1. Introduction

Polarimetric imaging has proven to be useful for retrieving information that is normally invisible to standard intensity imaging. This capability has been fruitfully applied to several fields such as industrial inspection, remote sensing, or biomedical imaging [1–8 ].

Polarimetric imagers often use polarization modulation devices based on liquid crystal variable retarders (LCVR), which are fast and reliable [8–11 ]. However, LCVR control the polarization state of light only at one given nominal wavelength, and performance loss might be observed if imaging is performed at other wavelengths, due to the wavelength dependence of the LCVR (this phenomenon will be referred to as polarimetric chromatic aberration in the remainder of the article). If the light source that illuminates the scene has a broad spectrum (for example, a white light source), it is thus necessary to insert a narrowband spectral filter in the imaging path [12, 13]. However, spectral filtering significantly decreases the amount of light entering the system and thus the signal-to-noise ratio of polarimetric images. A way to circumvent this issue is to achromatize the polarization modulators. However, this comes at the price of higher complexity and cost, and this may not be needed if the objective is to improve target detection performance by increasing the target/background discriminability (or contrast).

Recently, this issue has been investigated for passive polarimetric imagers, where the light scattered by the scene is analyzed by a chromatic polarization modulator called Polarization State Analyzer (PSA) based on LCVR. It has been shown [14] that despite the loss of polarimetric accuracy due to the apparition of polarimetric chromatic aberrations, the contrast is improved by allowing more wavelengths to enter the imager - or in other words, by broadening the spectrum of the light entering the system. Moreover, there is a configuration of the PSA that maximizes this improvement. In other words, the spectral bandwidth should now be considered as a further parameter to optimize polarimetric imagers.

In this Article, we address the same problem in the case of active polarimetric imagers where the illumination is also controlled by a chromatic polarization modulator called PSG (Polarization State Generator). This issue is important in practice since active polarimetric imagers are known to provide better contrast than passive ones thanks to the control of the polarization of illumination. The problem is more complex since broadening the spectrum now impacts both the PSA and PSG, that must be jointly optimized in order to increase target/background discriminability. We demonstrate that there exists configurations of the PSA and PSG for which the increase of light flux overcomes the polarimetric property mismatch of the system components caused by spectral broadening. We investigate the theoretical foundations of this result through numerical simulations and validate the results on images acquired with a real-world active polarization imager.

The Article is organized as follows. Section 2 presents the theoretical framework on which the Article relies. Through a numerical study, Section 3 investigates the behavior of typical polarimetric scenes as the spectral bandwidth of a polarimetric system is increased and how the PSA and PSG configurations can be set to optimize target/background discriminability. These results are then validated in Section 4 through an experiment performed with a division of time polarimeter (DOTP).

2. Theoretical framework

Let us consider an active polarimetric system that observes a scene composed of two regions with different polarimetric properties, one referred as the “target” and the other as the “background”. The polarimetric properties of both regions are defined by 4 × 4 Mueller matrices denoted as M _t and M _b for the target and the background respectively. The polarization state of the light illuminating the scene is determined by its Stokes vector S generated by a Polarization State Generator (PSG), and the polarization state of the light scattered by the scene is analyzed through a Polarization State Analyzer (PSA) with a tunable eigenstate noted T.

Moreover, it is known that the polarimetric properties of a scene, as well as the characteristics of the PSA and the PSG depend on the wavelength of interest. Furthermore, the PSG and PSA are usually controlled using a set of parameters (such as a set of voltages in the case of Liquid Crystal Variable Retarders [LCVR], for example). Therefore, in the following, the Stokes vectors will be considered homogeneous to a light flux and will be parametrized as

After passing through the PSA, the light is then acquired by a quantum detector (a CCD or CMOS camera, for example) to obtain an image of the scene. The signal can then be expressed in number of photo-electrons and written for each pixel as

The statistical average of the signal defined in Eq. (4) for both regions is:

However, Eq. (10) is valid only for a specific wavelength. Consequently, if we remove the filter and increase spectral width, one has to take into account the integration over the whole illumination spectrum. Therefore, the previous optimal configuration does not apply anymore and the complete expression of the contrast should be taken into account:

3. Numerical study

In this section, we illustrate the impact of spectral broadening on the value of the contrast between the target and the background through simulations involving ideal components for both the set-up and the scene of interest. Section 3.1 presents the model of the simulated scenarios, and Section 3.2 then presents the results obtained for different scenes.

3.1. Description of set-up used in the simulation

The simulation scenarios are based on an existing set-up described in [13]. The illumination part is made of a cold halogen white source of spectrum I ₀(λ) followed by a PSG composed of two liquid crystal variable retarders (LCVR) and one polarizer as described in the previous section. The light is then analyzed by a PSA fashioned in a similar way and the image is obtained using a quantum detector characterized by its quantum efficiency η(λ).

The eigenstates of the PSG and PSA are given by Eq. (2). These eigenstates depend on the birefringence induced by the LCVR of the devices which is a function of the wavelength and of the control voltage V applied to the LCVR. For our simulation, we choose to model the voltage dependence based on the response of Meadowlark Optics LCVR and the wavelength dependence is assumed to follow a law of the form a + b/λ ² + c/λ ⁴ where (a, b, c) ∈ ℝ³ [18, 19]. Figure 1 gives an example of the retardance-voltage function for three values of the wavelength. We also assume that the transmission of the LCVR does not change with the wavelength. For the study, we limit the working voltage range between 1.5 V and 8.0 V.

 

Fig. 1 Relationship between the retardance induced by a LCVR and the voltage applied to the device. The three curves show this relationship for three different wavelengths: 450 nm (blue), 550 nm (green), 650 nm (red).

Download Full Size | PDF

The spectral dependence of the camera and the source characteristics are simulated through η(λ) and I ₀(λ) respectively. For simplicity, both parameters are gathered into the single function ρ(λ) as explained in Eq. (5). Figure 2 shows the evolution of the three parameters according to the wavelength: η(λ) (plain gray curve) is modeled from technical data of a Sony ICX414AL sensor, I ₀(λ) (dashed gray curve) is fashioned from an idealized spectrum of a xenon tungsten halogen lamp, and ρ(λ) (black plain curve) is calculated as the product of the two previous functions.

 

Fig. 2 Spectral response of the camera (η(λ) - gray plain curve) and spectrum of the source (I ₀(λ) - gray dashed curve), used in the simulations. The impact of both parameters is modeled by a single function ρ(λ) = η(λ)I ₀(λ) (black plain curve).

Download Full Size | PDF

3.2. Contrast calculation

To study the impact of spectral broadening on the contrast, we consider two scenes where the target and the background are composed of ideal retarders and diattenuators. Therefore, their Mueller matrices are of the form ${\mathbf{M}}_{•}=\mathbf{R}\left({\varphi }_{•}^{\lambda },{\alpha }_{•}\right)\mathbf{P}({\mathbf{d}}_{•})$ where $\mathbf{R}\left({\varphi }_{•}^{\lambda },{\alpha }_{•}\right)$ denotes the Mueller matrix of a pure retarder with an orientation of α _• and inducing a phase delay of ${\varphi }_{•}^{\lambda }=2\pi A_{•}/\lambda$ with A _• being a constant, and where P(d _•) is the Mueller matrix of a pure diattenuator with d _• being the 3-dimensional diattenuation vector of the considered region of the scene.

In this study, we consider two scenes. In the first scene, referred to as Scene 1 in the following, the target and the background are composed only of pure retarders with different orientations and retardances (the diattenuation vector d _• is null for both regions). In the second scene, referred to as Scene 2, the target and the background are composed of a diattenuator followed by a retarder. The retarders have the same properties as in Scene 1. The diattenuation vectors are now different from the null vector and are different for the target and the background. Table 1 summarizes the value of the parameters chosen for the two scenes of the study.

Table 1. Parameters defining the two scenes considered in the simulations.

View Table | View all tables in this article

3.2.1. Contrast calculation for pure retarders

We consider in this section a scene composed of two pure retarders as target and background (Scene 1). Let us first consider a situation where a narrowband filter centered at 550 nm is used in the imaging system. In this case, using Eq. (11) and (10), one can find the configuration that optimizes the contrast. As stated in Section 2, this contrast depends only on θ ₁. Figure 3 shows the values of the contrast C_λ (θ ₁, ${\mathit{\theta }}_{2,\text{opt}}^{\lambda }$). Since for a given wavelength, there is a bijective relationship between the retardance and the voltage in our working range, we represent the contrast as a 2-dimensional map of ( ${\delta }_1^{\lambda }$, ${\delta }_2^{\lambda }$) for λ = 550 nm. One can see that an infinite number of configurations lead to optimal contrast (bright yellow regions), as these configurations form a continuum on the contrast map. This phenomenon is due to the particular nature of the target and background and is explained in Appendix A.

 

Fig. 3 Contrast map at λ = 550 nm, as a function of ( ${\delta }_1^{\lambda }$, ${\delta }_2^{\lambda }$) the retardances induced by the PSG with τ ²/σ ² = 1 for Scene 1.

Download Full Size | PDF

Now, if we implement one of these configurations on the system and broaden the spectrum of the illumination, two antagonist effects will impact the value of the contrast. On the one hand, the increase of light flux in the imaging path will tend to improve the contrast. Indeed, if the scene and the polarimetric response of the set-up components were independent of the wavelength, the contrast in Eq. (13) would be of the following form

 

Fig. 4 Contrast as a function of bandwidth of the system with τ ²/σ ² = 1 for Scene 1.

Download Full Size | PDF

On the other hand, due to the spectral dependence of the scene and of the components of the imaging system, the chosen configuration ( ${\mathit{\theta }}_{1,\text{opt}}^{\lambda }$, ${\mathit{\theta }}_{2,\text{opt}}^{\lambda }$) optimal for 550 nm is no longer optimal for the other wavelengths. For example, let us assume that we chose the configuration corresponding to the red dot in Fig. 3. Figure 5 shows the “location” of this configuration for other wavelengths (450 nm, 600 nm and 750 nm). We can see that this configuration does not correspond to an optimum anymore for these three wavelengths. This mismatch will lead to a contrast loss when the system integrates the signal over the complete bandwidth. This decrease can be more or less important depending on the configuration initially chosen inside the continuum. This phenomenon is shown in Fig. 4 where the blue area represents the values of the contrast defined by Eq. (13) as a function of the bandwidth, for all the possible configurations which are optimal at 550 nm. We can see that globallyin this case, the increase of intensity overcomes the loss in polarimetric accuracy since the contrast increases with the bandwidth. However, for the worst case configurations, one can observe a loss of contrast as the bandwidth increases. This is due to the important mismatch between the properties of the set-up at the wavelength chosen for the initial optimization and the other wavelengths of the bandwidth.

 

Fig. 5 Contrast maps as a function of ( ${\delta }_1^{\lambda }$, ${\delta }_2^{\lambda }$) the retardances induced by the PSG with τ ²/σ ² = 1 for Scene 1 and different wavelengths: (a) 450 nm, (b) 600 nm (c) 750 nm. The red dot corresponding to the same configuration as indicated in Fig. 3.

Download Full Size | PDF

Let us now optimize the configuration (θ ₁, θ ₂) in order to maximize the contrast of Eq. (13). To do so we perform an exhaustive search by calculating the contrast C _{Δ λ} for all values of the retardances induced by the LCVR. These values are generated using the voltage-retardance relationship illustrated in Fig. 1 by scanning the voltages applied to the LCVR that compose the PSG and the PSA with a resolution of 65 mV. This leads to retardances induced by the LCVR ranging approximately from 0 to 800 nm. For each value of the bandwidth we are then able to find a specific optimal set of retardances. The results appear in the red dashed curve of Fig. 4. We now observe a monotonous increase of the contrast, meaning that we are always able to find a configuration for which the loss of polarimetric accuracy is more than compensated by the increase of light flux.

3.2.2. Contrast calculation in the presence of diattenuation

We now consider a scene with diattenuation (Scene 2), and as before, we first consider the situation where a narrowband filter centered on 550 nm is used in the imaging system. Figure 6 shows the values of the contrast C_λ (θ ₁, ${\mathit{\theta }}_{2,\text{opt}}^{\lambda }$) as a 2-dimensional function of ( ${\delta }_1^{\lambda }$, ${\delta }_2^{\lambda }$) for λ = 550 nm. The impact of the presence of diattenuation on the scene can clearly be seen as the continuum observed in Section 3.2.1 has now disappeared and only one global maximum remains (red dot).

 

Fig. 6 Contrast map at λ = 550 nm, as a function of ( ${\delta }_1^{\lambda }$, ${\delta }_2^{\lambda }$) the retardances induced by the PSG with τ ²/σ ² = 1 for Scene 2. The red dot indicates the position of the maximum.

Download Full Size | PDF

Now if we implement on the set-up the configuration leading to this maximum and increase the spectral bandwidth, the two antagonist effects mentioned previously should still impact the value of the contrast. The evolution of the contrast as a function of the system bandwidth is displayed in Fig. 7. The green line represents the evolution of the contrast in the case where the scene and the system characteristics are independent of the wavelength and the blue curve represents the evolution of the contrast defined in Eq. (13). It is worth noticing that in this case the increase of light always overcomes the loss of polarimetric accuracy since the contrast monotonously increases as the spectral width broadens. The loss of contrast compared to the C ₀Δλ ² case remains important, nevertheless.

 

Fig. 7 Contrast as a function of bandwidth of the system with τ ²/σ ² = 1 for Scene 2.

Download Full Size | PDF

However, if we search for the configuration (θ ₁, θ ₂) that maximizes the contrast of Eq. (13) for each value of the bandwidth, one can see that the contrast can be further increased (red dashed curve in Fig. 7).

3.3. Discussion

In this numerical study, we have analyzed the impact of spectral broadening on the contrast value for two different kinds of scenes composed of ideal components. Similar behaviors were highlighted.

On the one hand, for both scenes, the contrast is globally enhanced when we increase the bandwidth of the light entering the system, even if the set-up is optimized for a specific wavelength. This shows that the increase of intensity overcomes the loss of polarimetric accuracy in the contrast calculation. However, for situations where several configurations of the PSG and PSA lead to the same optimal contrast for a given wavelength, some configurations may result in a decrease of contrast for a particular bandwidth as the mismatch of the properties of the set-up between all the wavelengths becomes too high.

On the other hand, it is shown that optimizing the contrast by taking specifically into account the broadband characteristic of the set-up leads to a significant contrast enhancement. In this case, the bandwidth simply appears as a supplementary parameter that has to be taken into account to optimize the system.

4. Experimental study

4.1. Description of the experimental set-up

In order to verify the results of the numerical simulation, we performed a laboratory experiment with a division of time polarimeter (DOTP) in standard reflection configuration described in [13]. The system is an adaptive imager which can generate and analyze any polarization state defined on the Poincaré sphere. The light source is a cold halogen white light fiber source (Qioptiq, LQ 1100) which produces unpolarized light. The PSG as well as PSA consists of two nematic liquid crystal variable retarders (LCVR) (Meadowlark Optics) and a linear polarizer which are positioned as described in Section 2. The image is acquired using a CCD camera (AVT Stingray - F033). Figure 8 presents a schematic of the complete set-up.

 

Fig. 8 Schematic of the optical set-up: FS - white-light fiber source, L1, L2 - two lenses, AD, FD - aperture and field diaphragms, P1, P2 - linear polarizers, LC1...4 - liquid crystal variables retarders, F - filter in the monoband case, C - CCD camera.

Download Full Size | PDF

The scene of interest is composed of two different objects as shown in Fig. 9. The background of the scene is made up of an uniform metallic plate and the target to detect is a piece of birefringent material (translucent adhesive tape) placed over the plate. The standard intensity image of the scene is shown in Fig. 9(b) and, as one can see, the target cannot be distinguished on this image.

 

Fig. 9 (a) Scheme of the observed scene. (b) Standard intensity image, with an exposure time of 100 ms.

Download Full Size | PDF

4.2. Results and discussion

As in the numerical study, we first place a narrowband filter in front of the camera. The filter is centered on 640 nm with a bandwidth of 10 nm. We then measure the Mueller matrix of the scene, and using Eq. (11) and (12), we determine the optimal configuration for the central wavelength $\left({\mathit{\theta }}_1^{\text{opt}},{\mathit{\theta }}_2^{\text{opt}}\right)=\left(2.96,\hspace{0.17em}4.13,\hspace{0.17em}\hspace{0.17em}2.58,\hspace{0.17em}\hspace{0.17em}3.63\right)\hspace{0.17em}\text{V}$ and implement it on the system. We obtain the image of Fig. 10(a) for an exposure time of 100 ms. One can see that the contrast has been improved in comparison to standard intensity image (see Fig. 9(b)). However, if we strongly reduce the acquisition time to 2 ms, for example to increase the acquisition speed, we obtain Fig. 10(b). One can now see that the contrast has strongly decreased, due to the presence of high sensor noise. This can be quantified by computing the intensity distribution of the regions of interest. Figure 11.a and Figure 11.b show the distributions for the target (in blue) and the background (in red). The dashed blue line represents the value of the average intensity 〈i_t〉 over the target region and the dashed red one the average value 〈i_b〉 over the background region. The histograms were shifted in order to have min{〈i_t〉,〈i_b〉} = 0. The experimental contrast C _exp can then be calculated using ΔI = |〈i_t〉 − 〈i_b〉| and Eq. (6). We obtain C _exp = 4.57 × 10⁵/σ ² and C _exp = 90/σ ², for τ = 100 ms and τ = 2 ms respectively.

 

Fig. 10 (a) Intensity image taken using the 10nm-wide narrowband filter centered at λ = 640 nm and the PSG/PSA configuration optimal for 640 nm, with an exposure time of 100 ms. (b) Same image for an exposure time of 2 ms. (c) Intensity image taken without the filter using the PSG/PSA configuration optimal for 640 nm and an exposure time of 2 ms. (d) Intensity image taken without the filter after optimizing the PSG/PSA configuration optimal for the bandwidth and with an exposure time of 2 ms.

Download Full Size | PDF

 

Fig. 11 Intensity distribution of the target (blue) and the background (red) for (a) Figure 10.a, (b) Figure 10.b, and (c) Figure 10.c and (d) Figure 10.d. For better reading and comparison, the three histograms were shifted in order to have min{〈i_t〉, 〈i_b〉} = 0.

Download Full Size | PDF

If we now remove the filter without changing the configuration of the PSG and PSA, we obtain the image of Fig. 10(c). We can see that the visual contrast has improved even if the configuration we use is not optimal for the other wavelengths of the bandwidth. This is due to the increase of light flux in the system. It can be quantitatively confirmed using the histograms of Fig. 11(c) and by calculating the new contrast C _exp = 4.76 × 10⁵/σ ². Now, if we perform an exhaustive search by varying the voltages of the LCVR to find the configuration that optimizes the contrast without the filter, we find that the optimal configuration is $\left({\mathit{\theta }}_1^{\text{opt}},{\mathit{\theta }}_2^{\text{opt}}\right)=\left(6.17,\hspace{0.17em}2.77,\hspace{0.17em}\hspace{0.17em}5.44,\hspace{0.17em}\hspace{0.17em}3.87\right)\hspace{0.17em}\text{V}$ and we obtain the image of Fig. 10(d) with the histograms of Fig. 11(d). We can see that the optimal configuration is different from the previous monoband case, and we have further enhanced the contrast to reach a value of C _exp = 7.11 × 10⁵/σ ².

Table 2 summarizes the results obtained for the different configurations for an acquisition time of 2 ms. We can see that increasing the bandwidth of the system allows to enhance the contrast and that by taking into account the broadening of the bandwidth we are able to find an optimal configuration that further increases the contrast and thus compensates the loss of polarimetric accuracy of the system.

Table 2. Summary of the different experimental configurations and results (exposure time = 2ms)

View Table | View all tables in this article

5. Conclusion

We have shown that in active polarimetric imaging, contrast can be increased by broadening the spectral width of the light entering the system. Indeed, the increase of light flux overcomes the contrast decrease due to the mismatch of the system component properties caused by the broadening. In addition, contrast can be further improved by optimizing the PSG and PSA configuration with respect to the bandwidth. In this case, the optimal configuration is in general different from that obtained in the narrowband case.

These results illustrate that for target detection applications, a DOTP system should no longer be seen only as a tool to estimate polarimetric properties but also as a tunable device allowing to acquire relevant information for optimal target detection. In this case, spectral width is simply a new parameter to optimize the polarization imaging architecture. These findings open new perspectives for high-speed polarimetric imaging in applications where contrast optimization is required.

This work has many perspectives. In the present system, the optimal configurations of the PSA and PSG in the broadband case have been found by exhaustive search. In order to decrease the computation time of the optimization, one needs to quickly estimate the Mueller matrix of the scene, which can be possible by using a system similar to the one described in [20] for example. Moreover, a polarimetric imager with high acquisition speed and high signal-to-noise ratio images can be interestingly applied to such fields as microscopy or industrial visual inspection. Finally, studying the impact of bandwidth increase on the precision of Mueller matrix estimation is also a very interesting perspective since it would allow one to quantify to what extend a polarization imager has to be achromatized for a given application.