SkyPole—A method for locating the north celestial pole from skylight polarization patterns

Abstract

True north can be determined on Earth by three means: magnetic compasses, stars, and via the global navigation satellite systems (GNSS), each of which has its own drawbacks. GNSS are sensitive to jamming and spoofing, magnetic compasses are vulnerable to magnetic interferences, and the stars can be used only at night with a clear sky. As an alternative to these methods, nature-inspired navigational cues are of particular interest. Celestial polarization, which is used by insects such as Cataglyphis ants, can provide useful directional cues. Migrating birds calibrate their magnetic compasses by observing the celestial rotation at night. By combining these cues, we have developed a bioinspired optical method for finding the celestial pole during the daytime. This method, which we have named SkyPole, is based on the rotation of the skylight polarization pattern. A polarimetric camera was used to measure the degree of skylight polarization rotating with the Sun. Image difference processes were then applied to the time-varying measurements in order to determine the north celestial pole’s position and thus the observer’s latitude and bearing with respect to the true north.

During the last few decades, many high-precision geolocation tools have emerged thanks to Global Navigation Satellite Systems (GNSS) (1). However, GNSS-based position estimates are sometimes unreliable because of the multipath reception involved (2) and their sensitivity to jamming and spoofing. To overcome this problem, inertial navigation systems (INS) have been used to obtain position and orientation by implementing dead reckoning methods (3). However, INS suffer from a loss of precision in time due to the accumulating sensor measurement errors. Navigation issues have also been successfully handled by some animals by processing scarce information using simple navigational methods. The authors of neuro-ethological studies have described the use of skylight polarization patterns by insects such as desert ants for navigation purposes (4). Several studies have also established that migratory birds, such as Indigo Buntings (5, 6) and Savannah Sparrows (7), calibrate their magnetic compass at night by observing the dynamic movements of stars around the celestial pole (8). On similar lines, migratory songbirds calibrate their magnetic compass during the daytime on the basis of skylight polarization patterns (9, 10). Skylight polarization can be described by Rayleigh’s single scattering model (11), in which sunlight is assumed to be scattered by small particles present in the Earth’s atmosphere. Based on this model, cheap and reliable GPS-free polarization-based navigation methods have been developed. For instance, by mimicking the Cataglyphis desert ant, a polarization-based optical compass can provide a reference local bearing defined with respect to the polarization pattern in the sky (12, 13). However, like ordinary INS, polarization-based INS also suffers from integration drift. To overcome this problem, direct geopositioning can be achieved based on skylight polarization patterns without any need for GPS. By estimating the Sun’s position from the skylight polarization pattern and combining this information with complex calculations (solar ephemeris), the observer’s position can be computed (14–16). However, animals do not have access to ephemeris, and their use of polarization as a local or global reference for navigation purposes has not yet been completely elucidated.

In line with previous assumptions about the perception of polarized light by animals, we have adopted the suggestion put forward by Brines (17) that the temporal properties of the skylight polarization pattern can be regarded as a strong useful navigation cue. We developed a bioinspired method to find the geographical north bearing and the observer’s latitude, requiring only skylight polarization observations.

Skylight is mostly linearly polarized and characterized by two parameters: the angle of linear polarization (AoLP) and the degree of linear polarization (DoLP). The present method consists in comparing skylight DoLP images taken at different moments, in order to find the north celestial pole (NCP). The geographical north bearing and the observer’s latitude can then be deduced from the NCP coordinates.

Numerical Simulations of the SkyPole Method

When sun rays enter the atmosphere, Rayleigh scattering occurs, resulting in an observable pattern of skylight polarization depending on the Sun’s position. As described by the single scattering Rayleigh model, skylight polarization is mostly linear, and two quantities describing this phenomenon can therefore be observed, namely the direction of linear polarization and the DoLP. In this study, we focused on the DoLP, which is defined by the ratio between the polarized light intensity and the total light intensity. In the Rayleigh single scattering model, the pattern of DoLP can be described as follows (11):

$$\begin{array}{c}\hfill DoLP(\gamma )=\frac{1-{cos}^2\left(\gamma \right)}{1+{cos}^2\left(\gamma \right)},\end{array}$$ [1]

where γ is the scattering angle (cf. Fig. 1A).

Fig. 1.

(A) Scattering angle γ, azimuth α_P of a point P, and altitude θ_S of the Sun S. Parameters are presented in the ENU frame, namely East, North, and Up frame, centered on the observer O. The colored patterns stand for the skylight DoLP as described in Eq. 1. Dark blue corresponds to near-zero DoLP and yellow to maximum DoLP values (1 in theory, less in reality). (B) The trajectory of the Sun in the ENU frame, centered on an observer O located at latitude ϕ. NCP is the north celestial pole. The Sun moves on a plane perpendicular to the observer–NCP vector. (C) Invariance axis on the celestial sphere. Comparison between simulated and analytical sets of solutions. The green circle is the radial invariance circle; the red circle is the plane invariance circle computed from analytical calculations (cf. SI Appendix). The colored half sphere is the simulated absolute difference between two DoLP patterns associated with the Sun’s positions S₁ and S₂ at two different times. Dark blue corresponds to near-zero values. The red dot is the NCP. (D) Method for finding the NCP based on the skylight’s DoLP pattern. In the first row are the DoLP patterns taken at four different times. Absolute differences between DoLP patterns were then computed, giving the second row. A thresholding step was then applied to those images, and the results are presented in the third row. Last, binary images were summed, and the NCP was then located at the intersection between the radial invariances.

Eq. 1 describes two noteworthy properties of the DoLP pattern: its radial symmetry about the solar vector and the plane symmetry about the plane perpendicular to the solar vector, including the reference point O (Fig. 1A). Therefore, as depicted in Fig. 1C, when the Sun has shifted from an initial position, the resulting shifted DoLP pattern remains unchanged on two axes due to the symmetry of the DoLP function. From now on, we will refer to these invariances as radial and plane invariance. Similar patterns in the magnetic field might be seen by migratory birds to find north (18). We computed the difference between the images of the two patterns of DoLP taken at two distinct moments (with time intervals ranging from 30 to 60 minutes) in order to display the radial and plane invariance axes (Fig. 1C). Since the Sun rotates around the NCP (the south celestial pole as perceived by an observer in the Southern hemisphere, cf. ref. 19 for further details), the scattering angle γ is constant at this point, and therefore, the DoLP at the NCP will remain constant at all times of day. The NCP is located on the radial invariance circle, and this is the only visible point on the celestial sphere which is present at all times on this axis (cf. SI Appendix). The NCP can therefore be found at the intersection between the radial invariance axes resulting from time-varying DoLP image differences. Theoretically, only three views of the sky’s DoLP pattern are necessary for computing the position of the NCP.

Results

The SkyPole algorithm was tested on data recorded with a polarimetric camera equipped with a fisheye lens. The camera was placed on the roof of a laboratory in Marseille, France (43.286987°N, 5.4032786°E) giving a quasi-hemispherical view of the sky dome. The SkyPole algorithm described in Fig. 1D was applied to DoLP images of the sky in order to locate the NCP and assess the accuracy of the method. As the NCP is located in the sky at an azimuth equal to that of the geographical north and an altitude equal to the observer’s latitude, the latitude and north bearing measured were compared with the ground truth values obtained from the solar ephemeris. As shown in Fig. 2, the results obtained by applying the SkyPole method to experimental data ((20)) were consistent with the simulated results presented in Fig. 1D. We also estimated the NCP’s position with an azimuth of the camera ranging from 0 to 170° in steps of 10° (cf. Fig. 3). A mean absolute azimuth error of 2.6° and a mean absolute latitude error of 3.8° were obtained.

Fig. 2.

SkyPole algorithm applied to experimental data for finding the NCP. Preprocessing of the first row of DoLP images consisted in filtering the images obtained using a circular averaging filter. Details of the following steps are presented in Fig. 1D.

Fig. 3.

NCP coordinates computed with the SkyPole algorithm (Fig. 2) from experimental data versus ground truth NCP coordinates. α_cam is the azimuth of the camera with respect to the north. α_NCP is the azimuth of the NCP with respect to the azimuth of the camera. θ_NCP is the altitude of the NCP, which is also equal to the camera’s latitude. Δα_NCP and Δθ_NCP are the azimuth and altitude error, respectively, of the NCP measured with respect to the ground truth values. n_meas is the number of measurements for each error interval.

Discussion

The method presented here for locating the NCP requires only visual observations of the skylight polarization patterns. No knowledge of time, date, or ephemeris, and no estimates of the actual or initial position are required for this purpose. The image-processing steps do not rely on large computational resources. The efficiency of our method has been proved in simulations and under experimental conditions, thus confirming the validity of the theoretical model (cf. SI Appendix).

The accuracy of our algorithm may also seem rather too low for some geolocation applications. It is worth noting, however, that this algorithm has been kept as simple as possible, and that more sophisticated data processing algorithm would no doubt greatly improve the accuracy. Most of the errors in the NCP position estimates can be explained by two different sources of noise: skylight noise and camera noise. Skylight noise originates from multiple scattering or non-Rayleigh scattering in the atmosphere, such as Mie scattering, for instance. In this study, in order to reduce the influence of this kind of noise, we used data with low noise levels, corresponding to a clear blue sky or an only slightly overcast sky. In future studies, special emphasis should be placed on image filtering in order to reduce the influence of noise. In addition, in the case of both simulated and measured observations, each NCP position estimation process required several hours of observation. In this study, the aim was to find the NCP using only the radial invariance of the difference between skylight DoLP patterns. However, the point visible at the intersection between plane invariance and radial invariance gives the sky’s maximum DoLP value, which gives direct information about the Sun’s elevation.

Materials and Methods

The SkyPole method was tested with data collected from a Rayleigh sky simulation. We also tested our method with experimental data obtained with a calibrated Lucid Vision Lab PHX050S-QC, division of focal plane polarimetric camera. Our data were first preprocessed with a circular averaging filter. Next, we implemented the absolute difference in DoLP between several moments of time. We then applied a threshold-based binarization algorithm. Next, in order to obtain the NCP, several binary images were added. The NCP was located by searching for the maximum valued point, namely the intersection between all the radial invariance axes (cf. Fig. 1D). Last, the coordinates of the point were transformed to north and latitude coordinates and compared with the ground truth values. Additional experiments and data are available here: https://osf.io/fcsgk/.

Conclusion

In this paper, a method is presented for finding the NCP in the daytime by measuring the skylight polarization patterns. This method gives an estimate of the bearing toward the Geographical North Pole and the latitude of the observer.

Our minimalistic SkyPole algorithm could suggest a hypothesis in terms of possible visual processing steps used by animals to navigate. With a relatively long acquisition time, SkyPole may not be sufficient for autonomous navigation. However, SkyPole might be implemented on board future geolocation systems to provide navigation information in GPS-denied environments since it requires only skylight polarization observations.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Code and data images have been deposited in Open Science Foundation.