which indicated good performance from the matrix figures of merit, results to have significantly worse performance. In addition, results show that combinations with the retarder at 90° have the lowest precision in the recovery of polarization. This behaviour is also supported by the increase of spatial variation of the parameters when the QWP is at 90° with respect to its counterpart at 30°.
3.1.3 Polarizer calibration method
The last calibration method uses the same set-up in ECM presented in Fig. 5 but leaves aside the use of the QWP in the calibration. Two attempts for the angles ${\alpha _{_i}}$ are tested to see the influence of using a higher number of samples. The figures of merit for the obtained calibration matrices of this set-up are gathered in Table 5.
Table 5. Figures of merit ${{\boldsymbol \kappa }_2}({{{\boldsymbol A}_{\boldsymbol{PCM}}}} )$, ${{\boldsymbol \kappa }_2}({{{\boldsymbol G} }} )$ and EWV obtained from PCM for the two combinations of the samples. ${{\theta }_{{L}{{P}_{i}}}}$ stands for the angles of the LP
The unique difference between the sets is adding two ${\alpha _{_i}}$ more in the sample of the calibration. From the figures of merit, no improvement is remarkable and the values are near the ideal values indicating a good choice of the samples a priori.
The recovery of the Stokes vectors for the test sample using the instrument matrix of the set N = 4 is presented in Fig. 7 for comparisons between methods. The associated errors and deviations from theoretical values of all parameters are compiled in Table 6. It is appreciated both in the figure and the table that the recovery errors are larger than in the previous methods.
Table 6. Maximum absolute errors for the Stokes parameters (S1, S2 and S3), the DOP, DOLP, DOCP and AOLP together with their associated spatial variances using the PCM
The PCM errors are higher than the errors in DRM and similar to ECM, both in the Stokes parameters and in the DOP. The AOLP error is up to 6.56°, close to the ones in DRM and ECM. In this case, the orientation angles used in the LP sample seem to have significantly influenced the results of polarization recovery since introducing two more states in the sample leads to a worse recovery. Therefore, having a higher number of samples (${\alpha _{_i}}$) during calibration does not assure a better result. Only studying the parameter of the optical axis angle of the LP will lead to the optimization of the calibration for the system.
No results in the literature have been found to compare the provided results with the behaviour of PCM in other architecture systems for full-Stokes recovery. As far as we know, it is the first study in passive imaging of this method.
3.2 Active imaging mode
In active imaging, only the last two methods (ECM and PCM) are used since they provide the two instrument matrices needed for the estimation of the Mueller matrix (${\boldsymbol {\rm A}}$, ${\boldsymbol G}$). Both calibrations are performed in the same way as explained in their respective sections in passive imaging. They used the same RPS in the PSG for illuminating the sample and the DoAP is used as PSA. The samples used in the calibrations are Set I, in the case of ECM, and set of N = 4, in the case of PCM.
In this mode, the Mueller matrix of the achromatic QWP with the fast axis at 30° is chosen as the test sample. The set-up for measuring the Mueller matrix is the one in Fig. 5. The calibration matrices from the sets that outperformed in Stokes vector retrieval in both methods are employed in these experiments and can be consulted in Appendix A. The ideal Mueller matrix of the test sample and the experimental Mueller matrices for the two methods are displayed in Fig. 8. The associated figures of merit for quantifying the error in the calculation are the deviation error matrix, the mean absolute error, the percentage average error per coefficient and the mean-variance, previously described in subsection 2.5. The first three parameters are calculated with the mean of each map of the Mueller matrix components. The last one is calculated from the Gaussian fitted distributions of each map, like in Stokes’ recovery. All of them are compiled in Table 7.
Fig. 7. (Top) Average recovered Stokes parameters corresponding to a fixed LP at 0° followed by a QWP rotated between [0,180]° using PCM with N = 4 set. (Bottom) Related errors of the previous parameters.
Fig. 8. (Top) Theoretical Mueller matrix of a QWP, with a fast axis at 30°. Recovered Mueller matrix (bottom left) using ECM with the sample set I and (bottom right) PCM using the set N = 4.
Table 7. Figures of merit of the experimental Mueller matrices obtained through ECM and PCM instrument matrices
For active imaging mode, the QWP at 30° is chosen to show the capacity of these methods in recovering full polarization information, which means including elliptical components. The experimental Mueller matrices seem to follow the ideal matrix of the retarder looking at the maps and values in Fig. 8. The mean value of each Mueller matrix coefficient in each of the maps is presented. The associated errors $({\varepsilon _{\rm M}},{{\Delta }_{\rm M}})$ in Table 7 show that the ECM outperforms PCM with lower global mean error and percentage error per coefficient. In addition, the deviation matrix ${\mathbf{\Delta M}}$ predicts the hypothetical Stokes vector entering the system if the calibration matrix was perfect. So, the closest ${\mathbf{\Delta M}}$ is to the identity matrix, the lower deviations in the Stokes estimation. Table 7 shows the ECM method is closer to the identity matrix sustaining the conclusion the ECM works better. The same is observed in the variance parameter ${{\boldsymbol \delta }_{\boldsymbol M}}$ proving that the PCM matrices propagate more noise in the recovery process.
These previous results depicted a specific elliptical state for a deeper analysis of the Mueller matrix components maps. In the following, a complete study of the evolution of the Mueller matrix when the fast axis of the achromatic QWP varies between 0° and 180° is displayed in Fig. 9. The theoretical values agree with the experimental data from the DoAP system using the ECM matrices. The error bars are higher in the circular-related components but below the 10% of error, aligning with the errors shown in Table 7.
Fig. 9. Theoretical Mueller matrix values (red dots) and recovered Mueller matrix mean values (blue line) of an achromatic QWP rotated from 0° to 180° using ECM with the sample Set I.
When employing the calibration matrices recovered from the PCM, the experimental Mueller matrix exhibits a more unstable recovery, as depicted in Fig. 10. The errors are larger than the ones in ECM demonstrating the superiority of the ECM for the calibration of an active imaging system.
Fig. 10. Theoretical Mueller matrix values (red dots) and recovered Mueller matrix mean values (blue line) of an achromatic QWP rotated from 0° to 180° using PCM with the sample N = 4.
In literature, ECM has been used for calibrating the division of time narrowband Mueller polarimeters [19]. The results are similar to the ones presented here. They chose Set III for calibration but no quantitative data is provided from Stokes or Mueller components. In addition, the previous study of a rotating QWP sample is presented in [31] using the ECM. Their results agree with the ones presented here but use a monochrome illumination. The PCM was studied in [12] and compared to the ECM. They arrived at the same conclusions for Mueller matrix acquisitions, although a punctual commercial polarimeter was employed.
4. Conclusion
In recent years, polarimetric imaging is emerging as a useful tool for very different applications. Many architectures in the optical design of cameras sensitive to polarization are being developed. However, only very few studies regarding the calibration of these systems are available, in specific for DoAP polarimetric cameras. These types of devices require careful calibration for recovering an accurate polarization signal, considering the radiometric and a possible geometric calibration for improving the pixel matching between intensity maps.
In this article, the evaluation of three general calibration methods (DRM, ECM and PCM) for imaging polarimeters using a DoAP architecture is addressed in detail. Moreover, the results are compared with systems in the state of the art using these methods, despite they do not provide a detailed study of the calibration procedure. Knowing in advance the experimental advantages or disadvantages of each method provides useful information when using imaging polarimeters. The following conclusions can be generalised since they are supported by the literature as discussed in the previous section.
When using the DoAP camera for recovering the Stokes vector and advanced parameters, the DRM has the simplest and fastest calibration set-up for recovery. It can be done using only four optimal RPS in the calibration without needing samples, making it the best choice when considering only passive imaging. The main drawback is that cannot be employed in precise Mueller matrix recovery.
The ECM has four possibilities of calibration sets and the analysis concludes that the combination of samples in Set I gives comparable accuracy results to DRM. The ECM has better mean results than PCM, just using an additional sample type during the calibration and not requiring a priori knowledge of the polarimetric model of the system to be calibrated.
The novel PCM method is developed for Stokes imaging and applied to this type of polarimeter for the first time. It has a medium set-up complexity and its results can make it feasible for full Stokes recovery since errors in Stokes recovery do not become much larger than in ECM. The PCM can be used in Stokes vector estimation when there is not available a good retarder for calibration, or if there is no possibility of a good alignment of the retarder. It provides a simple and low-cost set-up in comparison with ECM since no retarder is needed. Some future work can be done in this line by studying the relation between the number of required samples and the optimal parameters of the LP sample for enhancing calibration performance.
When using the DoAP for a complete determination of the polarization properties of the scene and so using active imaging mode, we conclude the best estimation is obtained using the ECM. The PCM can be used in Mueller matrix estimation when no retarder sample is available and the tolerance in results can be around 10%. Although the detailed system model was introduced and diverse angles of the LP were tested, the errors were still more than double the errors in ECM. Future work can be done on the optimization of the LP orientation in the PCM to reduce their influence on the calibration results.
To conclude, the desirable method for a DoAP system relies on the constraints of the experiment. On the one hand, the DRM can be applied when time and budget are compromised only being able to recover with confidence the Stokes vector of the scene. In the rest of the circumstances, the ECM can be performed only once and the system can be used in both passive and active imaging modes without extra calculations for complete and accurate retrieval of the polarization properties of the scene.
Appendix A
This appendix provides the instrument matrices used in the article for polarization recovery: the theoretical instrument matrix of the DoAP system and the experimental matrices calculated utilizing the DRM, ECM and PCM, respectively.
Funding
Ministerio de Ciencia e Innovación (PDC2021-121038-I00, PID2020-119484RB-I00); Agència de Gestió d'Ajuts Universitaris i de Recerca (2020FI_B2 00068); European Social Fund; Universitat Politècnica de Catalunya; Banco Santander.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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