1. Introduction

Polarization imagers can reveal contrasts invisible to human eyes and to conventional cameras [1–5]. Polarization imagers based on division-of-focal-plane (DoFP) cameras have received much attention due to their integrated structure and their ability to measure linear polarization in real time [6–8]. Since commercial DoFP cameras are based on microgrids of linear polarizers, they can only measure the linear Stokes vector [8–13]. Hence, to measure the full Stokes vector, one has to place a varying retarder in front of them. In this article, we will consider a continuously rotating retarder with fixed retardance, and we will call this setup RRDOFP (rotating retarder and DoFP camera). With this setup, one has to perform at least two image acquisitions with two different orientations of the retarder [7,14,15] to measure the full Stokes vector.

When using a continuously rotating retarder, one has to choose the number and the duration of camera acquisitions within a cycle of the retarder. In a seminal paper, the optimization of these parameters was investigated in the case of a rotating-retarder-fixed-polarizer (RRFP) polarimeter based on a standard camera [16]. The main conclusion was that these parameters have a significant influence on the estimation precision of the Stokes vector, and that their optimal value depends on whether the variance of the noise that affects the measurements is time-dependent or not. Our first goal in this article is to perform a similar type of investigation in the case of RRDOFP imaging polarimeters. We will show that for these devices also, the optimal number and duration of acquisitions depends on the nature of the dominant type of noise, and that the precision of Stokes vector estimation with RRDOFP is slightly higher than with RRFP.

Moreover, if the number $K$ of acquisitions is larger than a certain value, the intensity measurements become redundant and it has been shown in the case of RRDOFP, that polarimeter parameters, such as the retardance of the rotating retarder, can be estimated together with the Stokes vector [6,17]. In the present article, we apply this approach to the estimation of the starting angle of the retarder and show that this starting angle need not be known precisely and can be autocalibrated in both RRFP and RRDOFP setups. This property significantly facilitates the synchronization of the rotating retarder with the camera acquisitions. We investigate the precision and domain of feasibility of this autocalibration and show the RRDOFP setup has more attractive properties. These results are important since they show that the RRDOFP architecture is a preferable alternative to RRFP architectures with the advent of cheap and efficient DoFP cameras.

2. Modeling RRFP measurements

In this section, we concentrate on the basic RRFP architecture. We determine the model of intensity acquisitions as a function of the integration time, in the presence of time dependent and time-independent noise sources.

2.1 Modeling the intensity measurements

We assume that a retarder and a linear polarizer are placed in front of a standard camera. The linear polarizer has fixed orientation and the retarder is continuously rotating at constant angular speed $d\theta / dt = v$. All temporal variations can thus be easily converted into angular variations by multiplying time intervals by the angular speed $v$. The camera measures the light flux at $K$ different moments in time which correspond to $K$ angular positions of the retarder, denoted $\{\theta _1,\ldots ,\theta _K \}$, over half a period of the rotation (since the polarimetric effect of the retarder has a periodicity of $\pi$ radian). Each of these measurements is integrated over a time interval $\Delta t$, which corresponds to the angular interval $\Delta \theta = v \Delta t$. Moreover, we assume that there is a "latency/waiting time" $\delta t$ between two successive camera acquisitions, due for example to pixel readout time. This corresponds to an "angular" latency $\delta \theta = v \delta t$, which represents the angular interval swept by the retarder during the time interval $\delta t$.

The different angular values and intervals are connected by the following relations:

 

Fig. 1. The relationship between the angular integral $\Delta \theta$ and the angular latency $\delta \theta$: The number of the angular positions of the retarder is $K = 5$ in this example.

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Let us define the incident photon flux on a pixel of the sensor when the retarder is at angle $\theta$ as:

2.2 Modeling the estimation variance

The expression in Eq. (6) is ideal. In practice, we have to take into account the measurement noise, and Eq. (6) becomes:

The Stokes vector is estimated from these noisy intensity measurements with the pseudo-inverse estimator:

From the above equations, it is seen that the estimation covariance matrix depends on the measurement matrix $\mathbb {W}$, and Eq. (9) can be written as follows:

The estimation precision of the Stokes vector can be globally expressed by a single parameter, the "Equally Weighted Variance", defined as

$\mbox {EWV} =\mbox {trace} \left [ {{\Gamma ^{\hat S}}} \right ]$. It is a reasonable and widely used quality criterion for Stokes vector measurements, since it represents the sum of the variances of the four elements of the Stokes vector estimate. According to Eq. (12), the EWV can be written as:

2.3 Definition of the optimization criterion

Our objective is to optimize the acquisition parameters ($K$, $\Delta \theta$, $\delta \theta$). There are some degrees of freedom to define the optimization criterion, depending, for example, on the relative importance of the different elements of the Stokes vector for the application at hand. However, in all cases, it must be a sort of signal to noise ratio (SNR) between the square norm of the true value of the Stokes vector and its estimation variance due to noise sources. In this article, we choose the ratio of the measured intensity of the Stokes vector over the EWV as the optimization criterion:

We will call this criterion "precision contrast". It depends on the parameters $K$, $\Delta \theta$, $\delta \theta$, in particular through $\mathbb {W}$, and on the input Stokes vector.

In Eqs. (14) and (15), it is seen that the third additive term of the EWV depends on the polarization state of the input Stokes vector. In consequence, the EWV varies around a fixed value (given by the first two terms) when the polarization state changes. This can be shown mathematically by computing the "average" level of EWV, averaged over the whole Poincaré sphere, in the following way [20,21]:

In consequence, to simplify optimization, it is legitimate to optimize the following expression of the contrast, where we have suppressed the third term of the EWV:

Let us now consider two particular cases of this equation. In the first one, we assume that the additive noise sources are negligible with respect to signal dependent Poisson noise (thus, photon noise), that is, $\sigma _r \simeq 0$ and $d_\theta \simeq 0$. This situation is frequently encountered in modern cameras. In this case, Eq. (17) becomes:

Let us now assume on the contrary that the readout noise is dominant with respect to the time-dependent noise sources (including photon noise corresponding to the signal, dark and background noise). In this case, Eq. (17) becomes:

The main difference between $\mathcal {C}_{poi}$ and $\mathcal {C}_{add}$ is their dependence on the integration angle $\Delta \theta$. We will see in the following that this incurs significant differences in terms of optimal acquisition parameters.

3. Contrast optimization

The equations derived until now are very general since they depend only on the measurement matrix $\mathbb {W}$ corresponding to the $K$ different measurements performed during a half rotation of the retarder. In this section, we derive the actual expressions of the precision contrast for the RRFP and RRDOFP setups. Then, we determine the measurement parameters that optimize the estimation contrast in the presence of signal independent additive noise (Eq. (19)) and of signal dependent Poisson noise (Eq. (18)). We finally discuss and compare the results obtained with these two setups.

3.1 Expression of the measurement matrix in RRFP and RRDOFP setups

Let us consider a general class of polarimetric imagers composed of a polarizer and a retarder that may have variable orientations. The analysis vectors, which are the rows of the measurement matrix $\mathbb {W}$, have the following expression [6]:

In this equation, $c_\delta =\cos \delta$, $s_\delta =\sin \delta$, where $\delta$ is the retardance of the retarder, and

In the particular case of RRFP setups, the direction of the polarizer is fixed. Hence the values of $\varphi _k$ are all equal and can be assumed, without loss of generality, to be equal to $0^\circ$. Hence, Eq. (22) becomes:

In the RRDOFP configuration, the expression of the vector $\textbf {w}_k$ is given by Eq. (22), where $\varphi _k$ can be equal to 0, 45, 90, or 135$^\circ$. The matrix $\mathbb {W}$ has $4K$ rows.

3.2 Precision contrast optimization with the RRFP setup

It is to be noted that the optimization of the number of acquisitions with the RRFP setup has been addressed in Ref. [16] albeit with a slightly different optimization criterion. However, in this section, we devote some effort to analyzing it in order to prepare the comparison with the RRDOFP setup that will be the object of Section 3.3.

For the RRFP setup, estimation of the full Stokes vector is possible only if the number of measurements is $K \ge 5$ [16]. In this case, it is easily shown from Eq. (24) that:

The most easily available type of retarder is quarter wave plate (QWP), that is, $\delta = 90^\circ$. In this case, the matrix in Eq. (25) simplifies and one has:

Let us first assume that the dominant noise source is additive, so that the expression of the precision contrast is given by Eq. (19). Substituting Eq. (26) in this equation, one obtains the following expression of the precision contrast:

The question is to find the parameter values optimizing this contrast. In practice, the minimal latency between the measurements $\delta \theta$ is fixed by camera technology. For a given value of $\delta \theta$, one thus has to optimize the number of acquisitions $K$ or, which is equivalent, the integration angle $\Delta \theta$ since these three parameters are linked by Eq. (1). We have plotted in Fig. 2(a) the variation of $\mathcal {C}_{add}$ as a function of $K$ for different values of $\delta \theta$ equal to $0^\circ , 2^\circ , 4^\circ , 6^\circ$ and $8^\circ$. It is seen that all the curves decrease. This means that for any value of $\delta \theta$, the optimal number of measurements is minimal, that is, equal to $5$. To interpret this result, it is important to remember that in the case of time independent additive noise, each single measurement adds the same amount of noise, whatever its duration. Thus, increasing $K$ increases the total amount of collected noise. Figure 2(a) shows that this effect is dominant compared to the decrease of the function $f(\Delta \theta )$ due to the reduction of $\Delta \theta$ with the increase of $K$. Moreover, it is noted that the contrast decreases as $\delta \theta$ increases, which is normal since as the latency increases, the total amount of light acquired by the camera, which is proportional to $\Delta \theta$, decreases.

 

Fig. 2. $C$ as a function of $K$ in terms of different $\delta \theta$ with RRFP setup: (a) $C_{add}$, (b) $C_{poi}$, and with RRDoFP setup: (c) $C_{add}$, (d) $C_{poi}$.

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Let us now assume that the dominant noise source is Poisson shot noise, so that the expression of the contrast is given by Eq. (18). Substituting Eq. (26) in this equation, one obtains:

We have plotted in Fig. 2(b) the variation of this contrast as a function of $K$ for different values of $\delta \theta$. It is seen that when $\delta \theta = 0^\circ$, the contrast monotonically increases. Indeed, in this case, one has $\Delta \theta = \pi /K$ and Eq. (28) becomes:

Since $f(\pi /K)$ decreases as $K$ increases, the contrast increases with $K$. This means that in the absence of latency, the best strategy is to make as many measurements as possible in order to reduce $\Delta \theta$ and thus increase the coefficients $u$ and $v$ to values arbitrarily close their maximum value, which is $1$. This is in sharp contrast with what happens in the presence of additive noise, where the optimal number of measurements is minimal ($K=5$). This is due to the fact that in the case of Poisson shot noise, making more measurements does not incur any increase in noise variance, since the noise level only depends on the total number of photons collected by the camera, irrespective of the number of acquisitions in which it is split. However, when $\delta \theta$ is non zero, it is observed in Fig. 2(b) that the contrast no longer increases indefinitely, but reaches a maximum for a finite value of $K$. This is due to the fact that the total amount of lost acquisition time is equal to $K\cdot \delta \theta$ and thus increases with $K$. Hence, there is a compromise between the reduction of coefficients ($u$, $v$) and the decrease of the amount of collected signal as $K$ increases. This results in an optimal value of $K$ that decreases as $\delta \theta$ increases.

3.3 Precision contrast optimization with the RRDOFP setup

Let us now consider the RRDOFP setup. In this case, for each image acquisition, 4 intensity measurements are performed with micro-polarizer orientations at $0^\circ$, $45^\circ$, $90^\circ$ and $135^\circ$. The corresponding measurement matrix $\mathbb {W}$ thus has $4K$ rows. It can be shown that the estimation of the full Stokes vector is possible only if $K \geq 2$ [6,7]. However, as will be seen in the following, the case $K \ge 3$ is particularly interesting. In this case, it is easily shown from Eq. (22) that:

For completeness, the expression of this matrix for $K=2$ is given in Appendix B. If ones assumes $\delta = 90^\circ$, Eq. (30) becomes:

This expression is quite close to Eq. (26), except for a coefficient $1/4$ that is due to the fact that in DOFP measurements, 4 pixels are used to estimate the Stokes vector ${\bf S}$ instead of a single one for a standard camera. Hence, the evolution of the contrast with $K$ is quite similar to the case of the RRFP setup. We have plotted in Fig. 2(c) the variation of $\mathcal {C}_{add}$ as a function of $K$ for different values of $\delta \theta$. It is seen that, as in the RRFP case, the contrast monotonically decreases. This means that for any value of $\delta \theta$, the optimal number of measurements is minimal, that is, equal to $2$. If the dominant noise source is Poisson shot noise, we observe in Fig. 2(d) that when $\delta \theta = 0$, the contrast monotonically increases, and when $\delta \theta$ is non zero, the contrast reaches a maximum.

3.4 Summary and discussion

In the two previous sections, we have considered the separate cases of pure additive and pure Poisson shot noise sources to analyze more easily the behavior of RRFP and RRDOFP imagers. However, the equations we have derived make it possible to determine and optimize setups for any camera and noise characteristics. The main conclusion of this study is that the parameter that primarily determines the evolution of the contrast with the number of measurements is the nature of the dominant noise. If the latency $\delta \theta = 0$, the contrast strictly decreases with $K$ in the presence of additive noise, and strictly increases in the presence of Poisson shot noise. In the practical cases when these two types of noise sources are simultaneously present, there is an optimal value of $K$ that depends on the latency $\delta \theta$. We have shown that for a given type of dominant noise source, the RRFP and RRDOFP setups behave in a similar way as $K$ varies. However, the precision contrast is always larger in the RRDOFP case. It is interesting to notice that for given values of the system parameters, the ratio $R$ of the contrast obtained with the RRFP setup (that we denote $C_{\operatorname {RRFP}}$), and with the RRDOFP setup (that we denote $C_{\operatorname {RRDOFP}}$) is independent of the nature of noise since

We have plotted $R$ as a function of $K$ for different values of the latency $\delta \theta$ in Fig. 3. One can notice that $R$ is always greater than $7.5$. Part of this difference is explained by the fact that in the RRFP setup, Stokes vector estimation is performed with 4 pixels, so that 4 times more photons are used for estimation, which accounts for a reduction by a factor 1/4 of the EWV, and, thus, an increase by a factor 4 of the contrast. The remaining part of the factor may be interpreted as a better exploitation of the input light when it is analysed with four different polarizer directions simultaneously. The counterpart of this contrast increase is that the spatial resolution of the RRDOFP is divided by 2 in both directions with respect to the RRFFP setup. Moreover, RRDOFP images show artefacts if the spatial variations of the scene have so high spatial frequency that different Stokes vectors are present within the same super-pixel [22]. On the other hand, we will see in the next section that one advantage of RRDOFP is that if the starting angle of the retarder is unknown, it can be autocalibrated with better precision than with the RRFP setup.

 

Fig. 3. The contrast ratio $R$ as a function of the number of measurements $K$.

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4. Bias due to the lack of knowledge of retarder angles

The Stokes vector estimation method considered in the previous section relies on the perfect knowledge of the measurement matrix $\mathbb {W}$, and thus of the measurement angles $\theta _k$ of the rotating retarder. This requires a perfect synchronisation of the motor of the retarder and of the camera acquisition time. If these angles are imperfectly known, it may introduce a bias in the estimation of the Stokes vector. In this section, we evaluate the bias due to a lack of knowledge of the starting angle $\theta _1$.

Let us assume that the first measurement angle $\theta _1$ is imperfectly known. Its nominal value is $0^\circ$, and the corresponding measurement matrix is $\mathbb {W}_0$, but its actual value is $\theta _1$ and the corresponding measurement matrix is $\mathbb {W}$. Since the value of $\theta _1$ is unknown, one uses the matrix $\mathbb {W}_0$ to invert the intensity measurements acquired with the actual matrix $\mathbb {W}$. Thus, the statistical mean of the estimated Stokes vector is $\langle \widehat {\textbf S} \rangle = \mathbb {W}_0^{+} \textbf {I} = \mathbb {W}_0^{+} \mathbb {W} \textbf {S}$, and the estimation bias can be defined as [23]:

Using the expressions of $\mathbb {W}$ given in Eqs. (22) and (25), it is shown after cumbersome but elementary calculations that for the RRFP setup, if we assume $\delta = 90^\circ$, we have ($K \ge 5$):

On the other hand, for RRDOFP setup, if we assume $\delta = 90^\circ$, Eqs. (22) and (30) lead to ($K \ge 3$):

Comparing Eqs. (34) and (35), it is seen that the expressions of the bias in the two setups have some similarities. In particular, the bias on $S_3$ is identical for both setups, and it only depends on the actual value of $\theta _1$. On the other hand, the bias on $S_0$ is significant for the RRFP setup whereas it is null for the RRDOFP. The biases on $S_1$ and $S_2$ only depend on $\theta _1$ in RRFP, but are multiplied by a factor $u/(u+1)$ in the case of RRDOFP.

Since the bias is a vector, let us consider its squared norm in order to get a scalar representation of its global value. From Eq. (33), one has

The values of this parameter for the RRFP and the RRDOFP setup are obtained from Eqs. (34) and (35):

More than the absolute value of the bias, what is really important is its ratio with the estimation standard deviation of the Stokes vector. Indeed, if the bias is smaller than the standard deviation, it has no considerable influence. Hence, the parameter of interest is the ratio between the bias and the EWV:

If this parameter is larger that 1, it can be said that the bias has a significant influence on the estimation result. When the noise source is additive, it is easily shown that the parameter $\rho _r$ has the following expressions in the two setups:

It is seen that this ratio is proportional to the SNR, i.e., as the SNR increases, the bias becomes more significant with respect to the estimation variance. Moreover, when $\theta _1$ is small, this ratio is proportional to $|\theta _1|$. For illustration, we have plotted in Fig. 4(a) the value of $\rho _{\textrm {RRFP}}/(P\cdot \textrm {SNR})$ and $\frac {1}{4} \rho _{\textrm {RRDOFP}}/(P\cdot \textrm {SNR})$ (the factor "$\frac {1}{4}$" is because the estimation is accomplished in a $2 \times 2$ super-pixel) as a function of $\theta _1$ for $K = 5$ and different values of the latency $\delta \theta =0^\circ , 2^\circ , 4^\circ , 6^\circ$, and $8^\circ$). It is obvious that when the SNR is large, the ratio can take values much larger than 1. For example, for $K=5$, $\delta \theta = 0$ and $P=1$, ${\rho _{\operatorname {RRFP}}}$ is equal to 1 for $\theta _1=8.7^\circ$ if $\textrm {SNR}=10$ (low SNR), but for $\theta _1=0.87^\circ$ if $\textrm {SNR}=100$ (good SNR). This means that the bias is highly detrimental to Stokes vector estimation. In addition, in order to compare the performance of the two setups, we have plotted in Fig. 4(b) the ratio $4\rho _{\operatorname {RRFP}}/\rho _{\operatorname {RRDOFP}}$ as a function of $\theta _1$. One can see that it ranges from about 0.5 to 3: the RRFP and RRDOFP setups are thus similarly affected by bias.

 

Fig. 4. (a) Values of $\rho _{\textrm {RRFP}}/(P\operatorname {SNR})$ and $\frac {1}{4} \rho _{\textrm {RRDOFP}}/(P\operatorname {SNR})$ as a function of the starting angle $\theta _1$ in the case of different latency $\delta \theta$ and $K = 5$. (b) The ratio of the parameters for RRFP and RRDOFP setups as a function of the starting angle $\theta _1$.

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As a summary, we have shown that the lack of accurate knowledge of $\theta _1$ has a rapid detrimental effect on the estimation precision of $\mathbf {S}$, especially when the SNR is large. It is thus necessary to know it precisely. In the following, we investigate a method to estimate its value from the acquired data.

5. Autocalibration of the starting angle

In order to avoid the bias due to the lack of knowledge of $\theta _1$, one solution is to estimate the value of this angle from the data acquired by the polarimeter. This is possible if the number $K$ of intensity measurements is sufficient. In this section, we first review the theory of autocalibration of redundant full Stokes polarimeters developed in Ref. [6]. We then apply this theory to evaluate the feasibility and the precision of autocalibration of $\theta _1$ in RRFP and RRDOFP setups.

5.1 Theory of autocalibration of redundant polarimeters

The theory of autocalibration of redundant full Stokes polarimeters has been developed in Ref. [6]. Let $x$ be an unknown parameter on which the polarimeter depends (such as a retardance or an angle of the retarder). It has been shown that in the presence of additive Gaussian noise of variance $\sigma ^2$, the Cramer-Rao lower bound (CRLB) [24] of the estimation of $x$ has the following expression [6]:

Autocalibration is possible only if $\mathrm {CRLB}[x]$ is finite, that is, from Eqs. (43), if $\left \|\mathbb {Q}\mathbf {S}\right \|\neq 0$. Hence, the domain of feasibility of autocalibration, that is, the set of input vectors ${\bf S}$ for which autocalibration is possible, is determined by the properties of the matrix $\mathbb {Q}$, and, in particular, its rank. For example, in Ref. [6], it was shown that autocalibration of the retardance of the rotating retarder is never possible for the RRFP setup since $\operatorname {rank} [ \mathbb {Q}] = 0$. In contrast, for the RRDOFP setup, $\operatorname {rank} [ \mathbb {Q}] = 2$ and autocalibration is possible as soon as the degree of linear polarization (DOLP) of ${\bf S}$ is non zero. In the following, we use these equations to investigate the domain of feasibility and the precision of the autocalibration of $\theta _1$ in the case of RRFP and RRDOFP setups based on continuously rotating retarder.

5.2 Feasibility and precision of autocalibration in the RRFP setup

The rows of the measurement matrix of the RRFP setup are given in Eq. (24). Deriving this equation with respect to $\theta _1$, one obtains:

In other words, the second and third column of the matrix $\frac {\partial \mathbb {W}}{\partial \theta _1}$ are linear combinations of columns of $\mathbb {W}$. Therefore, the second and third column of the matrix $\mathbb {Q}$ are zero. Since, obviously, its first column is also zero, only its fourth column is nonzero and $\operatorname {rank}[\mathbb {Q}] = 1$. Hence, the null space of $\mathbb {Q}$ comprises all the purely linear states. In consequence, for autocalibration to be possible, the input Stokes vector must have a nonzero circular component $S_3$. More precisely, it can be shown from Eqs. (43) and (47) that the CRLB of $\theta _{1}$ is equal to:

Let us now investigate the influence of estimating $\theta _1$ on the estimation precision of the Stokes vector. Using Eqs. (12) and (25), it is easily shown that when $\theta _1$ is known, the variance (and thus, the CRLB) of the Stokes elements have the following expressions:

Moreover, from Eqs. (43) and (47), one has:

Interestingly, this expression is independent of $K$ and $\Delta \theta$. Hence, using Eqs. (45) and (48), one obtains the expression of the EWV when the parameter $\theta _1$ is autocalibrated:

As an interesting particular case, when $\Delta \theta = 0^\circ$ and $\delta = 90^\circ$, we have:

The first additive term corresponds to the EWV when $\theta _1$ is known, and the second to the excess of variance due to estimating it. This excess of variance diverges when $S_3$ tends to 0, that is, when the input Stokes vector becomes purely linear.

Let us now verify these results with Monte Carlo simulations. In order to implement autocalibration, one has to jointly estimate $\theta _1$ and ${\bf S}$ in the Maximum-Likelihood sense by determining [17]:

An important point has to be noticed. We have represented in Fig. 5 the function $\mathcal {F}(\theta _1)$ as a function of $\theta _1$ for $K=5$, the true value of $\theta _1=0^\circ$, $\textbf {S}=[1,0,0.1,0.1]^T$ and without noise. It is clearly seen that the curve reaches a minimum for the true value $\theta _1=0^\circ$ but also for $\theta _1=90^\circ$. It is easily seen that these two values of $\theta _1$ correspond to opposites values of the ellipticity. As consequence, depending on the starting point of the optimization, one may estimate a different sign of ellipticity. The same ambiguity is present in the RRDOFP setup. However, if the value of $\theta _1$ is approximately known within an interval of $\pm 45^\circ$ around the true value of $\theta _1$, so that the starting point of the optimization algorithm belongs to this range, the ambiguity disappears. In the subsequent Monte Carlo simulations, we will place ourselves in this situation.

We have plotted in Fig. 6(a) the square root of the CRLB (that we call "RCRB") of $\theta _1$ as a function of the SNR (defined in Eq. (41)). We have assumed that $K=5$, $\delta =90^\circ$, $S_0$=1, $\alpha =0^\circ$ and three different values of the ellipticity $\varepsilon = 5^\circ , 25^\circ , 45^\circ$. It is verified that, as indicated by Eq. (48), for a given value of the SNR, the RCRB is minimal when the ellipticity $\varepsilon =45^\circ$, that is, when the input Stokes vector is circular, and increases as the DOCP decreases, i.e., $\varepsilon$ decreases. It would diverge for $\varepsilon =0^\circ$. We have also represented, on the same graph, the standard deviation(STD) of $\theta _1$ estimated by Monte Carlo simulations on 1000 noise realizations using Eq. (53) (dotted lines with markers). It is seen that when the SNR is large, the STD fits the RCRB, which means that the RCRB is an accurate representation of estimation precision in this case. However, when the SNR gets below a certain value, STD progressively departs from the RCRB. It is also noticed that the SNR value for which this deviation occurs depends on the ellipticity of the input Stokes vector: it increases as $\varepsilon$ gets closer to $0^\circ$.

 

Fig. 5. Variation of $F(\theta _1)$ as a function of $\theta _1$ for the RRFP setup for $K=5$, a true value of $\theta _1=0^\circ$, $\textbf {S}=[1,0,0.1,0.1]^T$ and without noise.

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Fig. 6. RCRB and STD as a function of SNR for estimation (a) $\theta _1$, (b) $S_1$, (c) $S_2$, (d) $S_3$ with the RRFP setup. The input Stokes vector is such that $S_0=1$, $P=1$, $\alpha =0^\circ$, $\varepsilon =5^\circ , 25^\circ$, or $45^\circ$. .

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To illustrate the influence of this deviation on the estimation of the Stokes vector, we have represented the RCRB and STD of the Stokes vector components $S_1$ (Fig. 6(b)), $S_2$ (Fig. 6(c)), and $S_3$ (Fig. 6(d)). It is observed that the RCRB of $S_1$ and $S_3$ is independent of the value of the ellipticity (that is, of $S_3$), which is consistent with Eq. (50) and the fact that in this simulation, $\alpha =0^\circ$, so that $S_2=0$. On the other hand, for a given value of the SNR, the RCRB of $S_2$ increases when the ellipticity decreases. Moreover, the STD progressively departs from the RCRB when the SNR gets below a certain value, and the SNR value for which this deviation occurs increases as $\varepsilon$ gets closer to $0^\circ$. This divergence can occur for large values of the SNR: for example, it begins at $\operatorname {SNR}=100$ for $S_1$ when $\varepsilon =5^\circ$. In conclusion, the RRFP setup is not adapted to autocalibration of $\theta _1$ when the input Stokes vector is close to linear.

5.3 Feasibility and precision of autocalibration in the RRDOFP setup

Let us now consider the RRDOFP setup. In this case, Eq. (22) leads to:

It can be checked that none of last three columns of $\frac {\partial \mathbb {W}}{\partial \theta _1}$ belongs to the subspace spanned by the columns of $\mathbb {W}$, hence $\operatorname {rank}[\mathbb {Q}] = 3$. Moreover, it can be shown from Eq. (43) and (54) that the CRLB of $\theta _{1}$ is equal to:

Interestingly, this expression is independent of $K$ but depends on the angular interval $\Delta \theta$ through the parameter $u$. Hence, using Eqs. (45), (55), and (57), one obtains:

When $\Delta \theta = 0^\circ$ and $\delta =90^\circ$, we have $u=v=1$ and $A=2$. Hence,

In order to illustrate the obtained results, we have plotted in Fig. 7(a) the RCRB of $\theta _1$ as a function of the SNR for different values of the ellipticity. It is verified that, as indicated by Eq. (55), for a given value of the SNR, the RCRB depends on the ellipticity. However, this dependency is weaker than in the RRFP case and, most importantly, it remains finite for all values of $\varepsilon$. The STD of $\theta _1$ (dotted lines with markers) progressively departs from the RCRB when the SNR decreases, but this divergence occurs for much lower values of the SNR than in the RRFP case (smaller than 10). We have also represented the RCRB and the STD of the Stokes vector components $S_1$ (Fig. 7(b)), $S_2$ (Fig. 7(c)), and $S_3$ (Fig. 7(d)). It is observed that the STD departs from the RCRB for SNR values much smaller than in the RRFP case for all values of the ellipticity. In conclusion, with the RRDOFP setup , it is possible to estimate $\theta _1$ for all values of the ellipticity. The only limitation is that the value of the degree of polarization must be different from 0. Autocalibration of $\theta _1$ is thus much easier with the RRDOFP setup.

 

Fig. 7. RCRB and STD as a function of SNR for estimation (a) $\theta _1$, (b) $S_1$, (c) $S_2$, (d) $S_3$ with the RRDOFP setup. The input Stokes vector is such that $S_0=1$, $P=1$, $\alpha =0^\circ$, $\varepsilon =5^\circ , 25^\circ$, or $45^\circ$.

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It should also be noted that all the above results obtained with $\delta =90^\circ$ can be extended to arbitrary retardance values thanks to the general expressions of the precision contrast $C$, EWV, and CRLB given in this paper. Moreover, the results presented above assume a perfect knowledge of retardance value of the retarder. When the true retardance value is slightly different from the nominal value, the results of estimated Stokes vector and autocalibrated starting angle may have slight biases compared with those deduced by the perfect known retardance. If these biases are too large for the application at hand, it would be necessary to perform autocalibration of the retardance with the method described in Refs. [6] and [17]. Investigating the joint autocalibration of the starting angle and the retardance is an interesting topic for future research.

6. Conclusion

In this paper, we have investigated the optimization of RRFP and RRDOFP full Stokes imagers, and first determined the optimal number $K$ of intensity measurements through a cycle of the rotating retarder, taking into account time integration. We have shown that this number mainly depends on the dominant type of noise: the optimal value of $K$ is significantly smaller for additive noise than for Poisson noise. In practice, the actual noise is a mixture of these two types of noise sources, and the equations we have derived make it possible to compute the optimal value of $K$ in any application scenario. We have also demonstrated that in both setups, the starting angle of the retarder can be estimated together with the polarization state of the input light and that the precision and the feasibility domain of this autocalibration depends on the measured Stokes vector itself. More importantly, we have shown that the RRDOFP setup is significantly more efficient in this respect since, contrary to the RRFP setup, it makes autocalibration possible for any input Stokes vector with nonzero degree of polarization.

These results are important to optimize and facilitate the operation of polarization imagers based on rotating retarders. In particular, they show that the RRDOFP setup is a preferable alternative to the RRFP setup with the advent of cheap and efficient DOFP cameras.

Appendix A. Main statistical noise sources in the camera imaging model

The main statistical noise sources in that we consider in our imaging model are listed in the following Table 1 [19]

Table 1. The main statistical noise sources in the camera imaging system.

View Table

Appendix B. RRDOFP contrast and bias for $K=2$

For the RRDOFP setup, when the number of image acquisitions are $K = 2$ and the retardance of the retarder $\delta = 90^\circ$, one has:

(B1)$$\mathbb{W}^T\mathbb{W} = \frac{\textrm{1}}{\textrm{4}}\left[ {\begin{array}{cccc} \textrm{8} & 0 & 0 & 0\\ 0 & {{{\left( {\sqrt u + 1} \right)}^2}} & {\sin 4{\theta _1}\left( {u + \sqrt u} \right)} & 0\\ 0 & {\sin 4{\theta _1}\left( {\sqrt u - u} \right)} & {{{\left( {\sqrt u - 1} \right)}^2}} & 0\\ 0 & 0 & 0 & {4v} \end{array}} \right]$$

Moreover, the bias matrix for a value $\theta _1$ different from the nominal value $0^\circ$ is:

(B2)$$\mathbb{B} = \left[ {\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & {\frac{{2\sqrt u {{\sin }^2}2{\theta _1}}}{{\sqrt u + 1}}} & { - \frac{{\sqrt u \sin 4{\theta _1}}}{{\sqrt u + 1}}} & 0\\ 0 & {\frac{{\sqrt u \sin 4{\theta _1}}}{{\sqrt u - 1}}} & {\frac{{2\sqrt u {{\sin }^2}2{\theta _1}}}{{\sqrt u - 1}}} & 0\\ 0 & 0 & 0 & {1 - \cos 2{\theta _1}} \end{array}} \right]$$

Funding

National Natural Science Foundation of China (61775163, 62075161); Direction Générale de l’Armement; Agence Nationale de la Recherche (ANR-16-ASMA-0007-01 POLNOR).

Disclosures

The authors declare no conflicts of interest.

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