Optimal distribution of integration time for intensity measurements in degree of linear polarization polarimetry

Xiaobo Li; Haofeng Hu; Tiegen Liu; Bingjing Huang; Zhanjie Song

doi:10.1364/OE.24.007191

1. Introduction

Measuring the degree of polarization (DOP) of the light scattered or reflected by a scene arouses a great interest in various fields, such as biomedical imaging [1], and remote sensing [2,3 ]. These systems can, for example, reveal contrasts between regions of a scene which have same intensity reflectivity but different polarimetric properties. The measurement of DOP is generally based on the intensity measurements. In practice, the intensity measurements are always perturbed by the noise, which causes the retrieved DOP to deviate from the true value. Therefore, one needs to find the proper methods to make the retrieved value as close as possible to the true value.

Statistically speaking, the measurement accuracy of DOP is determined by the variance of DOP estimator. Lowing the variance of DOP estimator can be beneficial for the higher accuracy of the measured parameters (such as stress or concentration of the solution) [4,5 ], and it can be also beneficial for a higher image contrast in the applications of polarimetric imaging [6]. Up to now, various methods for reducing the variance of the estimator have been proposed including optimizing the number of intensity measurements [7], performing the maximum likelihood estimation of the DOP [8–10 ], etc. Recently, it is found that the distribution of integration time for intensity measurements could considerably influence the variance of the estimator in Stokes polarimetry [11]. While for the DOP polarimetry, the factor of distribution of integration time has not been considered to increase the measurement accuracy yet.

In this paper, we address the issue of measuring the degree of linear polarization (DOLP). which is achieved by performing two intensity measurements at orthogonal polarization states [12–15 ]. The intensity measurements are assumed to be disturbed by the additive Gaussian noise. Under the condition that the total integration time for two intensity measurements is fixed, we investigate the dependence of the variance of DOLP estimator on the distribution of integration time for two intensity measurements, and we deduce the analytical solution of the optimal distribution of integration time. In addition, we perform the real world experiments to verify the effect of optimizing the distribution of integration time on decreasing the variance of DOLP estimator.

2. Degree of linear polarization polarimetry

We consider a configuration of polarimetry, which consists of illuminating the scene with a linearly polarized beam, and measuring two light intensities at orthogonal states [12–15 ]. The first intensity $I_{/ /}$ corresponds to the light intensity with the same polarization state of the illumination, and the second one $I_{⊥}$ corresponds to the light intensity at the orthogonal state of the illumination. From these two intensity measurements, a parameter of interest can be calculated by:

P = \frac{I_{/ /} - I_{⊥}}{I_{/ /} + I_{⊥}} .

It needs to be clarified that P can be negative if, for instance, the polarization rotation effect exists. Therefore, the range of P is

P \in [- 1, 1]

. There are different terms for P given by Eq. (1). The parameter P is often improperly called the degree of linear polarization (DOLP) [12,13 ] or degree of polarization (DOP) [8,9,14 ], however, it is different from the standard definition of DOP or DOLP as

DOP = \sqrt{s_{1}^{2} + s_{2}^{2} + s_{3}^{2}} / s_{0}

or

DOLP = \sqrt{s_{1}^{2} + s_{2}^{2}} / s_{0}

. In particular, P given by Eq. (1) is also named as the orthogonal state contrast (OSC) in some works [14,15 ]. In this paper, we term P as DOLP for the simplicity for discussion, but one should keep in mind that it is a non-standard definition.

In the presence of additive noise, the intensities at two orthogonal polarization states measured by the detector can be expressed as:

{\begin{cases} I_{/ /}^{m} = t_{/ /} I_{/ /} + n_{/ /} \\ I_{⊥}^{m} = t_{⊥} I_{⊥} + n_{⊥} \end{cases},

where

t_{/ /}

and

t_{⊥}

are the integration times of the two intensity measurements, while

I_{/ /}

and

I_{⊥}

are the true values of the two light intensities at orthogonal polarization states. We assume that the two intensity measurements are perturbed by the additive noise

n_{/ /}

and

n_{⊥}

with zero mean and variance

σ^{2}

. Actually, the noise leads to the variation of the intensity measurements

I_{/ /}^{m}

and

I_{⊥}^{m}

with the standard deviation same to

σ

in a statistical sense. The estimators of the two intensities at orthogonal polarization states are retrieved based on Eq. (2) as:

{\begin{cases} {\hat{I}}_{/ /} = \frac{I_{/ /}^{m}}{t_{/ /}} = I_{/ /} + \frac{n_{/ /}}{t_{/ /}} \\ {\hat{I}}_{⊥} = \frac{I_{⊥}^{m}}{t_{⊥}} = I_{⊥} + \frac{n_{⊥}}{t_{⊥}} \end{cases} .

It can be seen from Eq. (3) that the intensity measurement $I_{i}^{m}$ , $i \in {/ /, ⊥}$ is divided by the integration time $t_{i}$ , and thus the integration time multiplicatively modulates the noise variation. For the estimators of the two intensity measurements, the average values of them are $〈 {\hat{I}}_{/ /} 〉 = I_{/ /}$ and $〈 {\hat{I}}_{⊥} 〉 = I_{⊥}$ . In addition, since the noise in each intensity measurement is statistically independent from the other, the variations of the two measured intensities are independent from each other. Therefore, the covariance matrix of the random variables $X = ({\hat{I}}_{/ /}, {\hat{I}}_{⊥})$ is a diagonal matrix, in which the diagonal elements of the matrix indicate the variances of ${\hat{I}}_{/ /}$ and ${\hat{I}}_{⊥}$ , respectively, and thus $Γ^{X}$ is given by:

Γ^{X} = [\begin{matrix} \frac{σ^{2}}{t_{/ /}^{2}} & 0 \\ 0 & \frac{σ^{2}}{t_{⊥}^{2}} \end{matrix}] .

If one denotes the total intensity as

I = I_{/ /} + I_{⊥}

, then according to Eq. (1), one can parameterize the two light intensities by

I

and

P

as

I_{/ /} = \frac{I (1 + P)}{2}, I_{⊥} = \frac{I (1 - P)}{2} .

The estimator of DOLP is thus given by:

\hat{P} = \frac{{\hat{I}}_{/ /} - {\hat{I}}_{⊥}}{{\hat{I}}_{/ /} + {\hat{I}}_{⊥}} = \frac{(I_{/ /} + \frac{n_{/ /}}{t_{/ /}}) - (I_{⊥} + \frac{n_{⊥}}{t_{⊥}})}{(I_{/ /} + \frac{n_{/ /}}{t_{/ /}}) + (I_{⊥} + \frac{n_{⊥}}{t_{⊥}})} .

It can be seen in Eq. (6) that the integration time multiplicatively modulates the noise variation, and then the noise variation modulated by the integration time is transferred to the variation of the DOLP estimator with the fractional algorithm.

3. Estimating the variance of DOLP by Delta method

Actually, when the perturbations in the intensity measurements are small, $\hat{P}$ given by Eq. (6) can be considered as an unbiased estimator [8]. In this case, one can deduce the estimator of DOLP by an approximate method.

The “Delta Method” approximates the variance of any parameter (e.g. $y = f (X)$ ), which is a function of one or more random variables $X = (X_{1}, X_{2}, ..., X_{N})$ with mean of $〈 X 〉$ and covariance matrix of $Γ^{X}$ . If the variations of $X$ around $〈 X 〉$ are sufficiently small and the function $f (X)$ is sufficiently “smooth” around $〈 X 〉$ , then [16]:

〈 y 〉 \approx f (〈 X 〉), VAR [y] \approx {[\nabla f (〈 X 〉)]}^{T} Γ^{X} [\nabla f (〈 X 〉)],

where

\nabla f

refers to the gradient of the function

f

. Based on Eq. (6) and Eq. (5), the gradient of

\hat{P}

respecting to

X = ({\hat{I}}_{/ /}, {\hat{I}}_{⊥})

can be thus expressed as:

\nabla \hat{P} (〈 X 〉) = \frac{2}{{(I_{/ /} + I_{⊥})}^{2}} {[I_{⊥}, - I_{/ /}]}^{T} = \frac{1}{I} {[1 - P, - (1 + P)]}^{T} .

It is noticed in the last term of Eq. (8) that the partial derivatives of $\hat{P}$ with respect to ${\hat{I}}_{/ /}$ and ${\hat{I}}_{⊥}$ are different, indicating the different influences on the DOLP estimator by different intensity measurements. The approximate variance of $\hat{P}$ can be deduced by substituting Eq. (8) and Eq. (4) into Eq. (7), which is given by:

VAR [\hat{P}] = \frac{σ^{2}}{I^{2}} [\frac{{(1 - P)}^{2}}{t_{/ /}^{2}} + \frac{{(1 + P)}^{2}}{t_{⊥}^{2}}] .

In particular, it can be seen from Eq. (9) that the noise in the two intensity measurements influence the variance of DOLP estimator with different magnitudes, which are characterized by the weights ${(1 - P)}^{2} / I^{2} t_{/ /}^{2}$ and ${(1 + P)}^{2} / I^{2} t_{⊥}^{2}$ , respectively.

Besides, it can be seen from Eq. (9) that for the polarimetry system with a given value of DOLP, the variance of DOLP estimator depends on the integration times of two intensity measurements, which makes it possible to modulate the variance of DOLP estimator by adjusting the distribution of integration time, while keeping the total integration time constant. In this case, the problem of optimizing the distribution of integration time to achieve the minimum variance of DOLP estimator can be expressed as:

\min_{t} VAR [\hat{P}] = \min_{t} {\frac{σ^{2}}{I^{2}} [\frac{{(1 - P)}^{2}}{t_{/ /}^{2}} + \frac{{(1 + P)}^{2}}{t_{⊥}^{2}}]},

with the constraint that the total integration time is fixed.

4. Optimizing the distribution of integration time by Lagrange multiplier method

Lagrange multiplier method can be applied for the problem of constrained optimization [17], and thus it is applied to solve the optimization problem given by Eq. (10). The constraint of this optimization problem is that the total integration time of intensity measurements at two orthogonal states is constant. We assume the total integration time is fixed to be 2, and thus the constraint is given by:

h (t) = t_{/ /} + t_{⊥} - 2 = 0.

Consequently, the Lagrange function can be defined by [17]:

L (λ, t) = VAR [\hat{P}] + λ h (t) = \frac{σ^{2}}{I^{2}} [\frac{{(1 - P)}^{2}}{t_{/ /}^{2}} + \frac{{(1 + P)}^{2}}{t_{⊥}^{2}}] + λ (t_{/ /} + t_{⊥} - 2),

where

λ

is the Lagrange parameter.

The solutions of integration times for the optimization problem in Eq. (12) should fulfill that the partial derivatives of $L (λ, t)$ equal to zero, which are given by:

\begin{matrix} {\begin{matrix} \frac{\partial L}{\partial t_{i}} = \frac{- 2 C_{i}}{t_{i}^{3}} + λ = 0 \\ \frac{\partial L}{\partial λ} = t_{/ /} + t_{⊥} - 2 = 0 \end{matrix} & i \in {/ /, ⊥} \end{matrix},

where we denote

C_{/ /} = σ^{2} {(1 - P)}^{2} / I^{2}, C_{⊥} = σ^{2} {(1 + P)}^{2} / I^{2}

, in order to facilitate the discussion. It is clear that the solution of Eq. (13) should follow:

\frac{2 C_{/ /}}{t_{/ /}^{3}} = \frac{2 C_{⊥}}{t_{⊥}^{3}} = λ .

Consequently, the optimal distribution of integration time satisfies:

t_{/ /} : t_{⊥} = \sqrt[3]{{(1 - P)}^{2}} : \sqrt[3]{{(1 + P)}^{2}} .

If the total integration time is assumed to be 2, then the optimal integration times of intensity measurements are given by:

t_{/ /} = \frac{2 \sqrt[3]{{(1 - P)}^{2}}}{\sqrt[3]{{(1 + P)}^{2}} + \sqrt[3]{{(1 - P)}^{2}}}, t_{⊥} = \frac{2 \sqrt[3]{{(1 + P)}^{2}}}{\sqrt[3]{{(1 + P)}^{2}} + \sqrt[3]{{(1 - P)}^{2}}} .

The theoretical lower limits for

t_{/ /}

and

t_{⊥}

are 0, while the upper limits for

t_{/ /}

and

t_{⊥}

are the total integration time, which is considered to be 2 in our theoretical analysis.

The optimal integration times for $I_{/ /}$ and $I_{⊥}$ as function of P in all case of $P \in [- 1, 1]$ are plotted in Fig. 1 . In particular, it is noticed in Fig. 1 that the equalized integration times is the optimal solution only if the measured DOLP equals to zero. In addition, with the increase of the value of DOLP, the optimal value of $t_{/ /}$ decreases while the optimal value of $t_{⊥}$ increases.

Fig. 1 The optimal integration times of intensity measurements at different values of $P$ .

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Actually, it can be analyzed based on Eq. (9) that by increasing the value of $t_{⊥}$ , the decrease of the variance term corresponding to $t_{⊥}$ can overcome the consequent increase of the variance term corresponding to $t_{/ /}$ , and thus the total variance of the estimator can be decreased.

Substituting Eq. (16) in Eq. (9), we can get the expression of the optimized variance of DOLP estimator as:

VAR {[\hat{P}]}_{o p t} = \frac{σ^{2}}{4 I^{2}} {[\sqrt[3]{{(1 + P)}^{2}} + \sqrt[3]{{(1 - P)}^{2}}]}^{3},

while for the case of equalized integration times, the variance can be calculated from Eq. (9) as:

VAR {[\hat{P}]}_{e q u} = \frac{σ^{2}}{I^{2}} [{(1 + P)}^{2} + {(1 - P)}^{2}] .

Consequently, we can obtain the degree of optimization for the variance as:

Ψ = 1 - \frac{VAR {[\hat{P}]}_{o p t}}{VAR {[\hat{P}]}_{e q u}} = 1 - \frac{{[\sqrt[3]{{(1 + P)}^{2}} + \sqrt[3]{{(1 - P)}^{2}}]}^{3}}{4 [{(1 + P)}^{2} + {(1 - P)}^{2}]},

which indicates the ratio of variance reduction by optimizing the distribution of integration time.

It needs to be clarified that $P \in [- 1, 1]$ , and it can be seen in Eq. (17) to Eq. (19) that $VAR {[\hat{P}]}_{o p t}$ , $VAR {[\hat{P}]}_{e q u}$ and the degree of optimization $Ψ$ are even functions. Therefore, we will discuss the case for $P \geq 0$ in the following, and the result for $P \leq 0$ is same to that at $| P |$ .

We have plotted the normalized variance of $\hat{P}$ , which is given by $VAR [\hat{P}] / (σ^{2} / I^{2})$ , in Fig. 2(a) according to Eqs. (17) and (18) , and it can be seen that the variance of $\hat{P}$ with optimal integration times decreases with the increase of $P$ , while the variance with equalized times increases with the increase of $P$ . In addition, it is noticed in Eq. (19) that the performance of variance reduction by optimizing the distribution of integration time fully depends on the value of DOLP, and we have plotted the degree of optimization $Ψ$ as a function of the value of DOLP in Fig. 2(b), and it can be seen that $Ψ$ increases with the increase of DOLP, which means that the variance reduction by optimizing the distribution of integration time for intensity measurements is more effective at higher value of DOLP. In particular, it can be seen from Fig. 2(b) that when the value of DOLP is higher than 0.5, the variance can be reduced by more than 25% by optimizing the distribution of integration time. However, it is also noticed in Fig. 2(b) that when the value of DOLP is low, the effect of variance reduction is not significant.

Fig. 2 (a) The normalized variance of $\hat{P}$ with equalized integration times and with optimal integration times as a function of P; (b) The degree of optimization $Ψ$ at optimal distribution of integration time as a function of P.

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According to the analyses above, we address an approach that can reduce the variance of DOLP estimator, by optimizing the distribution of integration time for intensity measurements at two orthogonal states, while keeping the total integration time constant. In addition, this approach is meaningful for any value of DOLP, especially for higher value of DOLP.

5. Validation on real world experiments

In order to illustrate the effect of optimizing the distribution of integration time on the DOLP estimator, we perform the corresponding real world experiments with the arbitrary DOLP polarimetry setup (see Fig. 3 ), which is inspired by [18].

Fig. 3 Schematic of experimental setup of DOLP polarimetry. QWP: quarter-wave plate; LP: linear polarizer; P-BS: polarizing beam-splitter.

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The light source is a He-Ne laser. After the laser passing through a quarter-wave plate (QWP) and a linear polarizer (LP), a polarizing beam-splitter (P-BS₁) is placed, splitting the incident linear beam in orthonormal polarizations with the light intensities of $I_{/ /}$ and $I_{⊥}$ . The polarized component reflected at the P-BS₁ is steered by means of a mirrors system, and recombined with the transmitted component after reflection at the P-BS₂. By removing blocker 1 or blocker 2, we can achieve the measurements of $I_{/ /}$ and $I_{⊥}$ , respectively. In this setup, the degree of linear polarization of the light received by the detector depends on the orientation of the linear polarizer $θ_{L P}$ , which is given by $DOLP = | \cos 2 θ_{L P} |$ , and therefore, the degree of linear polarization can be fully controlled by adjusting the orientation $θ_{L P}$ .

In the case of low light intensity, the dominant source of noise can be considered to be the additive Gaussian noise [19]. Therefore, we perform the experiments at low light intensity, in which the main source of noise could contain the dark current noise, readout noise and quantization noise, and we check that the variance of the measured intensity is almost independent of the illumination intensity and integration time. In addition, it is also checked from the histograms of the measured intensities that the PDF (Probability Density Function) of noise are Gaussian shape. Therefore, the statistical performance of the noise is close to that of the additive Gaussian noise in our experiments.

In this experiment, we set the total integration time to 200ms, and we set the true value of DOLP to 0.6 (which means $〈 \hat{P} 〉 = 0.6$ ) by adjusting the orientation of the linear polarizer. Therefore, in the case of equalized integration times $t_{/ /} = t_{⊥} = 100 ms$ , while in the case of optimal integration times, it can be calculated based on Eq. (15) that the optimal integration times are $t_{/ /} = 57 ms,$ $t_{⊥} = 143 ms$ . In order to analyze the statistical characteristic of the DOLP estimator, we perform 1 × 10⁴ realizations to calculate the variance of DOLP estimator and the corresponding degree of optimization at each set of integration times, and the results are shown in Table 1 . In addition, the theoretical results of degree of optimization are also presented in Table 1.

Table 1. The experimental variances of estimator at different distributions of integration time. The experimental and theoretical results of degree of optimization Ψ are also presented. The true value of $P$ is 0.6.

View Table

It is noticed that the lower value of $t_{/ /}$ and the higher value of $t_{⊥}$ will lead to the variance reduction of DOLP estimator, and the variance reduction is maximum when the integration times of intensity measurements are 60ms and 140ms respectively, which is close to the optimal solutions of 57ms and 143ms. The theoretical degree of optimization, which is calculated based on Eq. (19), is also shown in Table 1 for the purpose of comparison. By comparing the last two columns of Table 1, it can be seen that for $t_{/ /}$ ranging from 40ms to 120ms, the experimental results are well consistent with the theoretical results, and thus the possibility of decreasing the variance of DOLP estimator by optimizing distribution of integration time for intensity measurements is demonstrated theoretically and experimentally.

Besides, it is noticed that when the value of $t_{/ /}$ is greater than 120ms, there is a considerable deviation of the experimental result from the theoretical result. This is because the degree of optimization also depends on the statistics of the noise. Since $I_{/ /}$ is relatively higher than $I_{⊥}$ , and with the great value of $t_{/ /}$ , the measured intensity $I_{/ /}^{m}$ could reach a high value according to Eq. (2). In this case, the statistics of the noise could deviate from additive Gaussian noise to Poisson shot noise, which is the dominant source of noise when the photon flux is large enough [19].

In order to experimentally investigate performance of variance reductions at different values of DOLP, we also perform experiments at various values of DOLP by adjusting the orientation of linear polarizer (LP) in the setup shown in Fig. 3. We perform the measurements of the values of DOLP at equalized integration times and at optimal integration times to calculate the degree of optimization $Ψ$ . In this case, we perform the experiments for the values of DOLP ranging from 0.1 to 0.9, and the results are shown in Fig. 4 . In addition, the theoretical values of $Ψ$ are also presented in Fig. 4 for the purpose of comparison. It can be seen from Fig. 4 that the experimental result of $Ψ$ increases with the increase of P, and it is also found that the experimental results consist well with the theoretical ones, which quantitatively demonstrates the correctness of our theoretical analyses in Section 4.

Fig. 4 The experimental and theoretical results of the degree of optimization $Ψ$ at optimal distribution of integration time as a function of $P$ .

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6. Conclusion

In conclusion, we consider the DOLP polarimetry perturbed by the additive Gaussian noise, and we have shown that the variance of DOLP estimator depends on the distribution of integration time for intensity measurements in the DOLP polarimetry. In particular, the equal distribution of integration time cannot lead to the minimum estimation variance in any case. In this paper, we obtain the closed-form solution of the optimal distribution of integration time in an approximate way by employing Delta method and Lagrange multiplier method. In addition, we also perform the real world experiments, and it is found that by optimizing the distribution of integration time for the intensity measurements, the variance of DOLP estimator can be decreased for any value of DOLP, and the variance reduction is more effective for the DOLP with a higher absolute value. In addition, it is found that the optimal value of $t_{/ /}$ decreases with the increase of DOLP while the optimal value of $t_{⊥}$ increases. The experimental result agrees well with the theoretical deduction, which demonstrates the feasibility of the method proposed in this paper.

This work has many perspectives. For example, it will be interesting to apply this method to other types of noise, such as Poisson shot noise, by modifying the Lagrange function in Eq. (12) according to the corresponding function of the noise variance. In addition, the idea of decreasing the variance by optimizing distribution of integration time proposed in this paper can be extended to other types of polarimetry systems, such as Mueller polarimetry system, in which the algorithm of optimizing the 16 integration times can be considered as an extension of that for Stokes polarimetry discussed in [11]. However, there are also some limitations for the method proposed in this paper, which include: 1) one must at least approximately know the value of DOLP in advance, in order to calculate the optimal measurement times; 2) this method is only available for the division of time polarimeter systems.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61405140 and No. 61379014), the National Instrumentation Program (No. 2013YQ030915), the Natural Science Foundation of Tianjin (No. 15JCQNJC02000), and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars. Haofeng Hu acknowledges the Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA) and the China Scholarship Council (CSC).

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$t_{/ /}$	$t_{⊥}$	Variance (Experimental)	Degree of optimization (Experimental)	Degree of optimization (Theoretical)
40ms	160ms	8.74 × 10⁻⁴	27.2%	26.5%
60ms	140ms	8.11 × 10⁻⁴	32.4%	35.6%
80ms	120ms	9.09 × 10⁻⁴	24.3%	25.5%
100ms	100ms	1.20 × 10⁻³	0%	0%
120ms	80ms	1.70 × 10⁻³	−41.7%.	−51.1%.

Optimal distribution of integration time for intensity measurements in degree of linear polarization polarimetry

Abstract

1. Introduction

2. Degree of linear polarization polarimetry

3. Estimating the variance of DOLP by Delta method

4. Optimizing the distribution of integration time by Lagrange multiplier method

5. Validation on real world experiments

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (19)

Optics Express