Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise

Passive Polarized Vision for Autonomous Vehicles: A Review

This review article aims to address common research questions in passive polarized vision for robotics. What kind of polarization sensing can we embed into robots? Can we find our geolocation and true north heading by detecting light scattering from the sky as animals do? How should polarization images be related to the physical properties of reflecting surfaces in the context of scene understanding? This review article is divided into three main sections to address these questions, as well as to assist roboticists in identifying future directions in passive polarized vision for robotics. After an introduction, three key interconnected areas will be covered in the following sections: embedded polarization imaging; polarized vision for robotics navigation; and polarized vision for scene understanding. We will then discuss how polarized vision, a type of vision commonly used in the animal kingdom, should be implemented in robotics; this type of vision has not yet been exploited in robotics service. Passive polarized vision could be a supplemental perceptive modality of localization techniques to complement and reinforce more conventional ones.

Optimized Stokes imaging for highly resolved optical speckle fields, Part II: optimal acquisition and estimation strategies

In this second paper of a three-paper series focusing on Stokes polarimetry of optical speckle fields resolved at the individual speckle grain scale, a theoretical study based on numerical simulations is presented in order to establish the optimum sensing, estimation, and processing strategies that guarantee the best precision, accuracy, and robustness for Stokes polarimetry in this specific context. In particular, it is demonstrated that the so-called state of polarization analysis by full projection on the Poincaré space (SOPAFP) approach can be optimized in order to ensure best estimation performance. These numerical simulations also make it possible to establish that the SOPAFP approach provides better results in terms of robustness to residual experimental imperfections of the setup when compared to classical Stokes polarimetry approaches.

High-accuracy reconstruction of Stokes vectors via spatially modulated polarimetry using deep learning at low light field

Polarization measurement is generally performed in scenes with a low signal-to-noise ratio (SNR) such as remote sensing and biological tissue detection. The spatially modulated polarimeter can satisfy the real-time measurement requirements in low SNR scenes by establishing the mapping between photon spatial distribution and polarization information. However, accurately measuring the polarization state under low-light illumination becomes highly challenging owing to the interference of background noise. In this paper, a deep learning method is proposed and applied to the high-accuracy reconstruction of polarization information at low light field. A reinforced two-layer deep convolutional neural network is designed to respectively extract global and local features of noise in this method. Accurate photon spatial distribution can be obtained by fusing and processing these features. Experimental results illustrate the excellent accuracy achieved by the proposed method with a maximum average value of the absolute measured error below 0.04. More importantly, the proposed method is well-performed for the reconstruction of Stokes vectors at low light fields of various levels without requiring changes to the model, enhancing its practicality and simplicity.

Error analysis and optimization for a full-Stokes division-of-space polarimeter

A generalized four-channel, full-Stokes division-of-space (DoSP) error propagation model and its version with a reference optical path are presented in this paper, covering all potential error sources such as the main detector noise, intensity fluctuations, and instrument matrix error. Based on the model, a classical division-of-amplitude polarimeter (DoAmP) structure consisting of a partially polarized beam splitter (PPBS), PBS, and wave plates is thoroughly evaluated. By optimizing the PPBS and azimuth of the wave plates, several optimal parameter configurations are identified where the condition number is 1.84, and the maximum wavelength deviation range is limited to (-3.4n m, 3.62 nm), where the degree of polarization and polarized angle errors do not exceed 0.03 and 0.3°, respectively, and the instrument matrix deterioration effect is minimal enough to be disregarded. In addition to the DoAmP structure, this error propagation model can be directly extended to other arbitrary four-channel DoSP structures such as division-of-focal-plane and division-of-aperture systems, which have guidance values for system structural design, error optimization, and discovering multi-wavelength compatibility of the instrument.

Performance analysis of a spectropolarimeter employing a continuous phase variation

The light emitted or reflected by a medium can exhibit a certain degree of polarization. Most of the time, this feature brings valuable information about the environment. However, instruments able to accurately measure any type of polarization are hard to build and adapt to inauspicious environments, such as space. To overcome this problem, we presented recently a design for a compact and steady polarimeter, able to measure the entire Stokes vector in a single shot. The first simulations revealed a very high modulation efficiency of the instrumental matrix for this concept. However, the shape and the content of this matrix can change with the characteristics of the optical system, such as the pixel size, the wavelength or the number of pixels. To assess the quality of the instrumental matrices for different optical characteristics, we analyze here the propagation of errors, together with the impact of different types of noise. The results show that the instrumental matrices are converging towards an optimal shape. On this basis, the theoretical limits of sensitivity on the Stokes parameters are inferred.

Local optimized Stokes polarimetry for specific polarization states

In this Letter, we propose a locally optimized Stokes polarimetry. Focusing on the effect on polarization measurements by Poisson noise, the studies establish a new, to the best of our knowledge, optimization function combining the equally weighted variance with the condition number. This method considers both the stability and the precision of polarization measurements; by trading an increase in the condition number by 2.48%, we realize a decrease in equal-weighted variance by 19.1% near the north pole. The advantages of this local optimization method are demonstrated based on Monte Carlo (MC) simulations and experiments of continuous polarization state modulation. Finally, an imaging demonstration using a 4 µm pathological section implies the potential of this new local optimization method in improving polarization measurements and applying it to more biomedical research.

Liquid-crystal based drift-free polarization modulators: Part II. Ultra-stable Stokes and Mueller polarimeters

We have previously reported a new design for drift-free liquid-crystal polarization modulators (LCMs) based on liquid-crystal variable retarders (LCVRs). Here, we study their performance on Stokes and Mueller polarimeters. LCMs have polarimetric responses similar to LCVRs and can be used as temperature-stable alternatives to many LCVR-based polarimeters. We have built an LCM-based polarization state analyzer (PSA) and compared its performance to an equivalent LCVR-based PSA. Our system parameters remained stable over a wide range of temperature, precisely from 25°C to 50°C. Accurate Stokes and Mueller measurements have been conducted, paving the way to calibration-free polarimeters for demanding applications.

Polarization calibration assessment for a broadband imaging polarimeter based on a division of aperture architecture

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.

Estimation precision for a normalized response matrix in linear polarization calibration

The purpose of polarization calibration is to obtain the response matrix of an instrument such that the subsequent observation data can be corrected. The calibration precision, however, is partially restricted by the noise of the detector. We investigate the precision of the normalized response matrix in the presence of signal-independent additive noise or signal-dependent Poisson shot noise. The influences of the source intensity, type of noise, and calibration configuration on the precision are analyzed. We compare the theoretical model and the experimental measurements of the polarization calibration to show that the relative difference between the two is less than 16%. From this result, we can use the model to determine the minimum source intensity and choose the optimal configurations that provide the required precision.

Optimal nonlinear Stokes-Mueller polarimetry for multi-photon processes

In this Letter, we present an optimization model for nonlinear Stokes-Mueller polarimetry (SMP) to improve the precision in estimating the nonlinear Mueller matrix (MM) for two- and three-photon processes. Although nonlinear polarimeters can measure the polarization properties of multi-photon processes or materials, existing methods are suboptimal, leading to low measurement precision. Based on the model and its solution, we have designed a new measurement strategy to substantially reduce the estimation variance of nonlinear MM coefficients by approximately 58.2% for second-harmonic generation polarimetry and 78.7% for third-harmonic generation polarimetry. The model and measurement method can be directly applied to multi-photon processes to improve the precision of SMP.

High-performance polarization management devices based on thin-film lithium niobate

High-speed polarization management is highly desirable for many applications, such as remote sensing, telecommunication, and medical diagnosis. However, most of the approaches for polarization management rely on bulky optical components that are slow to respond, cumbersome to use, and sometimes with high drive voltages. Here, we overcome these limitations by harnessing photonic integrated circuits based on thin-film lithium niobate platform. We successfully realize a portfolio of thin-film lithium niobate devices for essential polarization management functionalities, including arbitrary polarization generation, fast polarization measurement, polarization scrambling, and automatic polarization control. The present devices feature ultra-fast control speeds, low drive voltages, low optical losses and compact footprints. Using these devices, we achieve high fidelity polarization generation with a polarization extinction ratio up to 41.9 dB and fast polarization scrambling with a scrambling rate up to 65 Mrad s^-1, both of which are best results in integrated optics. We also demonstrate the endless polarization state tracking operation in our devices. The demonstrated devices unlock a drastically new level of performance and scales in polarization management devices, leading to a paradigm shift in polarization management.

Static full-Stokes Fourier transform imaging spectropolarimeter capturing spectral, polarization, and spatial characteristics

A static full-Stokes Fourier transform imaging spectropolarimeter incorporating a liquid-crystal polarization modulator (LPM) and birefringent shearing interferometer (BSI) is reported. It can decode the polarization information at each wavelength along the spatial dimension of a two-dimensional data array. The LPM has a high-speed time-division architecture and employs two ferroelectric liquid crystals and two wave plates to produce four polarization states, providing full-Stokes polarimetric information with a high signal-to-noise ratio. The BSI comprises two birefringent crystal plates and generates an optical path difference with good linear distribution for broadband interference, allowing a fast and high-precision spectral recovery. The optimized design of LPM and BSI are introduced in detail. Subsequently, the signal reconstruction is verified through simulations and experiments. The proposed scheme is highly efficient, exhibits a higher spectral resolution, and constitutes a compact technical approach to realize high-dimensional optical measurement.

No-Reference Physics-Based Quality Assessment of Polarization Images and Its Application to Demosaicking

Assessing the quality of polarization images is of significance for recovering reliable polarization information. Widely used quality assessment methods including peak signal-to-noise ratio and structural similarity index require reference data that is usually not available in practice. We introduce a simple and effective physics-based quality assessment method for polarization images that does not require any reference. This metric, based on the self-consistency of redundant linear polarization measurements, can thus be used to evaluate the quality of polarization images degraded by noise, misalignment, or demosaicking errors even in the absence of ground-truth. Based on this new metric, we propose a novel processing algorithm that significantly improves demosaicking of division-of-focal-plane polarization images by enabling efficient fusion between demosaicking algorithms and edge-preserving image filtering. Experimental results obtained on public databases and homemade polarization images show the effectiveness of the proposed method.

Comparison of different calibration methods for Mueller matrix microscopy of cells

Mueller matrix (MM) imaging has demonstrated its potential application in much research, especially in probing delicate and complex biomedical specimens. Qualities of MM images are important for further quantitative characterization. In this paper, we compare the performance and imaging qualities of three calibration methods. Air, waveplate and cell specimen are selected as standard samples for comparison. In addition, we also propose two general MM imaging quality indices that can be used as quantitative evaluations for MM imaging systems and calculation processes based on real samples. The numerical calibration method turns out to give the best accuracy and precision, as well as the best image qualities.

Optimal configurations for different incident polarization states in linear polarization calibration

The purpose of polarization calibration is to measure the response matrix of an instrument and the deviation of noise to correct for subsequent flight measurements. The precision, however, is relative to the states of incident light. We investigate the influence of partially polarized light, in the presence of signal-independent additive noise or signal-dependent Poisson shot noise. We obtain the estimation precision for different numbers of the polarization state generators and analyzers in linear Stokes measurements. To reduce the influence of incident light, we suggest that the numbers of the polarization state generators and analyzers should be greater than or equal to 4. In particular, for an instrument including three polarizers oriented at 0°, 60°, and 120°, estimation precision is found to be dependent on the response matrix and incident polarization states.

Optimal polarimeter structures for estimating polarization degree, angle, and ellipticity in the presence of additive noise

In polarimetry, it is well known that measurement matrices based on spherical 2 designs optimize Stokes vector estimation in the presence of additive noise. We investigate the optimal matrices for estimation of the degree of polarization (DOP), the angle of polarization (AOP), and the ellipticity (EOP), which are nonlinear functions of the Stokes vector. We demonstrate that spherical 2 designs also optimize DOP and EOP estimation, but not AOP estimation, for which optimal structures consist of linear analyzers forming a regular polygon on the equator of the Poincaré sphere.

Optimal tradeoff between precision and sampling rate in DoFP imaging polarimeters

A linear division-of-focal-plane camera combined with a controllable polarization modulator constitutes a versatile full-Stokes imager with four possible sampling rate modes, depending on the number of acquisitions. Considering several polarization modulator architectures, we determine the parameter settings that minimize estimation variance in each sampling rate mode, so that precision, sampling rate, and acquisition time can be optimally and dynamically balanced to implement the imaging solution best adapted to a given application.

Fundamental precision limits of full Stokes polarimeters based on DoFP polarization cameras for an arbitrary number of acquisitions

As an emerging technology, division-of-focal-plane (DoFP) polarization cameras have raised attention due to their integrated structure. In this paper, we address the fundamental precision limits of full Stokes polarimeters based on a linear DoFP polarization camera and a controllable retarder in the presence of additive and Poisson shot noise. We demonstrate that if the number of image acquisitions is greater than or equal to three, there exists retarder configurations that reach the theoretical lower bound on estimation variance. Examples of such configurations are one rotatable retarder with fixed retardance of 125.26° or two rotatable quarter-waveplates (QWPs) in pair. In contrast, the lower bound cannot be reached with a single QWP or a single variable retarder with fixed orientation. These results are important to get the most out of DoFP polarization imagers in real applications.

Optimized design, calibration, and validation of an achromatic snapshot full-Stokes imaging polarimeter

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.

On the validity domain of approximations to estimation variance of polarization degree, azimuth, and ellipticity

We determine the validity domain of classical approximations to estimation variance of the degree of polarization (DOP), angle of polarization, and ellipticity (EOP), when the measurement matrix of the Stokes vector is a spherical 2 design and noise is additive, white, and Gaussian. We demonstrate that this domain of validity is quite large, so that these approximations can be used safely for back-of-the-envelope calculations. In the presence of strong noise, DOP and EOP approximations show, however, some limits. We thus derive other approximations with extended domain of validity that can account for the dependence of DOP and EOP estimation variance on actual DOP and EOP values. The obtained results are important for design and performance characterization of polarimeters.

Precision analysis of arbitrary full-Stokes polarimeters in the presence of additive and Poisson noise

Estimation variances of the main polarimetric parameters (Stokes vector, degree of polarization, azimuth, and ellipticity) are derived for arbitrary Stokes measurement matrices in the presence of additive and Poisson shot noise. The obtained theoretical expressions, which are rather involved, are checked extensively with Monte Carlo simulations and their physical meaning is interpreted. The great benefit of these formulas is to enable comparisons of polarimeter architectures on a quantitative basis. As an example, we compare the optimal architectures based on spherical designs with a suboptimal one that may be easier to implement.

Optimal ellipsometric parameter measurement strategies based on four intensity measurements in presence of additive Gaussian and Poisson noise

The two ellipsometric parameters of an isotropic sample can be measured with simplified polarimetry setups that acquire at least four intensity measurements. However, these measurements are perturbed by noise and the measurement strategy has to be optimized, in order to limit noise propagation. We determine two different measurement strategies that are optimal for both white Gaussian additive noise and Poisson shot noise. The first one involves a polarization state generator (PSG) with a single state of polarization and a polarization state analyzer (PSA) with four states. The second one involves both PSG and PSA having two states. The total estimation variances obtained with both strategies are demonstrated to be minimal, of equal values, and independent of the ellipsometric parameters to be measured. They are based on simple optical elements and could simplify and accelerate ellipsometric measurements.

Estimation precision of full polarimetric parameters in the presence of additive and Poisson noise

The final product of polarimetric measurements is often such polarimetric parameters as degree of polarization (DOP), angle of polarization (AOP) and ellipticity (EOP). Since these parameters are nonlinear functions of the Stokes vector, it is difficult to derive closed-form expressions of their variances. We derive approximate but accurate expressions of the estimation variances of DOP, AOP, and EOP in the presence of both additive and Poisson noise for optimal spherical design-based Stokes polarimeters. These original closed-from expressions provide a clear insight into the physical parameters that govern the estimation precision of each polarimetric parameter. They are validated through optical experiments on a real-world polarimeter. These expressions are important for designing and sizing polarimeters or polarimetric imagers aimed at different types of applications, and for assessing their performance.

Polarimetric precision of micropolarizer grid-based camera in the presence of additive and Poisson shot noise

Polarimetric cameras based on micropolarizer grids make it possible to design division of focal plane (DoFP) polarimeters. However, the polarimetric estimation precision reached by these devices depends on their realization quality, which is estimated by calibration. We derive the theoretical expressions of the estimation variance of such polarimetric parameters as an angle of linear polarization and degree of linear polarization as a function of the calibrated micropolarizer characteristics. These values can be compared with the variances that would be obtained with ideal micropolarizers in order to quantitatively assess the effect of manufacturing defects on polarimetric performance. These results are validated by experimental measurements on a real-world camera.

Statistics of normalized Stokes polarization parameters

The normalized Stokes parameters are formed from the ratio of the polarization components to the intensity component of light. Such ratio distributions are known to have an undefined mean and variance, and yet researchers in the polarization community work with these normalized parameters all the time. We show that while in theory the normalized parameters have a pathological probability density, in practice they are quite well-behaved. We provide expressions for their approximate densities and confirm the results with laboratory measurements.

Water surface-clutter suppression method based on infrared polarization information

Targeting star-like water surface clutter, a clutter suppression method based on infrared polarization information is proposed. First, the clutter is suppressed from a global perspective using infrared polarization imaging technology, and a basic clutter-suppressed image is obtained. Then, using the Reed-Xiaoli anomaly detection algorithm, the remaining clutter positions in the basic image are determined from the polarization intensity image and basic image. Finally, an image filtering algorithm is utilized to further suppress the remaining clutter in the basic image. In experiments, the proposed method can not only improve the signal-to-clutter ratio as much as 152%, but also preserve the target information and background texture features effectively, indicating clear superiority of our method over existing clutter suppression algorithms. Clutter suppression and target detail preservation can enhance observer understanding of a scene significantly, so this method is applied to the detection and recognition of targets on the water surface.

Stokes polarimeter performance: general noise model and analysis

We calculate the photometric Stokes parameter covariance matrices and SNRs estimated by polarimeters exposed to general noise distributions, such as mixed Poisson-Gaussian (PG) noise. The measurement model includes the effects of optical losses and detector quantum efficiency, enabling quantitative comparison of instruments that have different photometric efficiencies. We demonstrate this capability by comparing the performance of many common polarimeter configurations, including diattenuator-based systems, such as Azzam's four-detector polarimeter [Opt. Lett.10, 309 (1985)OPLEDP0146-959210.1364/OL.10.000309] and Kudenov's stacked photovoltaic polarimeter [Opt. Express24, 14737 (2016)OPEXFF1094-408710.1364/OE.24.014737]. Working with the full covariance matrix under mixed PG noise, we also show that instruments optimized under assumptions of Gaussian noise simultaneously exhibit optimal behavior under Poisson noise.

Airlight-imposed errors for space-object polarimetric observations from the ground

We discuss and characterize how polarimetric sensing is contaminated by various "airlight" phenomena, as well as unpolarized light from the target, when space objects are observed with a ground-based telescope. Estimates of the polarization state are limited by unpolarized target light regardless of sensor technology or estimator algorithm, and increased target brightness actually degrades estimation of the S₁, S₂, and S₃ Stokes parameters if the added light is unpolarized. Unpolarized airlight in the field of view has an identical degrading effect. Atmospheric scattering can significantly polarize airlight, so airlight polarization must be calibrated and subtracted from the estimated target polarization. We derive an expression for the mean-square Stokes estimation error when noisy, biased estimates for the airlight polarization state are subtracted from noisy, biased estimates of the target polarization state; this expression shows that target and airlight Stokes estimation noise and bias generally sum in the ms estimation error for airlight-calibrated target Stokes. While SNR for the estimate of a given Stokes parameter increases with the magnitude of that parameter, estimation bias also appears to be correlated with magnitude. We note that when the linear Stokes reference is not arbitrary, requiring a rotational transformation of the estimated Stokes vector, the SNRs of the S₁ and S₂ estimates vary with the rotation angle. Finally, we show that measured data can be used in numerical calculations described here to approximate the errors associated with Stokes estimation, with or without airlight calibration.

Multiple scattering of polarized light in birefringent slab media: experimental verifications and simulations

The effective scattering Mueller matrices were measured for backward and forward scattering by applying a narrow polarized light on a polyacrylamide slab gel, which was strained vertically to generate birefringence inside. Monte Carlo simulations were performed in conditions that were the same as possible. The measured and simulated matrices were simplified to the reduced ones. They agreed well in both original and reduced forms. While they approximately take reciprocal forms for backward scattering, they approximately satisfy matrix forms that correspond to a reciprocal position of the mirror image for forward scattering. The reduced matrices were factorized by the Lu-Chipman polar decomposition to obtain the polarization parameters. The polarization parameters were in good agreement between the measurement and simulation and showed characteristic features of anisotropic slab media with a birefringence axis parallel to the slab surface.

Fourier transform imaging spectropolarimeter using ferroelectric liquid crystals and Wollaston interferometer

A time-division Fourier transform imaging spectropolarimeter (FTISP) for acquiring spatial, spectral, and polarized information is presented. The FTISP employs two ferroelectric liquid crystals (FLCs) and a Wollaston interferometer. The fast axes of the FLCs are controlled to switch quickly without mechanical movement, enabling the polarization state analyzer (PSA) to modulate the full set of Stokes parameters rapidly. The interferometer combines a Wollaston prism with a retroreflector, enabling high interference modulation and facilitating optical alignment. The optimal design for the FLC-PSA and Wollaston interferometer, and the Fourier transform recovery for the polarized interferogram, are presented in detail. To verify the proposed FTISP, laboratory and outdoor experiments were conducted, and the experimental results demonstrate that the proposed FTISP offers much promise for spectropolarimetric measurement with the advantages of fast speed, high spectral resolution, and high signal-to-noise ratio.

Optimal Mueller matrix estimation in the presence of additive and Poisson noise for any number of illumination and analysis states

We investigate the optimal strategies for estimating the Mueller matrix with arbitrary numbers of illumination and analysis states, in the presence of signal-independent additive noise or signal-dependent Poisson shot noise. We demonstrate that the architectures that minimize and equalize the estimation variances for both types of noise sources are based on spherical designs of order 2 or 3, and we provide closed-form expressions of the estimation precision obtained with these optimal measurement strategies. The obtained results are important to design Mueller polarimeters in practice and assess their fundamental limits in terms of estimation precision.

Optimal configuration of static Mueller imagers for target detection

We investigate the target detection performance of static Mueller imagers that implement a fixed number of illumination and analysis polarization states. Using a maximin approach, we demonstrate that the optimal sets of measurement vectors consist in regular tetrahedra on the Poincaré sphere and that, in this case, the obtained target/background contrast has a very simple expression. We then derive a universal lower bound on the contrast ratio between the best channel of a static imager and a fully adaptive one, and in a special case of practical interest, we demonstrate that this ratio is bounded and always larger than 1/9. This is very important in practice since static imagers are much easier to build and operate. Our results show that they constitute a good alternative where ultimate contrast improvement is not necessary.

Binary classification of Mueller matrix images from an optimization of Poincaré coordinates

A new binary classification method for Mueller matrix images is presented which optimizes the polarization state analyzer (PSA) and the polarization state generator (PSG) using a statistical divergence between pixel values in two regions of an image. This optimization generalizes to multiple PSA/PSG pairs so that the classification performance as a function of number of polarimetric measurements can be considered. Optimizing PSA/PSG pairs gives insight into which polarimetric measurements are most useful for the binary classification. For example, in scenes with strong diattenuation, retardance, or depolarization certain PSA/PSG pairs would make two regions in an image look very similar and other pairs would make the regions look very different. The method presented in this paper provides a quantitative method for ensuring the images acquired can be classified optimally.

Performance comparison of pseudo-inverse and maximum-likelihood estimators of Stokes parameters in the presence of Poisson noise for spherical design-based measurement structures

Estimation of the Stokes vector from N>4 intensity measurements is usually performed with the pseudo-inverse (PI) estimator, which is optimal when the noise that corrupts the measurements is additive and Gaussian. In the presence of Poisson shot noise, the maximum-likelihood (ML) estimator is different from the PI estimator, but is more complex to implement since it is not closed-form. We compare in this Letter the precisions obtained with the ML and the PI estimators in the presence of Poisson noise when using measurement structures based on spherical designs. We show that, in this case, the gain in precision brought by the ML estimator is real but modest, so that in applications where processing speed is an issue, the PI estimator can be considered sufficient. This result is important in the choice of the inversion strategy for Stokes polarimetry.

Fundamental limits of target detection performance in passive polarization imaging

We quantitatively determine the target detection performance of different passive polarization imaging architectures perturbed by signal-independent detection noise or signal-dependent Poisson shot noise. We compare the fully adaptive polarimetric imager and the best channel of a static polarimetric imager, and in each case, we compare the use of a polarizer and a polarizing beam splitter as the polarization analyzing device. For all these configurations, we derive a closed-form expression of the target/background separability and quantify the performance gain brought by polarization imaging compared to standard intensity imaging. We show in particular that all the considered polarimetric imaging configurations but one require a minimum value of the polarimetric contrast in order to outperform intensity imaging. The only configuration that always performs better than intensity imaging uses a polarizing beam splitter in the presence of background shot noise. These results are useful in evaluating the fundamental limits of the gain brought by polarization imaging and determining, in practice, which type of imaging architecture is preferable for a given application.

Intrinsic coincident full-Stokes polarimeter using stacked organic photovoltaics

An intrinsic coincident full-Stokes polarimeter is demonstrated by using strain-aligned polymer-based organic photovoltaics (OPVs) that can preferentially absorb certain polarized states of incident light. The photovoltaic-based polarimeter is capable of measuring four Stokes parameters by cascading four semitransparent OPVs in series along the same optical axis. This in-line polarimeter concept potentially ensures high temporal and spatial resolution with higher radiometric efficiency as compared to the existing polarimeter architecture. Two wave plates were incorporated into the system to modulate the S₃ Stokes parameter so as to reduce the condition number of the measurement matrix and maximize the measured signal-to-noise ratio. Radiometric calibration was carried out to determine the measurement matrix. The polarimeter presented in this paper demonstrated an average RMS error of 0.84% for reconstructed Stokes vectors after normalized to S₀. A theoretical analysis of the minimum condition number of the four-cell OPV design showed that for individually optimized OPV cells, a condition number of 2.4 is possible.

Optimal design and performance metric of broadband full-Stokes polarimeters with immunity to Poisson and Gaussian noise

In this paper, the design, optimization and analysis of broadband full-Stokes polarimeters with immunity to both Poisson and Gaussian noise are presented. Different from the commonly-used optimization metrics such as, the condition number (CN), the equally weighted variance (EWV), or the polarimetric modulation efficiency (PME) for Gaussian noise, the optimally balanced condition for Poisson noise (BCPN) is, for the first time, proposed and used as a metric for the optimization of polarimeters. The numerical results show that the polarimeters optimized with the BCPN have immunity to both Poisson and Gaussian noise. The broadband polarimeters optimized from the BCPN are achromatic and have similar polarimetric modulation properties over the waveband, in contrast to the polychromatic polarimeters optimized from the CN, EWV and PME, whose polarimetric modulation properties vary with wavelength.

Equalized estimation of Stokes parameters in the presence of Poisson noise for any number of polarization analysis states

Estimation of the Stokes vector is based on projecting the input light on a number N of polarization analysis states. We address the optimization of the distribution of these analysis states on the Poincaré sphere in the presence of signal-dependent Poisson shot noise for an arbitrary value of N. We show that if this distribution forms a spherical 3 design, the Stokes vector is estimated with minimal equally weighted variance and with estimation variances of the last three Stokes parameters equal and independent of the input Stokes vector. We also demonstrate that in the presence of Poisson shot noise, the estimation signal to noise ratio is independent of N, whereas in the presence of signal independent additive noise, it is proportional to 1/N, which means that there is a precision loss in increasing the number of measurements.

Performance comparison of fully adaptive and static passive polarimetric imagers in the presence of intensity and polarization contrast

We address the comparison of contrast improvement obtained with a fully adaptive polarimetric imager and the best channel of a static polarimetric imager in the presence of both intensity and polarization differences between the target and the background. We develop an in-depth quantitative study of the performance loss incurred by a static imager compared to a fully adaptive one in this case. These results are useful to make a well-informed choice between these two polarimetric imaging architectures in a given application.

Comparison of different active polarimetric imaging modes for target detection in outdoor environment

We address the detection of manufactured objects in different types of environments with active polarimetric imaging. Using an original, fully adaptive imager, we compare several imaging modes having different numbers of polarimetric degrees of freedom. We demonstrate the efficiency of active polarimetric imaging for decamouflage and hazardous object detection, and underline the characteristics that a polarimetric imager aimed at this type of application should possess. We show that in most encountered scenarios the Mueller matrices are nearly diagonal, and sufficient detection performance can be obtained with simple polarimetric imaging systems having reduced degrees of freedom. Moreover, intensity normalization of images is of paramount importance to better reveal polarimetric contrast.

Optimal configuration of static polarization imagers for target detection

We determine the set of analysis states of a static Stokes imager that maximizes target detection performance for the least favorable target-background polarimetric configuration. By using a minimax approach, we demonstrate that the optimal choice consists of four analysis states forming a regular tetrahedron in the Poincaré sphere. We also show that the value of the contrast in the best of the four Stokes channels is, in the worst case, equal to one third of that provided by a fully adaptive polarimetric imager. Static Stokes imagers thus constitute an attractive solution in applications where limited loss of discrimination ability can be tolerated.

Optimal Frames for Polarization State Reconstruction

Complete determination of the polarization state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number κ of the corresponding instrument matrix. Optimization of redundant measurement frames with an arbitrary number of analysis states, m, is considered in this Letter in the sense of minimization of κ. The minimum achievable κ is analytically found and shown to be independent of m, except for m=5 where this minimum is unachievable. Distribution of the optimal analysis states over the Poincaré sphere is found to be described by spherical 2 designs, including the Platonic solids as special cases. Higher order polarization properties also play a key role in nonlinear, stochastic, and quantum processes. Optimal measurement schemes for nonlinear measurands of degree t are hence also considered and found to correspond to spherical 2t designs, thereby constituting a generalization of the concept of mutually unbiased bases.

Optimal distribution of integration time for intensity measurements in Stokes polarimetry

We consider the typical Stokes polarimetry system, which performs four intensity measurements to estimate a Stokes vector. We show that if the total integration time of intensity measurements is fixed, the variance of the Stokes vector estimator depends on the distribution of the integration time at four intensity measurements. Therefore, by optimizing the distribution of integration time, the variance of the Stokes vector estimator can be decreased. In this paper, we obtain the closed-form solution of the optimal distribution of integration time by employing Lagrange multiplier method. According to the theoretical analysis and real-world experiment, it is shown that the total variance of the Stokes vector estimator can be significantly decreased about 40% in the case discussed in this paper. The method proposed in this paper can effectively decrease the measurement variance and thus statistically improves the measurement accuracy of the polarimetric system.

Error analysis of single-snapshot full-Stokes division-of-aperture imaging polarimeters

Single-snapshot full-Stokes imaging polarimetry is a powerful tool for the acquisition of the spatial polarization information in real time. According to the general linear model of a polarimeter, to recover full Stokes parameters at least four polarimetric intensities should be measured. In this paper, four types of single-snapshot full-Stokes division-of-aperture imaging polarimeter with four subapertures are presented and compared, with maximum spatial resolution for each polarimetric image on a single area-array detector. By using the error propagation theories for different incident states of polarization, the performance of four polarimeters are evaluated for several main sources of error, including retardance error, alignment error of retarders, and noise perturbation. The results show that the configuration of four 132° retarders with angular positions of ( ± 51.7°, ± 15.1°) is an optimal choice for the configuration of four subaperture single-snapshot full-Stokes imaging polarimeter. The tolerance and uncertainty of this configuration are analyzed.

Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon

Recently, we introduced the basic concepts behind a new polarimeter device based on conical refraction (CR), which presents several appealing features compared to standard polarimeters. To name some of them, CR polarimeters retrieve the polarization state of an input light beam with a snapshot measurement, allow for substantially enhancing the data redundancy without increasing the measuring time, and avoid instrumental errors owing to rotating elements or phase-to-voltage calibration typical from dynamic devices. In this article, we present a comprehensive study of the optimization, robustness and parameters tolerance of CR based polarimeters. In addition, a particular CR based polarimetric architecture is experimentally implemented, and some concerns and recommendations are provided. Finally, the implemented polarimeter is experimentally tested by measuring different states of polarization, including fully and partially polarized light.

Measurement errors resulted from misalignment errors of the retarder in a rotating-retarder complete Stokes polarimeter

Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters, which employ uniformly spaced angles over 180° or 360°, are most commonly used to detect the state of polarization (SOP) of an electromagnetic (EM) wave. The misalignment error of the retarder is one of the major error sources. We suppose that the misalignment errors of the retarder obey a uniform normal distribution and are independent of each other. Then, we derive analytically the covariance matrices of the measurement errors. Based on the covariance matrices derived, we can conclude that 1) the measurement errors are independent of the incident intensity s0, but seriously depend on the Stokes parameters (s1, s2, s3) and the retardance of the retarder δ; 2) for any mean incident SOP, the optimal initial angle and retardance to minimize the measurement error both can be achieved; 3) when N = 5, 10, 12, the initial orienting angle could be used as an added degree of freedom to strengthen the immunity of RRFP Stokes polarimeters to the misalignment error. Finally, a series of simulations are performed to verify these theoretical results.

On the performance of the physicality-constrained maximum-likelihood estimation of Stokes vector

We address the estimation of the Stokes vectors taking into account the physical realizability constraint. We propose a fast method for computing the constrained maximum-likelihood (CML) estimator for any measurement matrix, and we compare its performance with the classical empirical physicality-constrained estimator. We show that when the measurement matrix is based on four polarization states spanning a regular tetrahedron on the Poincaré sphere, the two estimators are very similar, but the CML provides a better estimation of the intensity. For an arbitrary measurement matrix, the CML estimator does not always yield better estimation performance than the empirical one: their comparative performances depend on the measurement matrix, the actual Stokes vector and the signal-to-noise ratio.

Maximum likelihood method for calibration of Mueller polarimeters in reflection configuration

We address calibration of Mueller polarimeters in the presence of noise. We compare an extension of the eigenvalue calibration method (ECM) and a maximum likelihood (ML) method. The performances of these two calibration methods are investigated with numerical simulations and real experiments on a broadband infrared polarimeter. It is found that the ML method is superior to the extended ECM in terms of calibration precision and can be used at lower signal-to-noise ratio.

Noise properties of uniformly-rotating RRFP Stokes polarimeters

Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters are most commonly used to measure the state of polarization (SOP) of an electromagnetic (EM) wave. Most of commercialized RRFP Stokes polarimeters realize the SOP measurements by rotating a 90° retarder to N(N≥5)uniformly spaced angles over 360° and performing a discrete Fourier transform of data. In this paper, we address the noise properties of such uniformly-rotating RRFP Stokes polarimeters employing a retarder with an arbitrary retardance. The covariance matrices on the measurement noises of four Stokes parameters are derived for Gaussian noise and Poisson noise, respectively. Based on these covariance matrices, it can be concluded that 1) the measurement noises of Stokes parameters seriously depend on the retardance of the retarder in use. 2) for Gaussian noise dominated RRFP Stokes polarimeters, the retardance 130.48° leads to the minimum overall measurement noises when the sum of the measurement noises of four Stokes parameters (viz., the trace of the covariance matrix) is used as the criterion. The retardance in the range from 126.06° to 134.72° can have a nearly-minimum measurement noise which is only 1% larger than the minimum. On the other hand, the retardance 126.87° results in the equalized noises of the last three Stokes parameters. 3) for Poisson noise dominated RRFP Stokes polarimeters, the covariance matrix is also a fuction of the SOP of the incident EM wave. Even so, the retardance in the range from 126.06° to 134.72° can also result in nearly-minimum measurement noise for Poisson noise. 4) in the case of Poisson noise, N=5, 10, 12uniformly spaced angles over 360° have special covariance matrices that depend on the initial angle (the first angle in use). Finally, simulations are performed to verify these theoretical findings.

Performance of Maximum Likelihood estimation of Mueller matrices taking into account physical realizability and Gaussian or Poisson noise statistics

We address constrained estimation of the Mueller matrices from noisy measurements, taking into account the physical realizability. Physical realizability is enforced based on the positive semi-definite Hermitian coherency matrix, and the statistics of the noise is taken into account by employing Maximum Likelihood (ML) method. We consider two types of noise sources frequently encountered in optical imaging systems: additive Gaussian noise and Poisson shot noise. In both cases, we demonstrate reduction of estimation error by enforcing the physical realizability constraint, and superiority of the ML constrained solutions compared to empirically constrained ones. The ML constrained estimation method proposed in this paper provides a justified and effective way to exploit experimental measurements of Mueller matrices.