4.1.

Relationship of Angle of Linear Polarization to Surface Texture and Orientation

Matching the sky AoLP to expectations is an informative way to start GroundMSPI AoLP image analysis. The AoLP of a cloudless sky can be described by a single-scattering Rayleigh model where the polarization orientation is 90 deg from the scattering plane.¹⁹ When reporting AoLP images, the effect being demonstrated dictates which coordinate system is most insightful. For comparison, Fig. 1 shows a spring carnival on the UA campus in both scattering coordinates and meridian coordinates. At a given time the coordinate system can be readily identified from the color circle. When 0 deg AoLP is aligned with the zenith (i.e., the vertical direction of an image) the AoLP is reported in meridian coordinates and is denoted as ${\mathrm{AoLP}}_m$. When 0 deg AoLP on the color circle is tilted away from vertical, the AoLP is reported in scattering coordinates and is denoted as ${\mathrm{AoLP}}_s$.

Fig. 1

Spring carnival in (a) RGB radiance, (b)–(d) ${\mathrm{AoLP}}_s$ images in scattering coordinates, and (e)–(g) ${\mathrm{AoLP}}_m$ images in meridian coordinates for 470, 660, and 865 nm, respectively. For natural materials and man-made materials, which are rough, AoLP measurements usually match a cloudless sky. This value is constant throughout the day in scattering coordinates and is rendered in cyan. This value changes throughout the day in meridian coordinates. Time-lapsed movies of ${\mathrm{AoLP}}_s$ and ${\mathrm{AoLP}}_m$ (08:03 to 16:36 MST) are provided [Video 1, 36 MB, MOV (URL: https://doi.org/10.1117/1.OE.58.8.082419.1) and Video 2, 35 MB, MOV (URL: https://doi.org/10.1117/1.OE.58.8.082419.2)].

At first glance, the ${\mathrm{AoLP}}_s$ images in Fig. 1 are mostly cyan, corresponding to an angle perpendicular to the scattering plane. The sky and the grass are cyan. The road is blue, which is about 95 deg. Polarized light scattering from natural surfaces such as vegetation and bare soil are modeled as single-scattering events oriented perpendicular to the scattering plane.^10,26 This prediction is based on a microfacet model of the p-BRDF, which describes the polarizance of a material as Fresnel reflections from subresolution-sized microfacets that are randomly oriented.^15,27 The plane of polarization for a cloudless sky is also expected to be perpendicular to the scattering plane, which is observed in GroundMSPI AoLP images.

As supplements to Fig. 1, two time-lapsed videos are provided for the scene shown. These videos show the progression of AoLP during the day (08:03 to 16:36 MST) at the three GroundMSPI polarimetric wavelengths, 470, 660, and 865 nm. The solar illumination direction changes throughout the day to provide a range of scattering angles with a constant view direction. The video of AoLP with respect to the scattering plane, ${\mathrm{AoLP}}_s$ (Video 1), shows the sky and roughened surfaces to be predominantly cyan throughout the day. Cyan indicates ${\mathrm{AoLP}}_s=90\mathrm{deg}$. Also notice that the color circle rotates with time as the scattering plane changes with the varying solar illumination direction. The white tent and metal awnings in the foreground tend to deviate from ${\mathrm{AoLP}}_s=90\mathrm{deg}$ and the ${\mathrm{AoLP}}_s$ values are not constant throughout the day. Invariance of the AoLP to illumination angle can be readily identified in meridian coordinates.

The video corresponding to Fig. 1, panels (e)–(g), Video 2, shows the same scene with AoLP in meridian coordinates ${\mathrm{AoLP}}_m$. Notice that the sky and rough surfaces (e.g., grass and soil) share the same AoLP at a given time. Throughout the day, the ${\mathrm{AoLP}}_m$ value of these objects changes. The rough (i.e., microfacet) surface model predicts that there will always be some set of microfacets that are at just the right orientation to specularly reflect sunlight into the camera, and therefore follow the prediction of Fresnel’s laws in having polarization perpendicular to the scattering plane.

The meridian plane is defined by the view direction and zenith; it does not depend on the solar illumination direction and will remain constant when the view direction is constant. This reference frame allows for a useful look into smoother surface types such as the white tent and metal awnings. In particular, notice the white tent in the foreground. Once it is directly illuminated (after about 9 AM), the ${\mathrm{AoLP}}_m$ from most of the white tent in the foreground does not change with solar illumination angle and remains approximately perpendicular to the surface orientation. The exception to this trend is sun-specular reflections. Specular reflections are oriented 90 deg from the plane of incidence. The plane of incidence is defined by the illumination vector and the vector normal to the surface of the reflecting object. In the plane of incidence, the angle of incidence equals the angle of reflection for specular reflection. Therefore, when the view direction is aligned to a sun-specular reflection, the scattering plane equals the plane of incidence. The AoLP of sun-specular reflections is ${\mathrm{AoLP}}_s=90\mathrm{deg}$ and is rendered as cyan in scattering coordinates. Sun-specular reflections are saturated to white in the RGB image, which is difficult to identify on the white tent. The location of the sun-specular reflections is more obvious in the ${\mathrm{AoLP}}_s$ image; from 9:07 to 14:30 MST, a cyan sliver propagates across the right-side of the tent. For a given view and solar illumination geometry, there is a small range of surface orientations for which sun-specular reflections are observed. There is a white tent in the background that has the same AoLP trend but no sun-specular reflections are observed. The two metal shade awnings in the foreground exhibit an AoLP pattern similar to the white tents. At 8:41 MST, the awning on the right picks up sun-specular reflections for about 30 min. In off sun-specular directions the AoLP changes with the curvature of the metal awning and the angle of solar illumination until falling into the shade around 4:30 MST.

Figure 1 is the only dataset in this paper for which all three wavelengths of the same scene are shown. The AoLP is very similar within this wavelength range, with the notable exception that most regions appear noisier at 470 nm than at the two longer wavelengths. Also, there are a two notable objects for which the AoLP changes over the three wavelengths: the left side of the metal awning and the blue crowd-control barrier to the left of the awnings [see RGB radiance image in Fig. 1(a)]. Both of these objects also differ in DoLP, as can be seen in Fig. 2.

Fig. 2

Spring carnival in DoLP (a)–(c) for 470, 660, and 865 nm, respectively. Note that the DoLP of the sky is highest in 470 nm due to stronger Rayleigh scattering. A time-lapse DoLP movie (08:03 to 16:36 MST) is provided [Video 3, 21 MB, MOV (URL: https://doi.org/10.1117/1.OE.58.8.082419.3)].

Figure 2 shows the DoLP at three different wavelengths for the same scene as Fig. 1. The DoLP of the sky changes by almost a factor of 2 over this wavelength range. This observation is within the variations reported by comprehensive sky polarization studies.²⁸ The highest DoLP object on the ground is the black road. Some color trends with DoLP are observed: the orange brick has a relatively higher DoLP at 470 nm and the DoLP of the blue crowd-control barrier at bottom-left of the image is highest at 660 nm. These trends match Umov’s effect, which relates DoLP to albedo.²⁹

To contrast rough and smooth materials, Fig. 3(a) displays both in a single scene: a smooth metal sculpture directly in front of a brick building under clear sky conditions. In the RGB image of Fig. 3(a), sun-specular reflections are observed on the top half of the statue’s head and the shoulders. Figure 3(b) is a close-up of the spherical head of the metal statue and shows an AoLP that varies tangential to the surface on the bottom-half of the sphere and an AoLP that is 90° from the scattering plane for most of the top-half of the sphere. The top-half of the sphere are sun-specular reflections and the bottom-half of the sphere are specular reflections from the sky dome. In the shade of direct solar illumination, incident radiation comes from scattered skylight. Figure 3(b) shows the notable difference between sun-specular and sky-specular AoLP trends. On the bottom-half of the sphere for any given surface orientation and view geometry there is incident radiation from skylight illumination and the AoLP is perpendicular to the plane of incidence. Scattered skylight is an extended source illuminating from many angles. When both direct and skylight illumination are blocked in deep shade, such as the hole near the neck of the sculpture, the AoLP is noisy due to very low reflected polarized radiance.

Fig. 3

(a) RGB radiance, DoLP, and ${\mathrm{AoLP}}_m$ images of a metal sculpture with a spherical head that offers a range of surface orientations. (b) The polarization orientation on the statue’s head is rendered in black arrows to show a trend tangential to the surface.

Consistent with rough surface p-BRDF models, the AoLP of the cloudless sky is the same as the AoLP from the brick building, gravel ground, and cement base of the statue when these objects are directly illuminated. Prior work demonstrated agreement between GroundMSPI measurements of directly solar-illuminated rough surfaces and a microfacet polarized light scattering model, which predicts the AoLP is always perpendicular to the scattering plane.¹⁰ This p-BRDF model comprises a volumetric reflection term plus a specular reflection term of Fresnel-reflecting microfacets.²⁷

Three measurements at different times are shown in Fig. 4. Note that the base of the statue is cement and the AoLP is nearly insensitive to surface orientation and closely matches the AoLP of the brick building. The rough surface AoLP changes with solar illumination angle and matches the AoLP of the cloudless sky, which is perpendicular to the scattering plane. In the locations of sun-specular reflections, the top-half of the head and the middle-left part of the shoulders, the AoLP matches rough surface scattering (i.e., perpendicular to the scattering plane). However, the bottom-half of the sculpture’s head and the bottom of the sculpture’s body maintain nearly the same AoLP that is tangential to the surface and invariant to different solar illumination angles. In the locations of sun-specular reflections, the top-half of the head and the middle-left part of the shoulders, the AoLP matches rough surface scattering (i.e., perpendicular to the scattering plane).

Fig. 4

${\mathrm{AoLP}}_m$ and DoLP images at three time points and therefore three scattering angles. ${\mathrm{AoLP}}_m$ of the metal statue changes less throughout the day than rough surfaces such as the brick building, brick ground, and cement base of the sculpture. The only notable change in the DoLP at the different time points is the location of the bright sun-specular reflection on the top-half of the statue’s head.

This tangential relationship between AoLP and surface orientation is not unique to metallics. As shown in Fig. 5, nonmetallic smooth surfaces can also exhibit AoLP that is either perpendicular to the scattering plane or tangential to the surface orientation. For a plastic trash can and recycling bin, the AoLP direction is superimposed on the cropped area in Fig. 5(b) and exhibits the tangential relationship. Similarly, the AoLP from the windshield of the car in Fig. 5 is tangential to the surface. On the other hand, the AoLP from the hood of the car is perpendicular to the surface. It is reasonable to classify both of these materials as smooth. The relationship between AoLP and surface orientation of smooth objects also depends on if the dominant reflections are from the first surface of the material or if there is appreciable penetration depth, subsurface scattering, and refraction back out of the material.

Fig. 5

A car, trash bin, and recycling bin in (a) RGB radiance, DoLP, and ${\mathrm{AoLP}}_m$ images and (b) cropped to smooth plastic objects with polarization tangential to the surface rendered in black arrows to show tangential trend.

4.2.

Relationship of Angle of Linear Polarization to Albedo

The most well-known relationship between polarized light scattering and albedo is Umov’s effect,²⁹ which states that the DoLP is inversely proportional to albedo. In other words, dark objects have a higher DoLP. This effect is attributed to a difference in the dominant scattering process of dark, as compared to highly reflective, objects. Multiple interactions are more depolarizing than a single reflection. The high albedo of white objects yields a larger fraction of multiple interactions in the reflected fields as compared to dark objects. Multiple interactions are extinguished by dark objects due to their low albedo. Figure 6(a) is an RGB image of two adjacent cars, one black and one white. Figure 6(b) shows the DoLP image, and the inverse relationship between albedo and DoLP is notable.

Fig. 6

Adjacent black and white cars in (a) RGB radiance, (b) DoLP, and (c) ${\mathrm{AoLP}}_m$. The DoLP of the black car is greater than the DoLP of the white car as per Umov’s effect. The AoLP of the black and white cars are perpendicular to one another due to the differences in albedo and associated dominant scattering processes.

Umov’s effect explains the change in the polarized light scattering processes that makes albedo and DoLP inversely related. Reflectance from low albedo objects is dominated by first surface reflections and high albedo objects yield relatively more multiple interactions. In this work, an effect between AoLP and albedo is observed. The explanation is similar to Umov’s effect: a change in the polarized light scattering process for different albedos. The AoLPs on the roof of the black car and the roof of the white car are perpendicular to one another [see Fig. 6(c)]. The AoLP of the white car is perpendicular to the surface (i.e., $p$-polarized) due to its penetration into the material and refraction back out toward the camera. The black car polarized reflectance is due to single reflection off the top surface of the car and therefore the AoLP is parallel to the surface (i.e., s-polarized). An example of these different ray paths is depicted in Fig. 7. The low albedo of the black car results in high absorption and therefore low volumetric scattering. The surface reflections create polarized light reflected from the black car. The high albedo of the white car results in low absorption and therefore higher volumetric scattering. The white car also reflects light from the surface but a larger contribution to the polarized reflectance comes from backscattered refraction. Light penetrates the surface of the white car, scatters within the material, and some of this scattered light is refracted back out of the material. Light becomes partially polarized when refracted. The two dominant scattering processes from the black and the white cars yield perpendicular polarization states: one from reflection from the car surfaces and the other from refraction. This relationship between the AoLP and the geometry of a car has been predicted in Ref. 1, although no dependence on the color of the car has been indicated. It is interesting to note that changing the sheen of the black paint on a car had no effect on the AoLP, as noted by researchers studying the behavior of polarization-sensitive insects who were attracted to the car surfaces.³⁰

Fig. 7

The AoLP on the black and the white car roofs are perpendicular to one another [see Fig. 6(c)]. The AoLP of the white car is perpendicular to the surface (i.e., p-polarized) due to penetration into the material and refraction back out toward the camera. The black car polarized reflectance is due to single reflection off the top surface of the car and therefore the AoLP is parallel to the surface (i.e., $s$-polarized).

The albedo of a black and white car is both approximately spectrally invariant over the visible wavelength range. For many objects, including non-neutral colored cars, albedo is wavelength-dependent. Figure 8(a) is an RGB image of a red car. Figure 8(b) is the AoLP measurement for 470 nm (blue), and Fig. 8(d) is the AoLP measurement for 660 nm (red). Notice that the AoLP for 470-nm matches the black car condition. Here, the major contribution to polarization is due to first surface reflections since the 470 nm will be highly absorbed by the red car’s paint pigment. The red wavelength matches an AoLP for high albedo surface types with high internal scattering like that of the white car. The AoLP on the hood of the car at 660 nm is close to perpendicular compared to the AoLP measured at the 470-nm wavelength. In other words, the AoLP on the hood of the car changes according to the albedo of the material at that wavelength. The trend between albedo and DoLP is predicted by Umov’s effect: DoLP of 470 nm in Fig. 8(c) is higher than the DoLP of 660 nm in Fig. 8(e). The AoLP albedo dependence is only observed for direct solar illumination; the side of the car is in the shadow and the AoLP is the same over both wavelengths. The neutral-colored cars in the background do not have a spectrally dependent AoLP, even in direct illumination. This example demonstrates that the illumination, the albedo, and associated dominant scattering process need to be known to correctly interpret AoLP measurements and/or select an appropriate polarized light scattering model.

Fig. 8

Red car with neutral-colored cars in the background shown in (a) RGB radiance, (b)–(c) 470-nm AoLP and DoLP, and (d)–(e) 660-nm AoLP and DoLP. The AoLP of the red car’s hood and roof at 470 nm is perpendicular to the AoLP at 660 nm due to differences in albedo and associated dominant scattering processes. The AoLP and DoLP values of the neutral-colored cars in the background are spectrally invariant compared to the red car.

4.3.

Relationship of Angle of Linear Polarization to Illumination

In the outdoor environment, the illuminating source can be any combination of sunlight, skylight, and radiation reflected from nearby objects. The white metallic dome of an observatory is measured on a cloudy day (Fig. 9) and on a sunny day (Fig. 10). Clouds are depolarizing over most scattering angles. Note that the DoLP of the sky on the cloudy day in Fig. 9(b) is close to 0, but the DoLP of the sky on the sunny day in Fig. 10(b) is close to 0.20. In both cases, the AoLP trends with the surface orientation of the dome but is flipped 90 deg when comparing sunny to cloudy conditions. This is an interesting special case of a white metallic object under two illumination conditions that each yields different dominant polarized light scattering. The illumination conditions on two different days causes a 90-deg flip of the AoLP measured from the white dome. In the car example the dominant scattering behavior changes when the illumination is constant but the albedo is different. On the sunny day, the image is acquired late in the day (see Table 1) and no sun-specular reflections on the dome are present. The remaining contribution to the reflected polarized radiance is from subsurface interactions and polarized light that is backscattered and refracted out of the surface (see $p$-polarized ray path in Fig. 7). This dominant scattering behavior is similar to the white car in Fig. 6. The dome and white car illuminated on the sunny day also radiate first surface reflections (see s-polarized ray path in Fig. 7). However, the high albedo of the material causes the backscattering term to dominate. On a sunny day objects are illuminated by two sources: direct solar illumination and scattered skylight. Direct solar illumination is several orders of magnitude greater than scattered skylight.³¹ On the cloudy day (Fig. 9), direct solar illumination is blocked by the clouds and the dome is only illuminated by skylight. The dominant polarized light scattering is a specular reflection of the dome’s surface toward the sky at all surface orientations. This dominant scattering behavior, i.e., first-surface reflections, is similar to the black car in Fig. 6 and creates the same AoLP trend with surface orientation.

Fig. 9

On a cloudy day under diffuse illumination a white painted metal dome in (a) RGB radiance, (b) 660-nm DoLP, and (c) 660-nm AoLP. The DoLP ranges from 0.01 to 0.03 across the dome. The AoLP is perpendicular to the same object under sunny conditions reported in Fig. 10.

Fig. 10

On a sunny day under direct illumination a white painted metal dome in (a) RGB radiance, (b) 660-nm DoLP, and (c) 660-nm AoLP. The DoLP ranges from 0.07 to 0.08 across the dome. The AoLP is perpendicular to the same object under cloudy conditions reported in Fig. 9.

There is a black metal door on the left-hand side of the building that is in the shade of a doorway. The DoLP of this object is high and it is vertically polarized on both cloudy and sunny days. By contrast, the AoLP values of the windows in this scene are flipped 90 deg on sunny versus cloudy days. There is a square window to the left of the dome and a taller window to the right of the dome that have an AoLP, which matches the left side of the dome on each day. These AoLP differences in Figs. 9 and 10 signify a difference in the dominant scattering process when the object is directly illuminated by the sun or in the shade (i.e., illuminated by skylight). This example shows that relatively ordinary changes in illumination can cause a large change in the dominant polarized light scattering process and AoLP.

The cloudy day dome measurement is also an excellent example of GroundMSPI’s polarimetric accuracy. A white object is very depolarizing and on the cloudy day the DoLP in Fig. 9(b) is only 0.01 to 0.03 across the dome. The associated AoLP image in Fig. 9(c) still shows a clear trend with surface orientation even though this image looks noisier than Fig. 10(c) since the DoLP is higher on the sunny day.

Können’s text on polarized light scattering in nature predicted and observed an AoLP tangential to the surface for objects in the shade.¹ Figure 11 illustrates this effect since the ground surface and perpendicular side of the pedestal are both in the shade. The AoLP in these two adjacent regions is tangential to each surface and perpendicular to each other. Where these regions intersect, at the bottom of the pedestal and ground next to the pedestal, the AoLP is shifted about 5 deg to 10 deg in these small regions toward the polarization orientation of the adjacent surface. This AoLP shift is likely due to reflections from the nearby surfaces. This effect is not observable when the two perpendicular surfaces are farther apart. Note that in Fig. 11(b) the cement ground does not match the AoLP of the sky. In this scene the sky contains clouds and therefore the sky AoLP is not expected to match a single-scattering Rayleigh model.²²

Fig. 11

A cement cowboy sculpture on a pedestal in (a) RGB radiance, (b) 660-nm DoLP, and (c) 660-nm ${\mathrm{AoLP}}_m$. Notice the shadows on the pedestal and concrete ground. The AoLP in these shaded regions is tangential to the surface. A time-lapse movie (07:18 to 16:18 MST) is provided [Video 4, 17 MB, MOV (URL: https://doi.org/10.1117/1.OE.58.8.082419.4)].