Stereoscopic motion analysis in densely packed clusters: 3D analysis of the shimmering behaviour in Giant honey bees
© Kastberger et al; licensee BioMed Central Ltd. 2011
Received: 23 October 2010
Accepted: 8 February 2011
Published: 8 February 2011
The detailed interpretation of mass phenomena such as human escape panic or swarm behaviour in birds, fish and insects requires detailed analysis of the 3D movements of individual participants. Here, we describe the adaptation of a 3D stereoscopic imaging method to measure the positional coordinates of individual agents in densely packed clusters. The method was applied to study behavioural aspects of shimmering in Giant honeybees, a collective defence behaviour that deters predatory wasps by visual cues, whereby individual bees flip their abdomen upwards in a split second, producing Mexican wave-like patterns.
Stereoscopic imaging provided non-invasive, automated, simultaneous, in-situ 3D measurements of hundreds of bees on the nest surface regarding their thoracic position and orientation of the body length axis. Segmentation was the basis for the stereo matching, which defined correspondences of individual bees in pairs of stereo images. Stereo-matched "agent bees" were re-identified in subsequent frames by the tracking procedure and triangulated into real-world coordinates. These algorithms were required to calculate the three spatial motion components (dx: horizontal, dy: vertical and dz: towards and from the comb) of individual bees over time.
The method enables the assessment of the 3D positions of individual Giant honeybees, which is not possible with single-view cameras. The method can be applied to distinguish at the individual bee level active movements of the thoraces produced by abdominal flipping from passive motions generated by the moving bee curtain. The data provide evidence that the z-deflections of thoraces are potential cues for colony-intrinsic communication. The method helps to understand the phenomenon of collective decision-making through mechanoceptive synchronization and to associate shimmering with the principles of wave propagation. With further, minor modifications, the method could be used to study aspects of other mass phenomena that involve active and passive movements of individual agents in densely packed clusters.
Additional file 1: Shimmering behaviour in Giant honeybees. An experimental Giant honeybee nest (140 × 70 cm) attached to a balcony of a hotel in Chitwan, Nepal. On the right, parts of the comb had been removed by a honey hunter some weeks before. With the exception of the mouth zone  (bottom right), the bee curtain exhibited a quiescent structure of surface bees, with their heads up and the abdomens down. The movie was recorded with a HD camera with a frame rate of 50 Hz in parallel to the stereo cameras (see Figure 2 for experimental design). At the beginning of the sequence the trigger light (fixed by black adhesive tapes to the wall behind) was turned on for one second. The red spot (middle of the right side of the nest) was produced by the beam of a Laser vibrometer measuring the thoracic z-position of the selected surface bee. Right to the nest, a black-and-white striped dummy wasp was mounted on the cable-car device, and its moving speed and direction were computer-controlled. The dummy wasp provoked shimmering waves before it is seen in the image. Note that the waves originated at the right nest side above the mouth zone. The yellow number on the right bottom gives the time in seconds. The first two waves of this sequence refer to the episodes w1, w2 of Figure 6 (2.1 MB, MPG). (MPG 2 MB)
Challenges arose, firstly, from the requirement to individually track identified surface bees, hereafter termed "agent bees", in successive frames (ff) of a shimmering process, which is difficult because all agent bees are extremely similar in morphology, are densely clustered, and show rapid movements in 3D during their abdominal flipping. Secondly, an individual bee sensing an incoming wave front due to the movements of her neighbours is free to decide whether or not to participate, and if so, whether to participate strongly or weakly. It is critical to distinguish active "movements", i.e. abdominal flipping, from passive "motions" caused by the surrounding bee curtain. Individuals that participate weakly in shimmering are difficult to detect by automated analysis. Thirdly, data of the positional coordinates of agents are, for the external observer, stochastic and noisy. This is because collective behaviours in eusocial insects are determined by self-organization [13, 24–26], whereby patterns at a global level of a system emerge from numerous interactions among the lower-level components of the system, and rules specifying interactions among the systems' components are executed using only local information, without reference to the global pattern . Hence, the data describing the behaviour of the lower-level components (here: the agent bees) appear stochastic and noisy, although the global effect (here: shimmering) is clearly recognisable. Fourthly, high-resolution and high-speed cameras are essential to produce rich data sets with high geometrical resolution, detecting movements of single agents within fractions of a millimetre across an entire nest that can span up to 1.5 m in diameter. The equipment must deliver reliable, accurate data, even under harsh field conditions in the natural habitats of Giant honeybees where electronic equipment may fail due to high temperature, air humidity and solar irradiance. Lastly, Giant honeybees are among the most aggressive insects known . To avoid unwanted colony arousal, data recording must be non-invasive, keeping a distance of at least 2 m between equipment and nest.
We propose a method for the automated identification of individual bees within the densely packed clusters of the bee curtain surface, including the computation of the 3D locations of individual agent bees, following them over time and identifying the arrival of shimmering waves at those agents. The data obtained with this method can enhance our understanding of the generation and propagation of shimmering waves in Giant honeybees and of the contribution of nest members to intra-colonial communication.
Additional file 2: Illustration of the triangulation of a selected agent by stereo tracking. The upper row shows image sequences of the left and right camera regarding the wave episodes w1 and w2 of agent 58 (for information about cut size and location, see Figure 6 and 7; for further details about z-movement and abdomen flipping, see Figure 7 and 9); the red crosses in the images mark the thorax of agent 58. The bottom bar chart gives the z-values of the thoracic position of agent 58 in mm; positive values refer to directions away from the comb, negative directions towards the comb. The moving red line in the graph marks the momentary time position. Note that the waves in the episodes w1 and w2 spread from right to left, and that the participation in shimmering of the neighbours of agent 58 varied strongly (1.9 MB, MPG). (MPG 2 MB)
Additional file 6: Stereo imaging of shimmering. Detail of the experimental Giant honeybee nest (cf. to Additional File 1, Movie S1). For comparison, the image sequence refers to the same 15 s displaying the shimmering waves shown in Additional File 1, Movie S1. Left and right images of the black-and-white high-speed stereo cameras with frame and time information; images are displayed in inverted luminance values. Note that due to the perspective only the left camera shows the cable car dummy (2.5 MB, MPG). (MPG 3 MB)
Additional file 12: Difference image sequences. Detail of the experimental Giant honeybee nest (cf. Additional File 1, Movie S1 and Additional File 6, Movie S5). For comparison, the image sequence refers to the same 15 s displaying the shimmering waves shown in Additional File 6, Movie S5. Left, left-camera image; right, difference image giving the subtraction value of Δlumi,i-1 = lumi - lumi-1 between the actual frame f i and the preceding frame f i-1; numbers refer to frame and time (in ms) information; the left image shows inverted luminance values (2.5 MB, MPG). (MPG 3 MB)
Additional file 4: Visualization of the differential of z-movements of all selected agents by stereo tracking during a single wave episode. This film shows the differential of the positional information for all 505 agents (cf. Figure 6c,d,f). For all other details, see legend to Additional File 3, Movie S3 (0.2 MB, MPG). (MPG 193 KB)
Technical Section: Algorithm and Implementation
At the experimental nest, the individual and collective motions of surface bees were captured in a stereoscopic video sequence (Figure 2). Pairs of synchronous, frame-locked images recorded from different viewpoints (" stereo images") were obtained by two digital cameras (for technical specifications, see Additional File 5, Table S1) placed symmetrically ~2 m in front of the nest with a baseline of 1 m between the cameras. The cameras were mounted on a solid carrier and connected to a computer. To elicit shimmering waves, a striped dummy wasp made from Styrofoam was fastened with fine yarn to a cable-car device for computer-controlled movements at variable velocity (0.1-0.5 m/s) and direction (Figure 2). In the experiments described here, the dummy wasp was moved at an angle of 90° to the nest surface (shown on the right side of the experimental nest in the images, see Additional File 1, Movie S1 and Additional File 6, Movie S5).
with e img as the pixel error of stereo matching; m, the magnification at the given working distance. Under the given conditions (α = 30°, m = 0.3 mm/px; eimg = 1 px; equation 2) a resulting depth error of e z = 0.6 mm was achieved. In summary, stereo images were typically recorded over 15 s, capturing 900 ff per camera at a rate of 60 Hz and at a spatial resolution of 0.3 mm per px within a measurement volume of 700 × 500 × 150 mm3 (x,y,z) whereby the nest-specific axes were defined as x (= horizontal: left-right), y (= vertical: up-down) and z (= directions towards and away from the comb).
Here, is the mean of t, and is the mean of f in the template.
The goal was to identify individual bees at the surface of a densely structured matrix in a multi-layered nest (Figures 1 and 2). For that, template patterns were defined from representative agent bees with varying orientations and scales (Figure 4a); such templates included the abdomen and the thorax rather than the head which is often concealed by other surface bees. Each template type was matched against every position (u,v) in the image. Local similarity maxima, which were obtained by non-maximum suppression, represented successful matches. To compensate for overlaps, weak local maxima in the vicinity of 20 px around a more dominant one were eliminated. Consecutively, the segmentation routine was repeated with scaled and rotated templates to account for variations between the individual prototypes in alignment and size. The charts (Figure 4b,c) exemplify template matching using five templates with a variation in orientation of 23° (which represented a typical measure of the maximum deviation angle in the quiescent areas of the nest), and with a scale tolerance of 14% (corresponding to 9 px). The procedure recognized hundreds of surface bees in the bee curtain (Figure 2c and 4c) under quiescent conditions when they were inside the orientation limits and were not overlapped by other bees.
Stereo matching allowed us to compare the template region of an agent in the left image with the paired, frame-synchronized right image, identifying the best match for the correct correspondence. This process could have followed the same strategy as that in segmentation. However, the high degree in pattern similarity of neighbours and the differences in perspective made it difficult to assign unique correspondences. Figure 4d,e illustrates the challenges arising in stereo matching exemplifying rectified honeybee templates: the agent segmented in the left stereo image coded by a yellow frame was identified in the paired right images by the similarity criterion and selected out of nine candidates; finally, the template matching was successful for the agent bee in the red rectangle.
Additional file 9: Illustration of abdominal flipping. Illustration of abdominal flipping of bees in the neighbourhood of agent 58 (marked by a red full circle) during the wave episode w2 (see Figures 6, 7 and 9). Footage in frames and time in ms are displayed (frame rate: 60 Hz). For all other details, see legend to Additional File 3, Movie S3 (0.5 MB, MPG). (MPG 520 KB)
Additional file 10: Slow motion of abdominal flipping. Slow motion of abdominal flipping of bees in the neighbourhood of agent 58 (marked by a red full circle) during the wave episode w2 (see Figure 6,7 and 9). Footage in frames and time in ms are displayed (factor slow motion: 10.6). For all other details, see legend to Additional File 12, Movie S9 (2.2 MB, MPG). (MPG 2 MB)
Additional file 11: Mechanistic model of an abdomen-flipping Giant honeybee at the surface of a nest. Model, explaining the movements of the body associated with abdominal flipping according to the y- and z-curves in Figure 9 (not considering the x-movements). The left brown vertical bar denotes the two-sided comb with cells on both sides separated by a mid wall. The two short brown vertical bars between the model bee and the model comb represent the subsurface layers of the bee curtain to which the model bee clings with her six extremities (not shown). In this model, the position of head and thorax, and the distance between thorax and the nearest brown bar was kept constant throughout the flipping process. In the film, the abdominal flipping is simplified: the model bee raises the abdomen by 90°, which provokes y-movements (here displayed in vertical directions) and z-movements (here displayed in horizontal directions). Two phases of the flipping are illustrated. In the initial phase the model bee is pressed towards the subsurface layers, when she actively pulls her body upwards using her extremities, and in the second phase, she moves away from the comb and recovers her initial lower position. In this sketch, the changing interspaces between comb and the subsurface layers additionally illustrate the pressing to and moving away from the comb of the model bee. Numbers indicate frames and time in ms; the sketch slows down real-time abdominal flipping by a factor of 10.78 (0.5 MB, MPG). (MPG 515 KB)
Despite the introduction of the symmetry criterion in the tracking procedure up to 16% of the 400-600 identified agents were "lost" in the course of multiple waves. Nevertheless, we were able to compensate for these losses by restarting segmentation and stereo matching after each wave to retain stable numbers of agents throughout the evaluation path. We also minimized standard errors by the offset-correction of the positional coordinates of each agent. The arithmetic mean of the first six of the initial 30 frames prior to the arrival of a wave was subtracted from all data in a 90-frames episode (1500 ms). Offset correction also compensated for the residual spatial deflections with a time constant of 2-3 s which was characteristic for the descending process after a shimmering wave process caused by the receding motion of the bee curtain exemplified in Figure 5a-b and in the Additional File 3. Movie S3 shows the original data before offset-correction and prove that 3D stereoscopic imaging delivers the positional data without high-pass filter effects.
The term [h]× here denotes the 3 × 3 cross matrix of a three-dimensional vector h (see ). A least-squares solution to the homogeneous equation system was obtained by solving the linear equation system. To maintain accuracy under the given field conditions, the stereo imaging system was calibrated daily using a standard calibration method  to maintain geometric consistency.
Detecting the arrival of a wave at individual agents
Application Section: Behavioural Studies
To demonstrate the applicability of the method we focused on selected behavioural aspects regarding shimmering waves. Here, data of only a single spreading direction were analysed for waves seemingly spreading from the right side to the left side of the experimental nest (nwaves = 14; nepisodes = 4672; nagents = 605; when a shimmering wave hits an individual agent bee, a shimmering "episode" can be defined which includes the active and passive movements of this agent bee shortly before, during and after the arrival of the wave front). For all these behavioural examples, the statistical proof is based on synchronizing shimmering episodes by detecting the arrival of the wave at the immediate neighbourhood of the agent bees and on stereo tracking thoracic positions. In the first subchapter the primary motion components of selected agent bees are identified (Figure 5, 6, 7 and 8, see Additional File 2, Movie S2) which could be potential candidates to enforce the mechanical process of wave generation [31–33]. Second, for the same agents their (passively induced) motions were traced hundreds of milliseconds before the arrival of a wave, that is when the decision has to be made if and when to participate (Figure 8i-l) in a wave. Finally, we classified the modes of participation on the single agent level (Figure 6, 7 and 8d).
Identifying primary motion components of agent bees during shimmering
A shimmering wave arriving at an agent provokes characteristic positional changes (Figures 5, 6, 7 and 8, see Additional File 2, Movie S2). Regarding the y-components, 70 ms after wave arrival, as defined by the detection method described above (black arrows in Figure 8l), a strong upward movement occurred (blue arrows in Figure 8j); 20 ms later, the agent bee started to move in the z-direction, first she shifted slightly negatively, i.e. towards the comb, then strongly away from the comb (red arrows in Figure 8k); this biphasic z-motion is displayed more explicitly in single-agent data (cf. Figures 6 and 7e 2). It is important to differentiate between the weak, positive transient deflections which resulted from the oncoming wave due to the general swinging of the bee curtain in the vicinity of the agent bee, and a strong, positive z-deflection which indicated "active" participation in the wave through abdominal flipping. With higher response strengths (RS values) the positive y-deflections started earlier (Figure 8j) and the positive z-motions later (Figure 8k) due to a more pronounced preceding negative phase of the z-component (cf. Figure 6). The horizontal components were the weakest of all, and the data illustrate that the positive x-components corresponded strongly with the positive z-components (r = 0.39; P < 0.001; n = 4782 wave episodes at agent bees; Pearson Product Moment Correlation Test); this happened at lower RS values at a lower intensity and with increasing time lags (Figure 8h,i,k). The deflections regarding x-, y- and z-directions seem to be very complex because they are caused by the specific attributes of the functional architecture of the bee curtain regarding an agents orientation (head up and abdomen downwards) and the coupling (physical contact) between the colony members of the different layers [8, 13, 16, 17]. More details are given in the discussion part.
Motion detection at an agent bee prior to the arrival of a wave
To produce the typical cascadic shimmering waves that are visually recognized by external addressees , each surface bee needs to respond to an oncoming wave within a time window of less than 100 ms. For that, she has to decide whether or not to flip the abdomen, and if she does, she can raise her abdomen with gradual strength. Her decision is particularly linked to mechanical cues sensed tens of milliseconds prior to the arrival of the wave front. The possibility that movements of her immediate neighbours are also perceived visually cannot be excluded, but it would probably take too long to synchronize her abdominal flipping with the wave front only by visual input; the duration of senso-neural processes associated with image perception may last more than 150 ms  to prompt behavioural decisions [13, 16, 35]. However, the visual domain is important, in particular to perceive threatening stimuli such as predatory wasps hovering around the nest. Such visual patterns are key to initialize shimmering [13, 16] and to collectively drive its spreading direction . Mechanical cues, although less understood [13, 16, 35], must be essential for the wave spreading process. Surface bees are well endowed to sense mechanical cues, as they cling on to bees of the subsurface layer with their six legs, acting as a potent web of mechanoreceptors. In this paper, 3D stereoscopic imaging provided evidence for mechanoceptive cues, which were traced at individual agents in terms of small increments up to 300 ms before the wave front arrived (Figure 5e and 8i-k). Statistically proven for the investigated scenario of waves with high RS values (see Figure 7a for the z-component, test not shown here) spreading from the right to the left side, a wave typically pressed the agent bees slightly (< 0.2 mm) towards the nest, then shifted them into the direction of the oncoming wave, and lastly slightly downwards. However, recordings of agent 58 (Figure 5 and 6) illustrate in two successive episodes that the sensing of incoming waves could be even more complex (see Figure 5e, episode w1).
Participation of individual agent bees in shimmering
Applications: Physical Principles of Abdominal Flipping
We also explored the physics of abdominal flipping at the single bee level (Figure 6 and 7, see Additional Files 9, 10 and 11, Movies S6, S7 and S8) and the potential accompanying mechanical cues in colony-intrinsic communication . The thoracic z-motions during shimmering exhibit a prominent initial peak (Figure 8c) due to the abdominal thrusting. This is exemplified for agent 58, where we manually linked the thoracic position and the angular movement of abdominal flipping (Figure 6). Initially (ff rel = 1-6, defined as ff 15-20 in Figure 6a), the thorax moved progressively towards the comb by 1 mm in 100 ms, but then (ff rel = 7-9) it was strongly pushed away from the comb (see Additional File 2, Movie S2 and Additional Files 9, 10 and 11, Movies S6, S7 and S8) with a six-fold greater velocity.
Giant honeybees in the peripheral and outer layers of the bee curtain deposit a lot of watery honey in their abdomens , which become much heavier than the counterbalancing heads. Therefore, an abdomen-flipping bee may be simplified, in terms of classic mechanics, as a torsion pendulum (see Additional File 11, Movie S8) with an a-centric axis (positioned between thorax and the tarsi of the six legs which fix the thorax to the movable subsurface) and with two disparate masses (the upwards head and the downwards abdomen) on both sides of the rotation centre, oscillating in a (harmonic but damped) curved motion. During abdominal flipping mass inertia produces a reaction force, which initially presses the agent with the thorax towards the nest, but consecutively, the thrust of the flail-type body with the asymmetry of the pendular masses initiates a centrifugal force directed away from the comb. In the collective process of shimmering the centrifugal forces of the surface bees are locally synchronized, which lastly pulls the subsurface layers of the bee curtain concertedly away from the comb (see Additional Files 9, 10 and 11, Movies S6, S7 and S8). Thus, the centrifugal z-component in the second phase of abdominal flipping mainly drives the shimmering process.
Limits to image acquisition within the given stereoscopic approach
The above application was mainly limited by geometrical and technical constraints. Studying active movements and passive motions at a Giant honeybee nest, such as in shimmering behaviour, needs a thorough 3D analysis with the accuracy of a fraction of a millimetre. We aimed to measure the positional deflections at a resolution of 0.1 mm in all directions of space of hundreds of agent bees simultaneously. This required a stereo baseline of 1 m, which made the stereo system rather large and sometimes difficult to handle under challenging field conditions. To maintain adequate calibration, the cameras were mounted on an aluminium girder that fixed the position of the cameras to each other and enabled movements of the two cameras in a fixed position in front of the bee colony if required. However, we had to compromise the temporal resolution with the illumination conditions at the scene. A nest may be positioned in a dark corner of a veranda or in the bright sun on a tree. Longer exposure times will produce motion blur during fast movements as occur during shimmering. Varying illumination at the nest site was compensated for by locally applied algorithms such as normalized similarity measures. We avoided excessive sunlight, which would prevent reliable measurement by overexposing of nest regions. To satisfy our needs in terms of accuracy and measurement time, two frame-synchronized cameras (see specifications in Additional File 5, Table S1) with a resolution of 4 Mpx were used, with fixed lenses (without motorized zoom, focus or iris) to provide calibration stability, and with CMOS image sensors that provide flexible capturing of images of different size at different frame rates (ranging from 60 to more than 500 fps). Lastly, one of the main problems was the enormous data rate produced by the cameras during the experiments. At a frame rate of 60 images with 4 Mpx recorded by each camera, an uncompressed data flow of about 480 MB/s had to be managed. Due to the limitation of electric power in the field during our expedition, it was not possible to use hard disks to capture the images in parallel. Therefore, the data had to be stored in the RAM of the computer and written to hard disks later on, constraining acquisition time to RAM storage capacity (in our case: 8 GB RAM, 15 s acquisition time).
Simultaneous recording of hundreds of agents by stereoscopic imaging
Existing optical tracking methods [36, 37] mostly record single views (e.g. in CCTV cameras) of isolated agents in low quantities; if large quantities of agents are viewed, such as hundreds or thousands, motion priors are usually derived from the composite movement of "crowds" [30, 37]. These single-view methods are inappropriate for mass phenomena such as shimmering in Giant honeybees, where the precise positional 3D coordinates of individual agents need to be known independently of the global motion. We chose to use 3D stereoscopic imaging, which allows motion analysis of densely packed agents in all directions of space. For this, we developed a system with stabilized tracking performance and resolved ambiguities, which identifies the 3D movements of hundreds of agents simultaneously. So far, shimmering has been documented on traditional film [5, 38] in conjunction with classical image analysis, providing only 2D projections of individual bees participating in a 3D process. Laser Doppler vibrometry (LDV)  can be useful for some applications, as it also facilitates 3D information. However, it has the disadvantage of delivering data of only one single agent bee over time, and can only pick up transient changes as it is limited by high-pass filter effects.
Here, the stereo tracking method has been modified to measure local, comparably small movements of hundreds of densely packed agents simultaneously, in a flat, but nevertheless three-dimensional stratum. The method is applicable not only to insect clusters such as Giant honeybee nests, but also to processes where positional changes over time have to be monitored on the surface of a stratum of agents with high packing density (such as the growth of plants or cells in 3D, or for displaying deformations of any type of textured surface). For the application of stereoscopic imaging to monitor the dynamics of flocks of birds or fish schools, the methodological approach, in particular the tracking rules, must be adapted.
Perspectives of analysing shimmering stereoscopically
For the shimmering behaviour of Giant honeybees, the described stereoscopic method produced data that provide behavioural details that could not have been measured by other techniques. It provided evidence for a series of potential mechanoceptive cues perceived by hundreds of surface bees on arrival of a shimmering wave. For example, at the beginning of the experiment, agent 58 drifted slightly away from the comb (< 0.3 mm), starting 300 ms before it strongly participated in the shimmering wave (Figure 5e: episode w1, see the positive deflection at ff 102-115). These "pre-trigger" z-motions of the agents are apparently caused by the abdomen-flipping neighbours. Although small, such passive motions may enable the surface bees to estimate strength and spreading direction of an oncoming wave, and may function to trigger their active participation in shimmering. This response to the wave front is complex: first, the bodies were lifted upwards and pressed towards the comb, and also shifted towards the spreading wave front. Thereafter, the actively shimmering, abdomen-lifting agents pull the subsurface layers massively away from the comb. We propose that the resulting biphasic time course of the z-motion is a subtle colony-intrinsic signal [13, 16, 35] that provides mechanoceptive information about the momentary defensive state of the colony regarding temporal, spatial and directional patterns. The addressees of such signals are those colony members that are positioned in the subsurface layers and inactive in shimmering, and that are excluded from receiving visual cues.
Hence, 3D analysis of singular aspects of shimmering, such as the thorax positions of surface bees, enhances the understanding of the mechanical basis of abdominal flipping at the single bee level, and of the sensory basis for colony-intrinsic information involved in the spreading of shimmering waves. On the individual bee level, the stereoscopic method enabled a comparison of the mechanics of abdominal flipping with that of a torsion pendulum. On the colony level, it allowed an assessment of the mechanic basics of shimmering to explore potential cues for colony-intrinsic communication. In our example, the wave arrived from the right side and drew the agents against the spreading direction. The x-motions of the agents detected can be plausibly explained by the mechanical principle of shear forces tangential to the individual agents. This is analogous to wind waves that propagate along the interface between water and air [31–33]; as the wind blows, pressure and friction forces perturb the equilibrium of the water surface, producing waves in which the moving paths of particles near the water surface form circles (regarding monochromatic, linear, and plane waves in deep water [31–33]). Wind waves are therefore a combination of longitudinal (back and forth) and transverse (up and down) motions [23, 32, 33]. The positive x-motions of agents in horizontally spreading shimmering waves are supposed to correspond to the longitudinal moving of water particles [23, 32, 33] when directed "backwards".
However, the analogy of shimmering waves with wind waves is limited for at least two reasons. First, wind waves are generated  by energy transferred onto the water whereas shimmering is produced by the "active" agents in the bee curtain themselves. Second, in wind waves the restoring force is gravity , but in shimmering waves it is the concerted muscular activity of the bees in the layers of the bee curtain acting rectangular to gravity. The positive z-motion in shimmering denotes that abdominal flipping has recruited energy into the previously quiescent parts of the bee curtain, pulling its elastic mass away from the comb with successive damped oscillations (Figure 5e; see Additional File 2, Movie S2 and Additional Files 9, 10 and 11, Movies S6, S7 and S8). This finding questions the analogy to Mexican waves  in football stadiums: Although both shimmering and Mexican waves are generated by the energy of the participating agents, the audience in football stadiums does not lift the ground into the air, whereas the wave front during the shimmering process pushes the surface of the bee curtain to the outside direction.
So far, stereo imaging has been applied to remote sensing , close-range photogrammetry , material sciences [42, 43], medicine [44, 45] and tissue mechanics . Although this method can be scaled to a multitude of measurement ranges and corresponding accuracy specifications, to date 3D stereoscopic imaging has not been used for ethological purposes, particularly not for the descriptions of 3D motions of densely packed agents. We propose that it is a valuable tool to study collective behaviours in Giant honeybee colonies, but also, subject to additional adaptations of the method, for swarm behaviours in other insects [47, 48], fish [49, 50] and birds [51–54] and for special aspects of escape panic in humans .
We thank Dr. Madhusudan Man Singh from the Tribhuvan University, Kirtipur, Kathmandu, Nepal and three anonymous referees for comments that helped improving the manuscript. Financial grant was provided by the Austrian Science Fund (P 20515-B16).
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