In situ measurements of reef squid polarization patterns using two-dimensional polarization data mapped onto three-dimensional tessellated surfaces

Abstract

Reef squids belong to a group reputed for polarization sensitivity, yet polarization patterns of reef squid have not been quantified in situ. To quantify polarization patterns from video polarimetric data, we developed a protocol to map two-dimensional polarization data onto squid-shaped three-dimensional tessellated surfaces. This protocol provided a robust data container used to investigate three-dimensional regions-of-interest, producing data lineouts derived from the squid's geometry. This protocol also extracted polarimeter and squid body orientations and the solar heading from polarization images. When averaged over the solar heading, the ventral midline gave a low degree of polarization (2.4 ± 5.3%), and the area between the ventral and flank midlines had higher values (9.0 ± 5.3%). These averaged data had a large discontinuity in the angle of polarization (AoP) at the mantle's ventral midline (64 ± 55°), with larger discontinuities measured on individual squid. Ray-tracing calculations demonstrated that the AoP pattern was not related to the squid's surface-normal geometry. However, the AoP followed virtual striation axes on the squid's surface oriented 24° to the squid's long axis, similar in angle (27°) to the striations of birefringent collagen fibres documented in other squid species’ skin.

1. Introduction

Reef squids are charismatic animals active during the day in shallow coral reef habitats. Able to change colour rapidly (less than 30 ms), they have an extensive repertoire of patterning signals, ranging from solid dark brown to translucent, intermixed with bands, longitudinal bars, crossbars and flashing signals [1,2]. They communicate with conspecifics in coordinated schools and change visual patterns based on social contexts or for predator avoidance [3–5]. Despite their beautiful coloration, with olive-brown to translucent pearl skin speckled with bright blue and green patches of iridophores, they belong to a group that is well known for lacking colour vision [6–8]. Despite being colour-blind, cephalopods visually sense polarized light [8–10], where polarized light is another visual sense observed in many animals [11–13]. Polarized light refers to the vibration coherence of the electric vector in the propagating light wave. The percentage of light that is coherently linearly polarized is the degree of linear polarization (DoLP), and the angle of the electric vector relative to the observer's point of view is the angle of polarization (AoP). Cephalopods respond behaviourally to polarized stimuli (e.g. looming stimuli [14]) and demonstrate sensitivity to small changes in AoP (down to 1.05° [15]).

A propensity for visual communication combined with polarization sensitivity would imply that reef squids likely use polarization patterning for communication. Polarization patterns have significant differences from other patterning modalities. Pigment-based patterns are least like polarization patterns in that they have little influence from the viewing angle direction. Iridescent patterns are dependent on the object's geometry, environmental lighting and viewing angle, yet polarization patterns can be more variably dependent on these factors than iridescence [11,16]. Angle dependent signals like iridescence can direct signals to an intended receiver or make signals flash and shimmer with movement and could also be used to hide signals from an unwanted observer [17,18]. This complexity also presents signalling challenges for the same reason—that projecting one's signal to the intended viewer depends on many variables. This complexity may be why there are so few studies of in situ polarization patterning [13,19]. While studies show significant polarization reflections from cephalopod skin in laboratory settings [8,20–22], there are no in situ polarization characterizations of squid polarization reflectance. Reproducing natural solar polarization conditions in the laboratory is difficult, especially in ways suitable for squids to act naturally, thus in situ characterizations are essential for understanding their natural reflectance.

To describe squid polarization patterning in situ, we quantified polarization variation correlated with viewing position, solar heading and within body variation. We used a diver-operated RGB polarimeter to measure free-swimming diurnal reef squid (Sepioteuthis lessoniana) at Lizard Island, Australia to collect data for this measurement. Because polarization signals are best analysed in a three-dimensional context, we employed computer vision tools to form a correspondence between two-dimensional polarization video footage and a three-dimensional virtual model of the squid. Virtual three-dimensional environments are mainstream in many computer applications, and tools for manipulating and analysing virtual three-dimensional structures are now found in well-developed open-source programs. With these tools, analysing polarization information in three-dimensional virtual environments is now reasonably achievable. We developed a protocol to map two-dimensional polarization information onto a three-dimensional virtual squid model to analyse the data in three dimensions. This protocol organizes all data connected to this projection into a single entity called a data container. This data container was used to investigate three-dimensional regions-of-interest (ROIs) consisting of lines superimposed on the model's surface. From these lines, linearly ordered datasets, or lineouts, were used to visualize the data derived from the squid's geometry and not the image. Using this data container, we test two possible conjectures of the observed polarization patterns’ origins, one originating from surface-normal reflections and the other from geometrically organized skin structures that modify polarization.

2. Methods

2.1. The animals and the environment

Using our diver-operated polarimeter, approximately 10–20 individual reef squids (Sepioteuthis lessoniana) were filmed at Lizard Island, Australia, with a polarimeter, in sunny conditions and shallow waters (approx. 2 m deep) on 4 September 2015. The water was clear, with visibility greater than 10 m, and the squids were filmed sufficiently close to the camera (approx. 1–2 m) that water column light scattering was minimal. Reef squids are active during the day and stay relatively close to the surface near reefs and seagrass. The squids generally were filmed near the water's surface with the polarimeter aimed up (a positive pitch angle) toward the squid. The squids were filmed with a close field of view, and the schooling structure was not observed. The squids rarely displayed striped signals and did not display anti-predatory behaviour such as arm flailing or inking. We extracted over 500 images of approximately 10 squids at various viewing positions from approximately 60 min of video footage filmed between 8 : 56 and 10 : 19. Images were chosen to obtain as many different squid positions as possible with the squid close to the polarimeter, static and with no motion blur.

We report n-values (sample sizes) that combine measurements across a number of individual squid (e.g. no. measurements reported > no. individuals sampled). Given that we collected measurements in the wild on live squid behaving naturally, this type of sampling was necessary albeit not ideal. Ideally, we would collect polarization measurements completely surrounding (360°) each individual due to the angular dependence of polarization, but measuring non-restrained animals in the wild made this ideal impossible to obtain. However, it is worth noting that more comprehensive polarization datasets on restrained fish in the ocean have shown concordance between individual average measurements and the combined (all individuals) n-value [23] which suggests that the current methodology we adopted here might be an accurate polarization characterization of this squid species.

2.2. Polarization and polarization contrast measurements

To measure visual quantities involved with background matching, we used the diver-operated polarimeter described in Garcia et al. [24] (see electronic supplementary material) to measure the RGB radiance and polarization values (AoP and DoLP) underwater (figure 2). The noise errors for this camera are 1.2% and 0.18% for DoLP and AoP, respectively, for high polarization and intensity sources and less than 5% and 1.7% for DoLP and AoP, respectively, for high intensity (near saturation) and low DoLP (approx. 0.05) sources. We filmed a MacBeth colour chart to calibrate the colour balance at each measured habitat at the time of the measurement.

Figure 2.

(a) The squid-like STL mesh comprised a mosaic of triangles. We select groups of these triangles to form linear ROIs. For this study, we used four linear ROI placements identified by colour: blue, magenta, yellow and green. Each linear ROI comprised triangular elements with unique ID numbers. Overlays of the magenta ROI are shown on squid images in light blue, and the outline and contours of the STL mesh are in dark blue and red, respectively. (b) The angles associated with the calculations in the study.

The DoLP and the AoP characterize polarization signals derived from mathematical constructs, but their relevance to how an animal perceives polarization is still in development. How & Marshall [25] derive the formulation of a polarization contrast, called the polarization distance, for animals equipped with two orientations of polarization detectors. How & Marshall [25] state that the sensitivity of a polarization receptor is

$$R(\theta ,d)=\left[1+\left(\frac{d(S_p-1)}{S_p+1}\right)\cos (2\theta -2{\theta }_{\max })\right],$$ 2.1

where θ is the AoP, θ_max is the orientation of the photoreceptor, d is the DoLP and S_p is the sensitivity of each sensor (assumed to be 10 [25]). The opponency signal between two receptors is given by

$$P1=\ln \left(\frac{R_1}{R_2}\right).$$ 2.2

The signal difference between the receptor pair measuring the object and background is

$$PD(o,b)=\frac{(P{1}_{\mathrm{obj}}-P{1}_{\mathrm{bgd}})}{2(\ln 10)}.$$ 2.3

We calculated the polarization distance based on two orthogonal polarization detectors aligned horizontally and vertically. We measured the background polarization states from the water column near the squid in the polarization image. The techniques for behaviourally measuring the minimum detectable polarization distance have only been used with a few animals. The minimum detectable polarization distance was reported to be 0.022 for a species of fiddler crab [25].

2.3. Mapping two-dimensional polarization information onto three-dimensional virtual models of squid

We introduce methods for using three-dimensional virtual squid models to organize and analyse data (see figure 2b; electronic supplementary material). There are four main advantages for converting the two-dimensional polarization imagery onto a three-dimensional virtual fish model, including (i) increasing accuracy of ROI placement on specific parts of the squid where the squid skin was featureless, (ii) ROI placement flexibility post analysis, (iii) ability to measure the relative position of the squid with the solar direction and camera angle and (iv) ability to perform ray-tracing simulations on the polarization nature of the reflectance. We developed an analysis program called MIDOM (Mapping Image Data Onto Meshes, https://github.com/scorpionjeger/PATMOS_MIDOM) to map two-dimensional polarization data onto three-dimensional virtual models of squid. The three-dimensional virtual surface used in this analysis was a Standard Tessellation Language (STL) mesh, which is a powerful virtual environment analysis tool. Key anatomical points, including eyes, squid mantle apex and other distinctive parts, with their three-dimensional virtual surface position values, were used to quantify the virtual surface orientation. We employed both manual and automated methods to find the error minimized map between the squid image and the virtual surface (see electronic supplementary material). This process also returned the relative orientations of the squid and polarimeter (pitch, yaw, roll, etc.). As the squid generally keeps both eyes level, we set the roll to zero. We estimated the error for this process by introducing a uniform random distribution 10 pixels wide to the automated surface-image mapping process. The measured spread of the orientation outputs gave an average standard deviation of the fish heading and pitch to be 14° and 4°, respectively (see electronic supplementary material). With a projection of the virtual surface onto an image, we mapped image polarization data onto the virtual surface. With the polarization data mapped onto the squid's virtual surface, any part of the surface can be chosen, post-process, for a three-dimensional ROI.

With the polarization data mapped onto the squid STL mesh, we can choose any part of the STL for a three-dimensional ROI. From visual inspection (see figure 1), the polarization patterns vary most about the squid mantle's azimuthal direction and uniformity along with the axis. Hence, we use ring-shaped ROIs of SLT mesh triangles about the mantle axis for our analysis (see figure 2b). Since most images recorded the squid's mantle from below, we only used the mesh triangles along with the mantle's ventral half. The linear order of the ROI was mapped onto Cartesian graphs for analysis. The background water column's polarization parameters were measured for each squid from a square ROI, approximately the squid's width, placed close to the squid. The data from the ROI overlay represent the average and standard deviation of the values of all pixels within the projected ROI.

Figure 1.

The reef squid (Sepioteuthis lessoniana) measured on Lizard Island display polarization patterns in the AoP (in degrees) and the DoLP. The DoLP and AoP images are rendered as false-colour images from the polarimeter's green channel (legends on the right). The water column serves as the images' backdrop, which generally has uniform polarization values that are dependent on the solar direction. These images have the mantle apex facing away from the polarimeter. The DoLP and AoP are thresholded by displaying black on the image if the DoLP is less than 5%. The DoLP colour maps are binned between 5% and 40%. Results are displayed from the surface AoP calculation measuring the AoP from the plane of incidence with the squid-like STL mesh. These AoP calculations are strikingly different from the measurements contradicting the conjecture that the AoP polarization patterns originate from surface-normal reflections.

2.4. Polarization ray-tracing

The AoP reflection often derives its value from the geometry and shape of an object. Reflections from the object's geometry are dependent on the surface's normal vector, which vector is perpendicular to the surface at any point and defines the plane of incidence for the reflection. Most natural reflections, i.e. glare, are governed by Fresnel reflection rules that define the reflected AoP value as perpendicular to the plane of incidence [16]. Thus, the surface-normal geometry designates the AoP for specular Fresnel-governed surfaces. This AoP–geometry relationship is significant and is used in industry to reconstruct an object's geometry from the AoP alone [26,27].

To calculate the AoP of the virtual squid meshes, we developed a ray-tracing algorithm that calculated light propagation in virtual environments by tracing light rays from the polarimeter to virtual objects. Virtual rays projected from the polarimeter, representing a polarimeter's pixel and intersected the mesh using the python-based open-source Visualization Tool Kit (VTK). VTK calculated the propagating ray's intersection with the surface and the surface's normal vector at the intersection. The plane of incidence is calculated from the cross product of the normal vector and the propagation ray. The perpendicular direction of the plane of incidence is projected to the polarimeter's image, and the angle between this projected direction and the polarimeter's horizontal is the AoP of the calculation. These ray-tracing calculations used the same polarimeter and squid STL mesh positions derived from the measurements.

3. Results

To quantify polarization patterns varying about the axis of the mantle, we analysed three-dimensional linear ROIs. These three-dimensional linear ROIs began just above the right flank midline (arms facing forward), where the fin was attached to the mantle. The ROI looped around the mantle axis to the other side of the squid, passing the ventral midline. There were four linear ROIs used in this study (delineated by colour in figure 2b); however, because the variation between these ROIs was minimal (figure 5b,d), we used only the magenta ROI for the following analysis. Because the variation between these colour channels was minimal (figure 5a,c), we used the mid-intensity green channel for the analysis, unless otherwise stated. The average MIDOM calculated distance between the polarimeter and squid was 8.3 ± 3.5 body lengths (defined as the distance from the eye position to the squid mantle apex on the squid's axis) equivalent to 100 ± 42 cm if the squid length was 12 cm. Given the water's clarity and the squids, proximity to the camera, the water column polarization contribution to the squids' polarization values will be negligible.

Figure 5.

Column (a) shows ROI lineouts of the polarimeter's colour channels for DoLP and AoP (polarimeter channel colours correspond to the graph lineout colours). The data come from the magenta labelled ROI in figure 2 with the mantle apex facing away from the polarimeter and the polarimeter pitch greater than 20°. The red channel has slightly higher DoLP, and the AoP shows a discontinuity that favours longer wavelengths suggesting that achromatic birefringence may contribute to the AoP signal. Column (b) shows the ROI lineouts of the four linear ROIs presented in figure 2 with the lineout colours following the ROI coloration in figure 2. The mantle apex is facing away from the polarimeter, and the polarimeter colour channel is green. The number of measurements represented by each lineout is 114. The vertical lines represent the error of the distribution, and the x-axis of each lineout is slightly shifted for better visualization of the error.

Because polarization signals are often strongly influenced by the sun's relative angle, we investigated the contribution of the solar heading angle to the polarization variation. The polarimeter was not equipped with an inertial measurement unit to measure the polarimeter's heading. Hence, the solar heading was calculated using the background water column's polarization values using analytical techniques developed by Powel et al. [28] (see electronic supplementary material). We binned linear ROI data using the mantle apex direction with an angular interval of 90° pointing either toward or away from the sun while the polarimeter viewing angle was binned at 22.5° increments around the squid, as shown in figure 3. Many scattering (e.g. from ocean water [29]) and reflection (e.g. from dielectric surfaces [16]) scenarios have maximal DoLP reflections at or near the direction perpendicular to the illumination angle and minimal DoLP parallel to the illumination angle. The ROI elements between the flank midline and the ventral midline consistently had higher DoLP at azimuth angles nearing perpendicular to the solar direction (figure 3a,b). The flank midlines show low DoLP values, and the ventral midline is variable (figure 3a,b). The AoP followed the background AoP sinusoidal pattern except for the ventral midline that deviated drastically from this trend (figure 3c,d), especially when the mantle apex pointed toward the sun. The ventral midline also had the highest consistent polarization contrast (also called polarization distance; see electronic supplementary material) [25] than other linear ROI elements (figure 3e,f), hovering around the 0.022 minimum detection threshold cited for fiddler crabs [25]. These variations show that the squid's polarization reflection generally followed the expected trends for solar scattering except for the ventral midline.

Figure 3.

Lineouts of five locations on the magenta linear ROI (see figure 2a) that comprised the right (cyan) and left (blue) flank midline, the ventral midline (red), and two points (right, black; left, green) between the flank and ventral midlines. The mantle apex is pointing (a,c,e) away from the sun and (b,d,f) toward the sun with a ±45° interval (see figure 2b). The polarization values of each element of the linear ROI were averaged over 22.5° increments in terms of the polarimeter's azimuthal angle with respect to the solar position. The AoP, DoLP and polarization distance are shown, when present in the dataset, as are the locations of each line out on the STL mesh. The error bars represent the standard deviation of the distribution and the x-values of the data are slightly shifted for error bar visibility. The data came from the polarimeter's green colour channel.

To visualize within-body patterning variation, we mapped the three-dimensional linear ROI onto two-dimensional Cartesian graphs, evaluating the ventral linear ROI from the left to the right flank midline. Electronic supplementary material, figure S4, displays polarization data with polarimeter pitch binned between 20° and 50° and the mantle apex pointing away from the polarimeter as heat maps. The graphs' y-column was the solar heading of the measurement obtained from the polarization solar heading calculator. A qualitative assessment of this heat map showed that the AoP reflections had large discontinuities, or abrupt changes in the AoP trends, often flipping sign, near the squids' ventral midline. The DoLP had more signal between the flank midlines and the ventral midline, and the AoP difference from the background showed high signals near the ventral midline. The power of using this three-dimensional ROI is that we can average each ROI element over various parameters and analyse a data lineout that is dependent on the squid's geometry and not the image.

To quantify within-body polarization variation, we averaged over the polarimeter solar heading values and binned data according to polarimeter and squid orientations. The area between the ventral and flank midlines had higher DoLP values dependent on the polarimeter pitch, and the ventral midline had low DoLP. The averaged DoLP was minimal at the ventral midline and maximal between the flank midline and the ventral midline (e.g. 2.4 ± 5.3% and 7.0 ± 5.3%, respectively for polarimeter pitch greater than 20°, mantle apex opposite polarimeter, n = 110, figure 4a,b). The DoLP peaks increased in intensity as the polarimeter pitch decreased (figure 4a,b). When the polarimeter pitches were between 0° and 10°, the maximal DoLP value was 9.7 ± 5.0% (n = 100) and 11.0 ± 5.5% (n = 100), with the mantle apex pointing away from and toward the polarimeter, respectively (figure 4a,b).

Figure 4.

Linear ROI (magenta ROI, see figure 2) averages represent data with the mantle apex facing toward and away from the polarimeter. Line colour represents different pitch values of the polarimeter with red having 0° to 10° (n = 100 left column, 26 right column), green 10° to 20° (n = 118 left, 45 right) and blue 20° to 50° (n = 110 left, 73 right). The vertical lines represent the error of the distribution, and the x-axis of each lineout is slightly shifted for better visualization of the error. The ventral midline is ROI element 20, the right flank midline is element 4 and the left flank midline is element 37. (a,b) The DoLP, (c,d) the polarization distance, (e,f) the AoP and (g,h) the surface-normal calculated AoP from the squid-like STL mesh using the same orientation parameters as the measurement, showing stark differences from the measured AoP values.

The AoP signal generally had a sharp discontinuity along with the midline of the averaged linear ROIs. With the mantle apex facing away from the polarimeter, the AoP increased from zero to a positive number on the squid's right flank, discontinuously flipped sign at the ventral midline and increased to zero at the left flank (figure 4e,f). This AoP discontinuity averaged to 64 ± 55° (n = 110) when the mantle apex faced away from the polarimeter. For individual AoP measurements, with the AoP discontinuities thresholded above 100°, there were 27 measured discontinuities with a mean of 146 ± 22° when the mantle apex faced away from the polarimeter, and the polarimeter pitch was greater than 20° (electronic supplementary material, figure S2b). The AoP discontinuity did not significantly change as the polarimeter viewing pitch increased and was most dramatic when the mantle apex pointed away from the polarimeter. When the mantle apex pointed toward the polarimeter, the discontinuity was less pronounced (30 ± 38°, n = 73), with the greatest discontinuity occurring when the polarimeter pitch angles were greater than 20°. The discontinuity direction flipped sign in these graphs because the squid's perspective was flipped, and the ROI elements were on opposite sides of the image. This discontinuity may be less pronounced because there are fewer measurements for these positions. The squid mantle apex facing the polarimeter presents a blind spot for the squid toward the diver, and they may be more likely to avoid that orientation.

The colour channels of the polarimeter had subtle differences in the linear ROI average. The red channel had a slightly higher DoLP at the midpoints between the ventral and flank midlines (DoLP maximum of 8.5 ± 6.7%, 7.2 ± 5.4%, and 7.1 ± 5.1% for red, green and blue, respectively, n = 110, figure 5a). The AoP discontinuity favoured the red channel with discontinuity values of 73 ± 65°, 64 ± 55° and 50 ± 42° for red, green and blue, respectively (n = 110, figure 5c). Colour differences in the AoP and DoLP can result from the birefringence of squid tissues. Polarization alteration from birefringence often has significant spectral effects.

The polarization distance [25] is dependent on the differences between both AoP and DoLP to the background. While both background and squid maximum DoLP were 27%, the two differed in mean DoLP with DoLP signal from the squid being generally lower (mean DoLP = 7 ± 4%) than background (mean DoLP = 13 ± 5%), suggesting that the squid might silhouette against a higher DoLP background. Along with the ventral midline, the DoLP is very low (2.4 ± 5.3%) and the AoP is very different from the background (electronic supplementary material, figure S2d), providing conditions for significant contrast against the background. Indeed, the polarization distance values along the ventral midline were the most pronounced and were bordered by low polarization distance values where the DoLP was maximal between the flank midline and the ventral midline (figure 4c,d). The midline polarization distance averaged was 0.024 ± 0.012, with a maximum measurement of 0.066, which was above the 0.022 polarization distance threshold reported for fiddler crabs [25], and squid polarization sensitivity is likely higher [15].

3.1. Relationships between the angle of polarization and the squid geometry

The squids' AoP pattern near the ventral midline was repeatable and distinctive due to its large discontinuity. This study's geometry-based methods were specially equipped to address two conjectures on the AoP pattern's origins. The first conjecture claims that the pattern comes from surface-normal geometry-based reflections common with most objects, and the second claims that the pattern comes from geometrically organized skin structures that modify polarization.

The reflected glare from most smooth surfaces comprises continuous AoP gradients dependent on the surface curvature. The squid reflected smooth continuous gradients of AoP that appeared to be related to specular surface geometry, hence the reason for the conjecture that the observed patterns originate from surface-normal geometry. A squid is a complicated optical composite with a mix of translucence and surface-bound chromatophores (iridophores and melanophores), so it is unlikely to follow this trend. However, in some cases, the iridophores in squid skin appear to be aligned to approximate a specular reflection [30].

The three-dimensional linear ROI with the associated polarimeter parameters and squid STL mesh orientation for each measurement were inputted into the ray-tracing calculation to measure the AoP from the surface-normal geometry of the STL mesh. The calculation results showed an AoP gradient continuously varying from the squid flanks throughout the mantle's circumference (figure 1; electronic supplementary material, figure S2b). The calculated AoP value was zero at the ventral midline and reversed sign for each flank (figure 4g,h). These gradients contrasted with the squid's AoP measurements (figure 4g,h), which had a large discontinuity at the midline. The sign of the calculated gradient was opposite to what we observed in the squid, and there was no abrupt discontinuity in the midline of the calculated squid data (figure 4g,h). This calculation showed that the AoP ventral midline discontinuities and gradients did not originate from the squid's surface-normal geometry.

The shape can influence the AoP other than surface-normal geometry reflections through striated dichroic structures. Dichroism is a material property that absorbs light for off-axis polarization components and transmits on-axis polarization components (e.g. polarization film). The AoP aligns with the axis of the dichroic material. An object patterned with different stria of dichroic material would create an AoP pattern that follows the dichroic pattern, as in the antennal scale of the stomatopod Odontodactylus scyllarus [31]. The DoLP is generally high for dichroic materials.

A property similar to dichroism that depolarizes off-axis polarization components instead of absorbing them has been described in collagen fibres from fish skin [19,32], and these same collagen fibre arrangements have been found in squid skin [33]. These fibres are birefringent, a property that modifies polarization based on the AoP. Birefringence has a value and an orientation, and in collagen fibres, the birefringence is random in value but ordered in orientation, and off-axis polarization states become randomized and depolarized [19,32]. The remaining polarization states are aligned, parallel or perpendicular, with the structure's orientation. These birefringence structures, which we term ‘dichroic-like’, have a similar alignment effect for the AoP as dichroic material, but the DoLP is generally much lower. A histology study of the squids Lolliguncula breiis and Loligo pealei found a layer of birefringent collagen fibres wrapped around the mantle's outer surface, with the collagen axis angled 27° ± 1° to the long axis of the squid [33]. These structures could create off-axis depolarization as seen in fish skin [19,32] and would create an AoP pattern that would follow 27° to the squid's long axis depending on the perspective of the observer.

The STL mesh structure provided a unique way to investigate the effect of dichroic-like skin structures on the AoP. The geometric construct of the STL allows for straightforward calculations of various vector operations. For each triangular component, the normal vector and the values of each vertex are included in the STL data structure. From these values, the axis vectors of virtual dichroic-like structures on the STL surface can be rotated about the normal vector for each triangular element (figure 6a) to provide a means for comparing these axis vectors with the AoP of the image. For reference, the angle for this rotation, θ, is aligned with the mantle's long axis at θ = 90° and aligned along with the mantle's circumference at θ = 0°. We projected the axis of virtual dichroic-like structures from the STL surface to the polarization image, to find the deviation angle, ψ, between the projected axes and the AoP (figure 6a). These axis vectors rotated about the normal vector of each linear ROI element between θ = 0° and θ = 80°, to find the value that correlated most with the measured AoP distribution. To find the average deviation angle, we calculated the magnitude of the cross product of unit vectors representing the projected axis vector and the AoP and converted the average of this value back to an angle using sin⁻¹ x.

Figure 6.

Data representing the projection of striation lines on the squid's surface to simulate the effects of dichroic-like structures within the skin. (a) The striation line (blue) rotates around the surface-normal vector (red) with θ, the rotation angle, from the baseline vector. The baseline vector is aligned with the circumference of the squid mantle. θ = 90° is aligned with the squid long axis. These striations are projected onto the polarization image, and the angle between the measured AoP of the squid and the projected striation line is ψ, the deviation angle. (b) The average deviation angle over all linear ROI elements with the squid mantle apex facing away from the polarimeter and the polarimeter pitch greater than 20° is recorded and plotted against the rotation angle θ. The orange graph has the same rotation angle for each linear ROI element. The blue graph mirrors the rotation angle, θ, across the ventral midline (θ_R = −θ_L for the right and left sides). The blue and orange graph minimum value occurs at 114° and 109°, respectively.

In figure 6b, we calculated the average deviation angle for all linear ROI elements with the squid mantle apex facing away from the polarimeter and the polarimeter pitch greater than 20° (used because of the completeness of the dataset). Because there is an AoP discontinuity in the squid ventral midline, we postulated that the striations of the collagen might flip sign across the midline (θ_R = −θ_L for the right and left sides) due to the bilateral symmetry of the squid. We graphed the average deviation angle ψ against θ for distributions that both uniformly vary θ across the linear ROI and for values that mirrored θ across the ventral midline in figure 6b. The uniform θ distribution showed a minimization influence both above and below the squid's long axis at θ = 90°. This distribution was fitted to two Gaussian distributions resulting in two minima with similar amplitude centred at locations 68° ± 22° and 112° ± 19°. These locations are 22° from the long axis of the squid, showing a muddled search to find an agreement on the true minimum of the deviation angle. The mirrored distribution, following bilateral symmetry, gave a strong minimum (ψ_min = 24°) at θ = 114°, which was much lower than uniform distribution (ψ_min = 34°) (figure 6b). This minimum value is 24° from the long axis of the squid, similar to the 27° measured collagen angle in Lolliguncula breiis and Loligo pealei [33], providing strong evidence that the AoP gradients arise from dichroic-like birefringent structures on the skin that depolarize off-axis polarization states. The AoP discontinuity likely came from a mirrored collagen distribution from the bilateral symmetry of the squid. Also, slight chromatic variations in these collagen structures' birefringence could explain the subtle AoP discontinuity's variation within the polarimeter's colour channels.

4. Discussion

Cephalopod polarization signalling studies have focused on narrow stripes along cuttlefish and squid arms that reflect high DoLP. These groundbreaking studies, both optical and behavioural measurements, were performed in the laboratory. However, as recommended by polarization researchers, taking illumination measurements in the field is paramount for understanding the true nature of polarization signals because of the unique lighting found in the ocean both from collimated sunlight and the side welling polarized background light fields [13]. These polarized arm stripes are very difficult to measure in situ. They are narrow and are on appendages whose shape is poorly defined. Also, there are inherent edge artefacts with the polarimeter used that make studying narrow structures unreliable. We use these polarimeters (division of focal plane polarimeters), however, because they can instantaneously measure an image, essentially making them the only polarimeter that can measure dynamic in situ animals. The mantle of the squid is an ideal body part to quantify polarization patterns in situ due to its rigid shape; however, no previous study has focused on the potential for polarization signals in the mantle. Many visual signals, however, do occur in the mantle [5].

While the AoP patterns of the squid's ventral mantle in this study have low DoLP and are possibly just a byproduct of the squid's anatomy, the polarization patterns are strong enough to be detected by the squid and perhaps this low DoLP signal is advantageous for communication. This mantle polarization pattern is uniformly dichroic with a weak DoLP. Strong uniform dichroic patterns with high DoLP are relatively rare in biological patterning. Strong dichroic signals, from an anthropomorphic standpoint, might seem like an ideal polarization signal. Indeed, there are dichroic filters everywhere in our modern usage, from sunglasses to LCD screens which give clean high DoLP values. With polarization vision so widespread in the animal kingdom, why are there so few examples of strong animal-based dichroic patterns? This is likely not because of any difficulty in biologically producing such structures, because, for example, polarization vision in invertebrates relies on dichroic filters found in the eyes, and mantis shrimp produce dichroic filters with astaxanthin in their antennal scales [31].

There may be several signalling problems with strong dichroic signals in communication. Because the AoP is dependent on the relative angle between the sender and observer the visual sense of the AoP value of the signal may never be consistent enough for effective communication. This may be similar to why strong iridescence is rare in nature. Many animals have evolved to reduce the iridescence inherent in structural coloration so that the colours are vibrant without changing the hue significantly with viewing angle which the underlying structural colour components themselves are generally strongly iridescent [17]. Interestingly the strongest dichroic reflections in nature happen to be circularly polarized reflections in scarab beetles. These reflections can be up to 90% from most reflection angles and the polarization sense does not change with relative sender–observer viewing angle. This would make an ideal polarization signal because the signal is consistent over many viewing positions; however, there is no evidence yet that these beetles use circularly polarized light for communication [34].

The low polarization AoP signal on the ventral part of the squid could be an advantageous signal. The high polarization sensitivity of cephalopods might enable this weaker signal visible for conspecifics and invisible for predators. It is unclear, however, how AoP is sensed by animals, whether it is an actual visual sense or if the AoP is processed by the brain solely for navigation [35,13]. If polarized light is used as an orientation sense in squid for navigation, this polarization signal may give orientation information to squid conspecifics. Reef squids need to school for protection from predators, but schooling is also problematic because squid can be cannibalistic. The directional information from the AoP polarization patterns could give conspecifics additional visual information to avoid the strike zone of another squid.

4.1. Other uses of virtual surfaces

The utility of virtual surface models has been demonstrated in this work for patterning visualization, analysis and surface geometry related calculations. Virtual three-dimensional surface models have an increasing presence for other biological applications. For example, virtual surfaces have been used for mechanical simulations of fossilized objects (e.g. shark teeth [36]) or the hydrodynamics of extinct animals (e.g. plesiosaur necks [37]). Applications in visual ecology have used interactions between animations of three-dimensional virtual models and living organisms that have given insightful behavioural data. For example, virtual fish have been used to create animations to present to live fish for mate choice experiments [38]. These animated fish can display precise patterning and ornamentation differences to investigate visually important characteristics for mate choice. However, such animations cannot yet incorporate iridescence or polarization into the mate choice measurement paradigm. Intriguingly, researchers have created an entirely virtual three-dimensional environment from which ecological questions about vision can be simulated [39]. In the study of Bian et al. [39], animal ornamentation visibility could be measured in this virtual environment from any viewpoint. This process has the potential to include simulations based on an animal's visual system. Furthermore, mapping and tracking measurement paradigms are advancing in technological scope such that natural environments can be converted to a virtual environment. Three-dimensional behaviour data can be incorporated into this virtual environment, and the animal's visual field can be modelled. For investigations with reef squid, a possible future direction would be to extrapolate the polarization patterns found in this paper, along with future measurements, into schooling behaviour data within a virtual environment to investigate pattern visibility within school members under the varying optical conditions of shallow ocean waters.

5. Conclusion

Projecting two-dimensional polarization images onto three-dimensional meshes has many advantages over traditional ROI protocols. Standard ROI procedures require similar amounts of annotation, yet this procedure produced a data container equipped with powerful visualization and analysis tools. Three-dimensional linear ROIs derived from this data container seamlessly mapped data onto Cartesian graphs, concisely visualizing data related to the squid's geometry. This method enabled the quantification of specific regions on the squid mantle with no distinct markings and derived information about the object's and polarimeter's orientation and even environmental illumination. The mapping procedure automatically produced all the parameters needed to perform polarization ray-tracing calculations on the STL mesh and has crossover potential to other optical modelling (e.g. modelling how animals view iridescence in dynamic courtship displays).

From this analysis, we found that the ventral midline and the area between the ventral midline and the flank midlines had the most distinct and repeated polarization signals. When compared with the solar heading, most parts of the squid followed the background ocean's polarization patterns, except for those values close to the ventral midline. When the results were averaged over the solar heading values, the area between the ventral and flank midlines had higher DoLP values dependent on the polarimeter pitch, and the ventral midline gave very low DoLP resulting in a peak in the polarization distance. A large discontinuity of AoP values occurred at the ventral midline, with strong AoP gradients trending to the discontinuity. These AoP trends were unexpected because ray-tracing calculations of the surface-normal geometry gave strikingly different AoP gradients, with no discontinuity at the midline and the gradients trending at opposite directions. The AoP gradients did follow virtual striation lines on the squid's surface with the best fit of 24° from the squid's long axis, which is similar in angle (27°) to the striations of collagen fibres documented in the skin of other squid species [33]. This agreement in angle suggests that the AoP patterns follow the collagen strata on the skin due to collagen birefringence that is random in value but ordered in orientation, depolarizing off-axis polarization contributions.

Data accessibility

The data are deposited through the Texas Data Repository at https://dataverse.tdl.org/dataverse/SquidPatterns/.

Authors' contributions

P.B. designed the study; M.G. and V.G. designed and built the field equipment; M.G. prepared for and designed fieldwork; A.V. took field measurements; P.B., M.G. and S.B. wrote analysis programs; P.B. and T.H. analysed the data; P.B. wrote the manuscript; P.B., M.C., V.G., T.H., M.G., S.B. and A.V. critically edited and contributed to the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the NSF Division of Ocean Sciences grant no. 1636196 to P.C.B., the NSF Division of Ocean Sciences grant no. 1636028 to V.G., and ONR grant no. N00014-19-1-2400 to V.G.