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Traditional methods of discrimination for Stokes polarimetric imaging use grayscale images, in which the difference of the polarimetric properties is only reflected by the difference of grayscale. In this paper, we propose a method of colorimetric discrimination and classification for Stokes polarimetric imaging by the composed color polarimetric image, in which the objects with different polarization properties can appear in different colors. We show with real-world experiment that compared with the traditional method for the grayscale Stokes scalar image, the method proposed in this paper has a better performance for distinguishing objects with different polarization properties.
© 2017 Optical Society of America
1. Introduction
Polarimetirc imaging has the capability of gathering polarization information that is not visible in classical intensity images. Therefore, it is applied in many fields related to the purpose of discrimination, such as target detection [1,2], classification [3,4], remote sensing [5,6], and biomedical imaging [7,8]. Improving the ability of distinguishing different objects is a key issue for improving the effect of polarimetric imaging in such fields. Up to now, several methods have been proposed to improve the discrimination performance of polarimetric imaging [4,9,10]. However, previous works mainly focus on the grayscale polarimetric scalar image [1,2,11], in which the difference of the polarimetric properties of the objects is only reflected by the difference of intensity (or grayscale) of the scene, and the corresponding approach of optimizing the discrimination performance is to maximize the intensity (grayscale) difference of the objects in the image [2,12]. Compared with the grayscale, color has a better diversity, and therefore, if we can present different polarization information in different colors instead of in different grayscales, the difference of the polarization information for different objects can be more distinct. In particular, when there are multiple objects, the superiority of presenting polarization information in color can be more significant.
In this paper, based on the Stokes polarimetry, we propose a method of colorimetric discrimination and classification based on the color polarimetric image, which is a RGB image composed by three grayscale polarimetric images obtained at different eigenstates of PSA. In the composed color polarimetric image, the objects with different polarization properties can appear in different colors. In addition, the real world experiments of discrimination and classification for the color polarimetric image and for the grayscale polarimetric image are performed for the purpose of comparison, which verify the feasibility and the advantages of our method.
2. Discrimination for the grayscale Stokes scalar image
The typical Stokes polarimetry performs four intensity measurements to estimate a Stokes vector [2]. Let us denote S = (S0, S1, S2, S3)T as the Stokes vector to be estimated and I = (I1, I2, I3, I4)T to be the 4-dimensional vector representing the intensity measurements. The Stokes vector can be obtained by [3]:
where W is the 4 × 4 measurement matrix.The traditional method of discrimination for the Stokes polarimetric imaging system focus on the polarimetric scalar image [1,2,11], which is a grayscale image filtered by a certain eigenstate of polarization state analyzer (PSA) T [1]. The general way to realize the optimal linear discrimination for the grayscale Stokes scalar image is to maximize the intensity (grayscale value) difference between the objects by adjusting the state of PSA [2]. In this case, the eigenstate of PSA T is adjusted to the optimal state Topt to realize the maximum intensity difference between the objects, which is given by [2]:
where Topt refers to the optimal state of PSA, andΔI is the intensity difference between different objects with the Stokes vectors of S1 and S2 respectively.3. Discrimination for composed color polarimetric image
The approach of discrimination mentioned above is capable to distinguish different objects in some cases [2,3,12]. However, due to the polarimetric scalar imaging is in the form of grayscale image, the objects with different polarization properties can be only discriminated based on the difference of intensities, and thus the ability of discrimination is limited. If we extend the grayscale scalar image into the color one, which means presenting different polarization information in different colors, then the difference of the polarization information for different objects can be presented not only by intensity difference but also by color difference. In this case, the discrimination ability of the polarimetric image can be improved.
In this paper, we try to extend the grayscale polarimetric scalar imaging into RGB color imaging to improve the performance of discrimination, which is illustrated in detail in the following. First, we need to obtain three grayscale polarimetric scalar images at three different eigenstates of PSA. The three different eigenstates of PSA are denoted to be T1, T2 and T3, and thus the corresponding three grayscale polarimetric scalar images are:
where (x,y) denotes to the coordinate of a certain pixel in the image. After that, we assign the three grayscale polarimetric scalar images given by Eq. (3) to the R, G, B color space respectively, and then we can obtain a composed color polarimetric image of the scene. Since different objects correspond to different polarization states (Stokes vectors), they will have different values in R, G and B channels according to Eq. (3). As a result, the objects with different polarization properties in the scene will show different colors in the composed polarimetric color image.To achieve the optimal linear discrimination ability, we need to find three optimal eigenstates of PSA to maximize the color difference between the objects. In this paper, we consider a scene with three objects (denoted to be object a, b and c respectively) as an example to illustrate our method. The Stokes vectors at different pixels in each object are assumed to be uniform for simplicity. In this case, each object has a uniform intensity at a certain eigenstate of PSA, and thus each object in the composed polarimetric color image has uniform R, G, B values. Then for each object i, the trichromatic coordinates in RGB space can be expressed as:
where stand for the intensities in R, G, B channels of the color image for the object i, which are given by Eq. (3). For a scene with three objects (as the condition in our experiment), we can obtain the trichromatic coordinates Pa, Pb and Pc for the three objects a, b and c respectively in RGB color space, which are shown in the schematic of Fig. 1.
Fig. 1 The schematic of the coordinates of the three objects and the corresponding triangle in RGB color space.
In fact, the essence of optimizing the discrimination performance is to properly value and maximize the global color difference between the three objects. In this work, we consider the area of the triangle, which is shown by the green area in Fig. 1, as the criterion of the global color difference between the three objects. Therefore, when the area of the triangle reaches its maximum value, the discrimination performance between the three objects is considered to be optimal. In the case of optimal discrimination, the trichromatic coordinate of each object will keep a considerable distance from the other two, and thus there will be apparent difference in color for the three objects.
The area of the triangle can be given by the Heron’s formula as:
where La, Lb and Lc refer to the three sides of the triangle shown in Fig. 1, and q = (La + Lb + Lc)/2. In fact, the shape of the triangle is modulated by the eigenstate of PSA T, and thus the area R depends on the three eigenstates of T1, T2 and T3. The optimal states of T1, T2 and T3 should maximize the area of the triangle in Fig. 1, which fulfills the following algorithmFor the typical PSA, each eigenstate of T has two degrees of freedom (DOF). In practice, the two DOFs of T is modulated by different parameters for different configurations of PSA [13,14]. For example, T can be modulated by the voltages of the liquid crystal variable retarders [3,15], or by the orientations of the linear polarizer and the quarter wave plate [2,10]. Therefore, according to Eq. (6), we need to perform the global search for 6 parameters to find the optimal states of T1, T2 and T3 to maximize the area R given by Eq. (5), in order to optimize the discrimination performance in the composed color image.It needs to be clarified that in this method, the criterion for the discrimination performance is considered to be the area of the triangle rather than the perimeter of the triangle. This is because if we take perimeter as the criterion, it is possible that when the perimeter is maximized, the optimal triangle could have two long sides together with a considerably short side. As a result, the two objects corresponding to the two ends of the short side will have little difference in color in the composed color polarimetric image, which will lead to the poor discrimination performance. Besides, there are still some other criterions that can express the discrimination performance, including the Bhattacharya distance that introduced in Ref [16]. However, for employing the Bhattacharya distance as the criterion, one needs to know the statistics of measurements in advance to deduce the function of Bhattacharyya distance [16], which may not be available in practice, while the criterion of triangle area can be widely and easily applied in various cases.
4. Real world experiment of discrimination
The real world experiment is performed to verify the feasibility and advantage of our method. A He-Ne laser is employed as the light source to illuminate the scene. The PSA in our setup is consisted by a linear polarizer and a quarter wave plate at 633nm. The images filtered by the PSA are received by a CCD camera (AVT Stingray F-033B). The scene in our experiment contains two targets and a background, which refers to object a, b and c respectively, as shown in Fig. 2. In particular, the background (object c) is a superposition of white paper, a sheet polarizer, and a translucent adhesive tape. The two targets (objects a and b) are two pieces of translucent adhesive tape with different orientations placed on the background. The polarization property of the translucent adhesive tape is similar to a retarder [3], whose Mueller matrix depends on the its orientation, and thus the two objects will lead to different polarization modulations.
The intensity image of the scene is represented in Fig. 3. It can be seen that the two targets and the background are not easy to be discriminated, because they have similar reflectivity.
The incident illuminate beam is the directly come from the He-Ne laser, which is partially polarized. In this case, the average values of the Stokes vectors corresponding to objects a, b and c in the scene are measured to be:
In practice, the polarization properties of different pixels inside the object cannot be identical. For simplicity, the mean Stokes vectors of objects a, b and c given in Eq. (7) are employed to represent the polarization properties of the three objects, based on which the optimization of the three eigenstates of PSA (T1, T2 and T3) is performed. In our case, since PSA is composed by a rotating linear polarizer and a rotating quarter wave plate, the eigenstates of PSA T is depends on the orientations of the polarizer and the wave plate. We perform the global optimization by employing the shuffled complex evolution(SCE-UA) method [17] to find maximum of the constrained nonlinear multivariable function of Eq. (5). The optimal orientations of the polarizer and wave plate (θP and θR) for T1, T2 and T3 are found to be , and respectively. The corresponding polarimetric scalar images at T1, T2 and T3 are shown in Fig. 4, which are assigned to the R, G and B channels respectively. The composed polarimetric color image is also shown in Fig. 4. It can be seen in Fig. 4 that different objects are presented in different colors. In particular, object a appears to be orange, object b appears to be blue, and object c (the background) appears to be deep purple. In this case, one can easily distinguish the objects with different polarization properties thanks to the extension of the polarimetric image from grayscale to color by our method illustrated in Section 3.
Fig. 4 The polarimetric scalar images at the optimal state of (T)1, (T)2, (T)3) and the corresponding composed polarimetric color image.
The point clusters of the trichromatic coordinates corresponding to the pixels inside of the three objects at the optimal state of (T1, T2, T3) are shown in Fig. 5. When the ranges of R, G, B values are considered to be from 0 to 1, the average values of trichromatic coordinates for objects a, b and c at the optimal states of PSA are calculated to be, and respectively according to Eq. (3). The green triangles in Fig. 5 are formed by the average values of the trichromatic coordinates of objects a, b and c. It can be seen in Fig. 5 that with the optimal states of (T1, T2, T3), the clusters of objects a, b and c are well separated in the RGB color space, which guarantees that each object will have a distinctive color in the composed color image to benefit the discrimination.
Fig. 5 The point clusters of the trichromatic coordinates corresponding to the pixels inside of the three objects and the corresponding triangle in the RGB color space at the optimal state of (T)1, (T)2, (T)3).
In fact, the human vision based colorimetric classifier can perform the classification based on the color difference in the composed polarimetric color image shown in Fig. 4. The method of this colorimetric classification is to compare the colors of each pixel to the average colors of objects a, b and c, respectively, and then, for example, the pixel is classified to object a when its color is closest to the average color of object a. The algorithm of this colorimetric classifier is similar to calculate the distances between the coordinate of this pixel and the average coordinates of objects a, b and c in RGB space shown in Fig. 5 respectively, and then to find the class with the minimum distance. We performed this colorimetric classification, and the result is shown in Fig. 6, in which the pixels being classified to objects a, b and c are colored by red, green and blue respectively.
The values of the probability of detection PD and the probability of false alarm PFA [18] of the colorimetric classification for objects a, b and c are listed in Table 1. It can be seen in Table 1 that the colorimetric classification can achieve high values of probability of PD and low values of PFA. It needs to be clarified that the nonlinear classifiers [18], for instance support-vector machines (SVM) [19,20], could achieve a better performance, but at much greater computational cost.
Table 1. The probability of detection and the probability of false alarm for objects a, b and c
For the polarimetric scalar image in grayscale, one can also optimize its discrimination performance. However, for the grayscale image, there is only one-dimensional gray space instead of the three-dimensional RGB color space. Consequently, the approach of discrimination optimization cannot be based on maximizing the area of the triangle. In this case, we take the parameter D as the criterion of the discrimination performance given by:
where ia, ib and ic stand for the intensities (grayscales) of the objects a, b and c in the grayscale polarimetric scalar image respectively, and thus D represents the global intensity difference of the three objects in the polarimetric scalar image. Based on the criterion given by Eq. (8), the optimal state of PSA, which maximizes D, is found to be , and the optimal polarimetric scalar image of the scene in grayscale is shown in Fig. 7.The average grayscales of objects a, b and c in Fig. 7 are measured to be 0.193, 0.259 and 0.06 respectively. It can be seen in Fig. 7 that it is clear that there are three objects in the scene, because: 1) the grayscales of objects a and b are much higher than that of the object c (background); 2) objects a and b are not adjacent to each other. However, it is not easy to identify the different polarization properties of object a and object b, because the grayscales of them are close to each other. If objects a and b are adjacent to each other, it is even harder to identify whether there are two or three objects in the scene in such case.
Comparing Fig. 7 with Fig. 4, the advantage of discrimination in color polarimetric image is apparent, and we can always identify the three objects in the scene in Fig. 4 no matter what the positions of the three objects are. The method of colorimetric discrimination for polarimetric image benefits from the color diversity, which is particularly beneficial for the case of human vision. In addition, it needs to be clarified that if there are even more objects in the scene, the advantage of discrimination in color polarimetric image could be more apparent compared with that in grayscale polarimetric image, because the diversity in RGB color space is much richer than that in gray space.
It needs to be clarified that the magnitude of color differences in the composed color images by our method depends on the magnitude of polarimetric parameters (Stokes vector) difference between the objects. In order to better demonstrate the usefulness of the proposed color representation, we realize the example with greater color differences. The objects are three linear polarizers with different orientations sticked on the white paper. The corresponding composed polarimetric color image by our method is shown in Fig. 8, in which the polarimetric information of objects a, b and c is represented in pink, purple and green respectively. Comparing Fig. 8 with the composed color image in Fig. 4, it can be seen that the color difference in Fig. 8 is much more distinct, which is attributed to the greater difference between the Stokes vectors of objects a, b and c in Fig. 8.
Fig. 8 Composed polarimetric color image for the scene composed by three linear polarizers with different orientations sticked on the white paper.
5. Conclusion
In conclusion, the method of colorimetric discrimination by composing the color Stokes polarimetric image is proposed, which can present the objects with different polarization properties in different colors. Compared with the previous method of discrimination for the grayscale Stokes scalar image, the method proposed in this paper shows better performance in distinguishing objects with different polarization properties, which is attributed to making full use of the distinguishing capability of colors, and the output of this method is particularly beneficial for the case of human vision. The colorimetric classifier proposed in this paper enables discrimination or classification in multi-dimensional polarization space with minimal computational overhead by utilizing the ability of human vision to perceive multiple dimensions as color.
The method proposed in this paper can be also applied to discriminate and classifying the scene with more than three objects, and the superiority of this method can be more distinct in such case thanks to the rich diversity of the color. Besides, the method proposed in this paper goes beyond the particular case of Stokes polarimetric imaging, and it can be extended to other configurations of polarimetric imaging, such as Mueller polarimetric imaging [21,22].
Funding
National Natural Science Foundation of China (NSFC) (61405140, 61227010); the National Instrumentation Program (NIP) (2013YQ030915); Natural Science Foundation of Tianjin (NSFT) (15JCQNJC02000).
Acknowledgments
Haofeng Hu acknowledges the Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA) and the China Scholarship Council (CSC).
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