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In the framework of Stokes parameters imaging, polarization-encoded images have four channels which makes physical interpretation of such multidimensional structures hard to grasp at once. Furthermore, the information content is intricately combined in the parameters channels which involve the need for a proper tool that allows the analysis and understanding this kind of images. In this paper we address the problem of analyzing polarization-encoded images and explore the potential of this information for classification issues and propose ad hoc color displays as an aid to the interpretation of physical properties content. The color representation schemes introduced hereafter employ a technique that uses novel Poincaré Sphere to color spaces mapping coupled with a segmentation map as an a priori information in order to allow, at best, a distribution of the information in the appropriate color space. The segmentation process relies on the fuzzy C-means clustering algorithms family where the used distances were redefined in relation with our images specificities. Local histogram equalization is applied to each class in order to bring out the intra-class’s information smooth variations. The proposed methods are applied and validated with Stokes images of biological tissues.
©2006 Optical Society of America
1. Introduction
The relevance of polarization-encoded images comes from the rich set of physical information they carry about the local nature of the target. Hence, polarization-sensitive imaging systems are emerging as a very attractive vision technique that provides insightful understanding of the elements that constitute the object based on their polarimetric properties, i.e., birefringence, dichroism, depolarizing properties, transmittances, etc. This is particularly useful to probe the constituent elements organization in biological tissues. Many results concerning the use of this technique in bio-imaging are already available in recent literature [1–3].
The design of imaging systems, that can measure the polarization state of the outcoming light across a scene, is mainly based on the ability to build effective Polarization State Analyzers (PSA) in front of the camera that permit to acquire the Stokes vectors corresponding to each pixel in the image. These Stokes polarimeters yield four-dimensional images called “Stokes images” corresponding to the four Stokes parameters. Accordingly, polarization-encoded images have a multidimensional structure; i.e. multi-component information is attached to each pixel in the image. Moreover, the information content of polarization-encoded images is intricately combined in the polarization channels making awkward their proper interpretation. This induces the need for a proper tool that allows the analysis and understanding of polarization-encoded images.
Classical monochromatic displays of polarimetric images have been widely used in the in the past and remain the principal way to present such images [1, 4–6]. This makes difficult to grasp the whole information at a glance. This issue is of prime importance for end users in biology domain who are not aware of polarization and have to decide whether a sample presents or not any pathological feature.
Few work addressed a synthetic approach that provides a single preview that is practical to human observer in terms of physical interpretation of the image content. We note nevertheless the use of polarization to color mappings by Wolff [7] to display the output images of a liquid crystal polarization camera and in [8, 9] where such a mapping was used in the context of polarization-difference imaging to improve the detection of targets with low degree of polarization. In these references, the authors considered only the case of linearly polarized light which is perfectly justified by the use context: the most part of information is carried by the wave intensity and little to no circular polarization component is present in the wave.
However, in the case where the samples responses may exhibit circular polarization component, no informative visual representation has been proposed yet.
The objective of this paper is to provide a framework that permits the analysis and the visualization of information contained in polarization-encoded images. We propose hereinafter a physically based polarization-to-color mapping that permits to segment properly polarization-encoded images and allows for a workable preview that summarizes the variation of polarimetric properties over the pixels of the image.
The proposed method may be of great interest for cases where little intensity contrast is present while strong variations are present in the polarization channels.
2. Lab and HSV Color spaces
Among the wide variety of color models, the Lab and HSV ones present an interesting choice to represent the colors that are visible to the human eye. These color spaces provide three-parameter representations that decouple the brightness information from the color-carrying parameters (chrominance) which make them suitable for describing colors in terms that are practical for a better interpretation of the color maps. More details about color spaces can be found elsewhere in the literature[10]. Here the HSV and Lab color spaces are summarized for completeness.
2.1 Lab color space
In the CIELAB color model, each color is described according to three physiological parameters, namely L,a, and b:
• The luminance component L, is an achromatic value that gives a measure of the brightness quantity in the color, L varies from 0 to 100. L=0 stands for black and L=100 corresponds to white.
• The a parameter describes the greenness-redness of the color and varies from -127 to 127. Negative values turn to green while positive values turn to red.
• The b parameter describes the blueness-yellowness of the color and varies from -127 to 127. Negative values turn to yellow while positive values turn to blue.
This model has been created to serve as a device-independent, absolute model to be used as a reference. Since the Lab model is a three-dimensional space, it is represented by a three-dimensional sphere. Each axis of the sphere represents one parameter as illustrated in Fig. 2(a). An interesting feature of the model is that the first parameter is extremely intuitive: changing its value is like changing the brightness setting in a monitor. In this model, the color differences, which observer perceives, correspond to distances measured colorimetrically.
2.2 HSV color space
The HSV color space offer intuitive handling of colors. It decouples the intensity component from the color-carrying information (hue and saturation)[10]. Each color is again described according to three physiological criterions:
• The hue (H) is related to the color perception, it describes the color purity 0≤H≤360°.
• The saturation (S) gives a measure of the degree to which a pure color is diluted by white light 0≤S≤1.
• The brightness (V) is achromatic information and gives a measure of the light quantity in the color (bright to dark) 0≤V≤1.
These three components can be represented in polar coordinates by a cone where the set of all reachable colors are synthesized. Figure 2(b) shows the geometric representation that corresponds to the HSV model.
The main advantage of this model comes from the fact that each component can be related to a physical quantity that can be interpreted visually. In this way, the variations of the physical quantity related to channel V in the image are represented by variations in pixel’s brightness.
The HSV color space can also be visualized in cylindrical coordinates in a similar way to the conical representation of Fig. 2, the hue varies along the outer circumference of a cylinder, while saturation is given by the distance to the center of a circular cross-section.
Brightness again varies from top to bottom as shown in Fig. 2. Such a representation might be considered the most mathematically accurate model of the HSV color space; however, in practice the number of visually distinct saturation levels and hues decreases as the value approaches black. Additionally, computers typically store RGB values with a limited range of precision; the constraints of precision, coupled with the limitations of human color perception, make the cone visualization more practical in most cases.
Fig. 3. I1. S 0 (a), S 1 (b), S 2(c), and S 3 (d) images of a histological section of a bone died with red picosirius and imaged at 650 nm wavelength. The image at the upper left is to be compared with a conventional intensity image. The scale in the intensity channel (S 0) is in arbitrary units. In the three remaining channels, the scales are relative to the first one.
3. Images description
In order to illustrate the developed approach in this paper, we will use three different Stokes images of biological tissues acquired with a full Mueller polarimeter based on liquid crystals modulators. The operation and extensive description of the instrument can be found elsewhere in the literature[11].
The first Stokes image (I1) is that of a red picosirius died histological section of a bone imaged at 600 nm wavelength (Fig. 3). The second one (I2) represents a histological section of a healthy vessel (Fig. 4), while the third one (I3) is that of a histological section a pathological vessel (Fig. 5).
Fig. 4. I2. S 0 (a), S 1 (b), S 2 (c), and S 3 (d) images of a histological section of a healthy vessel. The image at the upper left is to be compared with the conventional intensity image. The scale in the intensity channel (S 0) is in arbitrary units. In the three remaining channels, the scales are relative to the first one.
Fig. 5. I3. S 0 (a), S 1 (b), S 2 (c), and S 3 (d) images of a histological section of a pathological vessel. The image at the upper left is to be compared with the conventional intensity image. The scale in the intensity channel (S 0) is in arbitrary units. In the three remaining channels, the scales are relative to the first one.
The images were acquired in backlight illumination condition. We were able to control the polarization state of the impinging light. Special attention was paid to ensure that the measurements are physically admissible. We didn’t address the tricky task of absolute calibration of the instrument, but the measurements are correct up to constant factor multiplier. The scale of S 0 channel is in arbitrary units and the other channels scales are relative to the first one.
By observing the three Stokes images, we notice that the information is spread out over the four Stokes channels making difficult a direct physical interpretation. This justifies the need for a proper one-view-representation that brings out the information content from the Stokes image. This will be addressed in the coming sections.
4. Colorimetric processing of Stokes images
4.1 Stokes parameters to color space mapping
The general polarization state of a light wave can be described by the so called “Stokes vector” (SV) S which fully characterizes the time-averaged polarization properties of radiation. It is defined by the following combination of complex-valued components Ex and Ey of the electric vector in two mutually orthogonal directions x and y as
where S 0 defines the total intensity of the wave, while S 1 describes the excess of parallel to perpendicularly polarized light, S 2 is the excess of +45° linearly polarized part over the -45° linearly polarized part, and S 3 the excess of the right circularly polarized component over the left circularly polarized one.
It is straightforward to show that
where the equality holds for completely polarized radiation.
We note further that the geometrical polarization parameters of the wave, i.e., degree of polarization (0≤DOP≤1), orientation(-π2≤ϕ<π/2), and ellipticity(-π4≤χ≤π/4), can be expressed in terms of S 0, S 1, S 2, and S 3 as follows
If one normalize the Stokes vector to S 0, i.e., S̄=S/S 0=[1 S̄1 S̄2 S̄3]t, it can be shown that the reduced vector [S̄1 S̄2 S̄3]t defines a single point that lies in a unit ball called the Poincaré Sphere as depicted in Fig. 6.
In order to process coherently the physical contents of Stokes images, one needs to handle all the channels at once by the processing algorithms. This can be done by using an adequate mapping of the Poincaré Sphere to the HSV or the CIELAB spaces and using algorithm devoted to color image processing. Depending on the chosen color space, two different mappings can be defined, namely, Poincaré to HSV color space mapping or Poincaré to LAB color space mapping.
Fig. 6. The Poincaré Sphere. Completely polarized states lie on the surface of the sphere. Partially polarized radiations lie inside the sphere.
In the following, we deal with the normalized Stokes channels. By proceeding this way, there is a necessary loss of information since a four-dimensional value is represented in a three dimensional space. However, we note that normalizing by intensity; operates a smoothing effect in the transmittance heterogeneity over the sample surface and frees from difficulties due to the non ideal illumination homogeneousness. Moreover, in polarimetric imaging of biological tissues, the samples show to present strong signatures over the four Stokes parameters and little contrast in the intensity channel (S 0). It was also observed that the best indicators to be used in the analysis of such images were the linear degree of polarization and the circular polarization part of the wave induced by birefringence properties of some constituents of biological tissues, e.g. collagen.
4.2 Poincaré Sphere to HSV space mapping (PHSV)
This mapping is defined using the following transforms:
It can be interpreted as follows: pixels brightness (V) reflect the handedness of the wave (right to left handedness are represented by dark to bright pixels), the saturation (S) can be interpreted easily as the degree of linear polarization of the wave DOP while the hue (H) represents the orientation of the polarized fraction of the light. Figures 7(a–c) give the HSV mapping results corresponding to the normalized Stokes image I1.
Fig. 7. Resulting color channels representation of the normalized Stokes image I1. (a) H channel, (b) S channel, and (c) V channel. (d) L channel, (e) a channel, and (f) b channel.
4.3 Poincaré Sphere to CIELAB space mapping (PLAB)
For this mapping, we use the following transforms:
where, (am,aM), (bm,bM) are the minimal and maximal values allowed for the a and b axes.
The interpretation of this mapping is straightforward since it corresponds to a one to one point’s correspondence between the LAB sphere and the Poincaré one.
The peculiar case where S̄3 channel is vanishing is treated by choosing V=1 or L=50. In this case, the two representations provide nearly the same color preview.
Figures 7(d–f) show the LAB mapping result for the normalized Stokes image I1. As expected, we observe that the brightness channels (L-channel and V-channel) contain the same information. Indeed, one is a scaled version of the other. On the other hand, the color carrying channels are more specific to the used color space.
4.4 Polarization-based clustering
The reason for using a clustering process is to classify the pixels of an image into different sets, where each one corresponds to a specific physical feature in the imaged scene. Segmentation of multi-components images can prove to be a difficult task when the physical properties are spread out over many channels for each pixel location[1, 4]. A coherent processing of the vector features is needed since a segmentation based only on the scalar values in each channel is useless. Here, a clustering procedure is proposed, based on the polarization analysis of the scene via the mappings introduced in the preceding section.
A closer look at the H and V expression in equations (4) shows that the physical contents of these two channels are independent. Consequently, these two channels can be clustered independently into k classes by using a fuzzy C-means algorithm. We note that the classical Euclidean distance is used when clustering the V-channel while the 1-distance (dist(H 1,H 2)=|H 1-H 2) is used to cluster the H-channel.
At this stage, the membership vectors of each pixel (x, y) are obtained and written as:
We combine now these values to form a two-dimensional feature vector as follows:
We apply now the fuzzy C-means clustering algorithm on the above feature vector to obtain the final result for the polarization-based clustering. In the following, the proposed algorithm will be called “HSV-means”[12]. Fig. 8(a) shows the HSV-means clustering results into 4 classes of normalized image corresponding to I1.
Fig. 8. 4 classes label maps obtained using our clustering algorithms with the I1 image. (a) corresponds to the result obtained with the HSV-means, and (b) corresponds to the result obtained with the LAB-means. Each gray level corresponds to one class.
The clustering of the PLAB image is performed differently. For this case, we use the geodesic distance in the fuzzy C-means instead of the classical distance, i.e.:
Let X 1=[l 1,a 1,b 1] and X 2=[l 2,a 2,b 2] be two pixels in the Lab image. We call δ1,λ1 and δ2,λ2 respectively the latitude and longitude of X 1 and X 2 defined as:
for i=1,2. µ0(x) is the Heaviside function, equal to 1 if x ≥0 and 0 elsewhere. Sign(x) is equal to 1 for x positive,-1 for x negative and 0 for x=0. Finally, the distance between X 1 and X 2 is defined as follows:
which is the great-circle distance between the two points. R is the radius of the Lab sphere. Consequently, the Lab image can be clustered into k classes by using a fuzzy C-means algorithm after substituting its classical Euclidian distance by the distance defined in Eq. (9). This variant of the C-means will be called the “LAB-means”.
Figure 8(b) shows the LAB-means clustering results into 4 classes of normalized image corresponding to I1.
4.5 Brightness channels equalization and colorimetric previews
Brightness variations inside each class are not well displayed in the brightness-channels images ( Figs 7(c) and 7(d)) since it reflects the variations over the whole image.
Here we employ a technique that uses the segmentation map obtained by the above-mentioned algorithm as an a priori information in order to allow, at best, a distribution of the information in the color space. This is done in the following way:
Once the label maps are obtained from the above mentioned algorithms, different masks corresponding to each class (Ck) can be used to extract sets of brightness values from the brightness-channels images. Histogram equalization is then performed over each set to redistribute uniformly the brightness values inside each class in order to reflect in the best way the intra-class variations. Each set in the Brightness channel corresponds to one class (Ck) in the label map. The new brightness values are finally assigned to the brightness-channel. Figure 9 shows the result of the intra-class histogram equalization on the Brightness channels. We can see clearly the advantages of this processing by observing the smooth variation of the information content inside each physical feature represented by different classes as compared to the images in Fig. 7(c,d).
Finally, the three colorimetric channels can be used to generate an RGB color image for display purposes. The derived algorithm is applied on the three proposed images (I1, I2 and I3). The final results are respectively presented in Fig. 10. The distribution of the colors in the resultant RGB images is a compact manner to represent the variation of the physical properties of the scene represented initially by four different channels, in one single image. In the resultant image each color corresponds to physical information that can be interpreted according to the chosen color space.
Fig. 9. Brightness-channels images after histogram equalization for each class. (a) Equalized V-channel and (b) equalized L-channel. These images have to be compared to those of Figures 7(c) and 7(d).
The whole processing can be summarized as follows:
• Normalization of the three last channels by the first one.
• Poincaré Sphere to color space mapping
• Segmentation of the colorimetric image using the specific fuzzy C-means algorithms.
• Equalization of the histograms corresponding to the pixels of each class in the brightness-channel.
• Replace of the old brightness-channel values by the new one obtained at step 4.
• colorimetric image to the RGB image transform
Display of the RGB image.
The color previews of the three considered biological tissues are presented in Fig. 10 showing clearly the interest of the proposed method. We observe also that none of the proposed coloring scheme prevails over the other: while the HSV view of the image I1 seems to be more precise that the LAB one, this is not the case for the image I3. We note however, at least for the considered cases, there is always one preview that enhances significantly the image display. Finally, we observe that the HSV coloring scheme is more sensitive to image noises. This is due to the fact that the HSV method relies on angle calculation for the Hue values.
5. Conclusion
In this paper, a new approach to interpret the physical content of polarization-encoded images based on colorimetric preview was introduced. This allows identifying the polarization properties at pixel level in a straightforward way. Indeed, the map of the polarimetric images to color spaces permits to synthesize the maximum of information in one color preview and to distinguish visually the distribution of polarimetric characteristics over the pixels in the image. Consequently, qualitative interpretation of the target properties in terms of physical contents is made easier. The main interest of the method is the use of the segmentation map in order to yield a coloring scheme that preserves the smooth variations of the physical content across the scene. The method was validated on three Stokes images of biological tissues. Future work will address two aspects: the most suitable choice of a color palette that relates the coloring scheme with an efficient qualitative interpretation of the polarization information content and the extension of the proposed coloring scheme to the Mueller images case.
Fig. 10. Color previews of the three images I1, I2 and I3. (a), (b), and (c) correspond respectively to the HSV color maps. (d), (e), and (f) correspond to the LAB color maps.
Acknowledgment
The authors are indebted to two anonymous reviewers and to the editor for relevant comments that helped to improve the manuscript quality.
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