Conceived and designed the experiments: YY PS MOF HUS. Performed the experiments: YY PS. Analyzed the data: YY. Contributed reagents/materials/analysis tools: YY PS MOF AB. Wrote the paper: YY.
The authors have declared that no competing interests exist.
A critical step on the way to understanding a sensory system is the analysis of the input it receives. In this work we examine the statistics of natural complex echoes, focusing on vegetation echoes. Vegetation echoes constitute a major part of the sensory world of more than 800 species of echolocating bats and play an important role in several of their daily tasks. Our statistical analysis is based on a large collection of plant echoes acquired by a biomimetic sonar system. We explore the relation between the physical world (the structure of the plant) and the characteristics of its echo. Finally, we complete the story by analyzing the effect of the sensory processing of both the echolocation and the auditory systems on the echoes and interpret them in the light of information maximization. The echoes of all different plant species we examined share a surprisingly robust pattern that was also reproduced by a simple Poisson model of the spatial reflector arrangement. The fine differences observed between the echoes of different plant species can be explained by the spatial characteristics of the plants. The bat's emitted signal enhances the most informative spatial frequency range where the species-specific information is large. The auditory system filtering affects the echoes in a similar way, thus enhancing the most informative spatial frequency range even more. These findings suggest how the bat's sensory system could have evolved to deal with complex natural echoes.
More than 800 species of bats perceive their surroundings through echolocation. They emit ultrasonic pulses and analyze the information conveyed in the echoes returning from objects in their surroundings. This enables bats to orient in space, to acquire food and to perfectly function in complete darkness. In the absence of light, echoes constitute a major part of the sensory world of bats. Understanding their characteristics can thus help to shed light on the echolocation sensory system. The goal of this work is to study the characteristics of natural plant echoes. Plant echoes are very abundant in the world of the bats and are used by them to find food sources and to navigate. We show that some of the features of the echoes can be explained from the physical properties of the plant they were sampled from (e.g., its leaf size and density). We then analyze the effects of the sensory system on the echoes and suggest that they improve the representation of the echoes in a way that enhances the information that is most reliable for the bats.
The understanding of natural stimuli is critical for understanding the sensory systems that evolved to process them. More than 800 species of echolocating bats continuously emit echolocation signals and analyze the returning echoes to perceive their surroundings
Although vegetation echoes, on which we focus in this work, are among the most common echoes bats constantly encounter, they have been scarcely studied
Since plants have complex shapes that cannot be described in terms of simple geometrical primitives vegetation echoes exhibit highly complex temporal patterns
The filtering characteristics of a reflecting object are known as its frequency response and mainly depend on its acoustical cross-section. In a plant covered with leaves, the leaves will reflect most of the energy emitted by the bats
The spatial arrangement of the reflectors relative to the receiver determines the temporal statistics of the echoes. The spatial arrangement of the branches also plays an important role since it defines the arrangement of the leaves. The analysis of the temporal statistics of the echoes is not as straightforward as that of the spectral statistics. There are many possible parameters that represent the temporal statistics. We chose to use parameters that are strongly related to the spatial distribution of the reflectors, i.e., to the structure of the plant. Additionally, in order to be comparable to the findings in natural images we use a similar method to the one used to characterize them
We base our analysis on the envelope of the impulse response [IR, see
A) The normalized echo train returning when ensonifying an artificial branch with four identical plastic leaved twigs 15 cm apart from a flat horizontal angle (see illustration above A. a.u – arbitrary units. B) The normalized IR of the echo train presented in A. C) The normalized envelope of B. Note the clear correspondence to the structure of the plant. The envelope presents three main peaks ∼0.9 ms apart corresponding to the first three twigs. The echo from the fourth twig is much weaker due to occlusion. Additional smaller peaks follow each of the main ones and correspond to the leaves on the twigs. D) The PSD of C shows a main peak around 14 cm, as expected, with following peaks at the higher harmonics at, e.g., 7, 3.5 and 1.8 cm. One peak however, at ∼5 cm (marked by an asterisk) deviates from this sequence and is probably a result of the periodic structure created by the leaves on the twigs. The energy (Y-axis) is normalized to a sum of one.
The basic assumption of our analysis is that the energy of a certain frequency in the PSD correlates with the amount of reflectors in periodic structures with corresponding spatial distances between them (
When averaging the spectra of a plant from many different orientations, particular distances that are very common in the plant such as distances between twigs or larger branches will create broad peaks of energy around the corresponding frequency in the spectra. These distances are common from any direction the plant is observed. Uncommon distances, however, will affect the energies in all frequencies and will be negligible in relation to the common ones. If no such common distances exist we should receive a flat spectrum. Naturally there could be periodic structures that are very salient from a single orientation, but these clearly do not constitute a reliable feature for characterizing a plant species.
Throughout the paper we shall refer to spatial frequencies according to the wavelength of the corresponding periodic structure of reflectors, which we will refer to as the
In an attempt to understand the vegetation properties that determine the temporal statistics of the echoes, we developed a computer model of the physical processes that underlie their creation. The model first produces a three-dimensional distribution of reflectors. Next, an echo is created as a superposition of the echoes returning from all of the reflectors with delays corresponding to their distance from the receiver (R), and with energies attenuated according to the geometric attenuation of point reflectors (1/R4). Modeling reflectors with a different cross-section would lead to the same temporal statistics but to a different frequency response.
Assuming that the spatial arrangement of the reflectors has the most important effect on the temporal statistics, we started with a simple model that only considers the reflectors distribution and does not take into account other possible factors such as leaf shape, orientation, occlusion etc.
The uniform Poisson model is the simplest reflector distribution we examined. It assumes that the reflectors are evenly distributed everywhere in the plant according to a Poisson distribution. Thus, if
Higher-order models can be created in a sequential fashion by using the reflectors of the previous order as centers for smaller Gaussian clouds. These models can be intuitively thought of as a representation of the arrangement of reflectors in a plant, starting from the gross skeleton, bifurcating several times and finally ending at the leaves.
The average normalized spectra of the echoes for each of the investigated four species is shown in
The spectra were first divided by the emitted signal spectrum to calculate the relative reflected energy at each frequency. The result was then normalized to have a sum of one.
* Natural plants: In the second analysis we examine the normalized spectra of the envelopes of the IRs. When presenting the averaged spectra on a doubly logarithmic plot, a surprisingly robust pattern emerges: all of the plant species we examined have an overall inverted sigmoid shape consisting of three approximately constant-slope domains in a spatial scale range corresponding to 1.7–50 cm, (
Spectra were normalized to have a sum of one.
* Plastic model: To verify our basic assumption that a peak in the spectrum of the IR envelope corresponds to a common inter-reflector distance, we examined the echoes from a plastic plant model consisting of four broad-leaved twigs in a row perpendicular to the impinging sound. The peak of energy in the mean spectrum of the plastic branch echo envelope over a variety of ensonification angles indeed corresponds to the distance between the four twigs (
Each branch was composed of a row of 4 artificial plastic twigs which were 8, or 15 or 20 cm apart. Notice the energy peaks around 8, 15 and 20 cm in the three graphs (marked with asterisks). Spectra were normalized to have a sum of one. We present the spectra on a linear scale here in order to emphasize the peaks of energy corresponding to the distance between twigs.
* Leaf density: To confirm that the differences between the sigmoid curves can be related to the complex spatial structure of the species, we examined the influence of the leaf density on the spectra of a
A) Mean normalized spectra of the envelopes of the echo trains of a
McKerrow et al.
To study the effect of the emitted signal we generated artificial random IRs with a uniform distribution of distances (in the range 0–1 m) between the glints. Such an IR represents a plant that has an equal number of reflectors in all distances from any point of ensonification. These IRs were then convolved either with the emitted bat-like signal described above or with an artificial signal with a spectro-temporal structure that is identical to the emitted one, but with a constant energy in all frequencies (we call this the ideal bat sweep). We then ran the artificial echoes created by this process through the same analysis as the natural ones (
Normalized spectra of the artificial IR-envelopes simulated with two different emitted signals aside the real spectrum of beech. Spectra were normalized to have a sum of one.
The temporal analysis presented so far was based on the envelope of the impulse response of the echoes. In order to have access to this information bats would have to be able to use a receiver that is equivalent to a non-coherent matched filter, yet such ability was never shown. In order to test the effect of the filtering of early stages in the auditory system on the statistics, we applied the commonly used auditory system model (see
A–C) Normalized echo spectra of two species (apple and spruce) after auditory system filtering compared to the spectra of the same species with no filtering. The system response is represented by the output of three channels with different center frequencies (Fc) representing the entire frequency range. (A)
The effects of biological filtering are quite salient: it increases the spectral energy at the large scales and attenuates the energy at the small scales in all filter-bank channels. With the cutoff frequency of the leaky integrator used by us (8 kHz) the energy becomes very small (less than 1% of the total energy) for scales smaller than ca. 5 cm. A lower cutoff frequency, as suggested by some authors
We tested different spatial distributions of point reflectors and compared the temporal statistics of the created echoes to those of the real data. We used the Kullback-Leibler (KL) divergence as a measure for goodness of fit. The smaller the KL divergence is, the more similar the distribution of the artificial data is to the observed one.
The inverted sigmoid shape that characterizes the natural spectra could be reproduced by the simple uniform model that treats a plant as a 3D array of point reflectors distributed with a characteristic distance between them (
A) Mean normalized power spectra of blackthorn and the best fitting uniform model for all scales (d1 = 16 cm). The shaded area depicts the standard deviation predicted by the model. B) A sample from the uniform 3D Poisson distribution of reflectors with a distance of d = 16 cm. C) Same as A, for the best fitting second-order model (d1 = 40 cm, d2 = 16 cm, r2 = 50 cm). D) A sample from the 3D distribution of reflectors created by the second-order model with the same parameters as in C.
Plant Species | Spruce | Blackthorn | Apple | Beech |
d1 (cm) | 20 | 20 | 16 | 20 |
KL divergence | 1.5 | 1.4 | 2.0 | 1.5 |
The minimum KL values and the corresponding distance parameter (d1) for the uniform model when fitting all scales or only large scales.
Plant Species | Spruce | Blackthorn | Apple | Beech |
d1 (cm) | 40 | 40 | 40 | 40 |
d2 (cm) | 4 | 4 | 8 | 4 |
r2 (cm) | 75 | 75 | 50 | 75 |
KL divergence | 0.9 | 1.2 | 2.0 | 1.4 |
The minimum KL values and the corresponding parameters (d1, d2, r2) for the second-order model when fitting all scales or only large scales.
The spectral echo statistics of the plant species we examined behaved as expected from the cross sections of their leaves (see
The temporal statistics of complex echoes of different plant species show both robust similarities as well as interpretable differences between them. On a doubly logarithmic plot the averaged power spectra of the IR envelopes of all plant species we examined had an inverted sigmoid curve (
The spectrum of a single plant echo envelope is much noisier than the average sigmoid curves presented in the results (
When classifying a large object such as a tree while flying past it, a bat could collect several echoes of the object taking into account the time needed to pass the object and an average time interval between emitted calls. Although it has been shown both theoretically
The analysis of an artificial IR with an equal number of reflectors in all distances suggests that the logarithmic decrease of energy at the small scales (<3 cm) mainly results from the convex energy distribution of our emitted signal. This bias results from the partial ability to deconvolve an echo with a signal of limited bandwidth which acts as a low-pass filter. A larger bandwidth, such as in the ideal bat sweep we simulated, reduces this bias. The differences between the first two domains (50–3 cm) however might still be attributed to the different domains of scale in the plant (e.g, distances between branches vs. distances between leaves). This implies that most of the information that is relevant for the bat in the echo, for instance for classification, is created by large scale structures.
FM Bats will also be subject to a bias of this sort since their down-sweep is never the ideal one. The matched filter that we used for this analysis is optimal to resolve temporally adjacent glints. Thus, even if bats do not perform a direct deconvolution of the echo (i.e., using a coherent or non-coherent matched filter) the signal they receive will be biased towards large scale structures. The extent of the bias will depend on the bandwidth of the calls they emit. Some bats species can manipulate the band-width of their calls by up to several tens of percents. In addition different bat species emit calls with different spectral modulations that might enhance or attenuate the relevant features of plant echoes' temporal statistics (see
The effects of the biological filtering of the auditory system are quite salient: it increases the energy at the large scales and attenuates the energy at the small scales in all filter-bank channels. With the cutoff frequency of the leaky integrator used by us (8 kHz) the energy becomes very small (less than 1% of the total energy) for scales smaller than ca. 5 cm. Interestingly the scale range that is amplified by the auditory system coincides with the range where information for classification should be of the highest relevance for the bat, due to the limited bandwidth of its emitted signal (see above).
The increase of energy in the large scales is visible in all filter-bank channels. Despite this consistent pattern, the different channels emphasize different features of the echo (
A very simple model that only considers the spatial arrangement of the plant reflectors was able to reproduce the observed typical sigmoid shape of the IR envelope spectra. This implies that the spatial arrangement of the reflectors is indeed the most important factor in shaping the temporal echo statistics when using a certain emitted call. Several findings suggest that the model was able to capture the basic relation between the structure of the plant and its echo statistics. Increasing the distances between reflectors an adding branch-like clusters improved the total fit of the real data and improved the large scales fit even more (compare KL values in
The optimization of the model parameters clearly revealed the different nature of apple trees in comparison to the other three species (
In previous work, we already showed that classification of plant species is possible based on both spectral and temporal features in their echoes
Our previous work found that spectral cues and low frequency temporal cues (energy distribution at certain time points along the echo) are advantageous for a linear classifier of the sort we were using. The findings in this work provide an explanatory basis for those conclusions. Here we clearly showed how the frequency response of a plant and the temporal modulations created by the large scale structures can be related to its physical structure.
In this work we demonstrated on the one hand the existence of an interpretable relation between the physical world of plants and the statistics of their echoes, and on the other hand found that the general temporal statistics of all plants follow a common pattern. The robustness of this pattern suggests that the sensory system could have evolved in a way that improves the extraction of relevant information.
Previous examinations of the power spectra of natural images
Here, we present the first analogous results on the statistics of natural plant echoes. These echoes should be of major importance to more than 800 species of echolocating bats, and might have influenced the evolution of their auditory system. Our results point in a similar direction to the findings for natural images. We find that the information that is enhanced by the bats' active sensor (i.e. the emitted signal) and moreover by the filtering of the early stages of their auditory system coincides with the most reliable domain. Depending on the angle of ensonification, the depth distance between two reflectors as measured from the echo will usually appear to the bat shorter than the actual distance between them. This effect is similar to objects that are far from each other in 3D but appear near in a flattened 2D image. In echolocation, the depth equals the actual distance only if both reflectors are aligned with the ensonification axis. Small scales are thus over-represented in the echoes reflecting both real small scale structures and large scale structures ensonified from a non-perpendicular angle. The small scale information is therefore less reliable when attempting to decipher the structure of the plant. Large scale structures however, represent real large scales and are expected to be less commonly observed in the echoes. In summary, large scale structures are most informative, since they convey information on the differences between plants and also most reliable since they convey real information about the spatial structure (these two features are related).
Both the emitted signal and the auditory system function as low-pass filters in the relevant frequency range, enhancing large scale information. Interestingly, the scale range they enhance corresponds exactly to the one that is most informative as just described. The border between the less and more informative structures (<5 cm) is a result of the model parameters and the emitted signal we used. In contrast to vision, in echolocation, this border can be actively altered by the bat. It can be shifted towards smaller scale structures by increasing the bandwidth of the emitted signal, by shortening the signal or by using higher frequencies and vice versa
All echoes were acquired using a bat-mimicking ultrasonic system consisting of a sonar head with three transducers (Polaroid 600 Series; 4-cm-diam circular aperture) connected to a computer. The sonar head was mounted on a portable tripod. Its central transducer served as an emitter (simulating the bat's mouth) and the two side transducers functioned as receivers (simulating the ears). Backscatter received from the emitted signal was amplified, A/D converted, and recorded by a computer. The emitted signal resembles a typical frequency-modulated bat call in terms of its duration and frequency content (
A) Field data acquisition setting. The grid of points denotes the 25 acquisition positions used for each specimen. B) Ensonification of a plastic model plant from a single elevation angle and 5 horizontal angles. C) Ensonification of a Ficus plant with decreasing leaf density from 36 angles around the plant. Examples of 100% and 5% leaf coverage are shown. C) Time signal and spectrogram of the emitted signal.
We recorded the echoes of four plant species, representing common species in the environment of the bats of central Europe. We recorded 50 specimens (always in the field) of each of the following plants:
Apple tree (
Norway spruce tree (
Common beech tree (
Blackthorn tree (
Each specimen was recorded from 25 different aspect angles on an equally spaced 5×5 grid centered at the horizon and the midline of the tree (
To understand the relationship between plant structure and corresponding echoes we conducted two indoor experiments:
As a very simple test of our assumptions we ensonified a plastic model plant that was composed of four identically leaved twigs mounted on a main stem (∼1.5 m long,
In the second experiment we ensonified a single potted plant of the species
All signal processing was performed with Matlab 7.0. To analyze the spectra of the echoes their power spectral densities (PSD) were computed using Welch's method with a 5 ms window and an overlap of 50%. The PSDs (referred to as spectra) were normalized to a sum of one so that the value for each frequency corresponds to the proportion of energy in this frequency. Since the spectrum of the emitted signal was always identical, differences between the spectra of different species imply structural differences and particularly differences in the cross section of the reflectors [see
In order to investigate the relationship between the spatial arrangement of the plant reflectors and the corresponding echo, all echoes went through the following analysis steps:
Cross-correlation with the emitted signal to approximate the IR of the plant.
A Hilbert transform for calculating the envelope of the IR.
The PSD of the envelope using Welch's method with an 8 ms window for the long outdoor echoes and a 4 ms window for the indoor echoes and a 50% overlap for both. The PSDs were observed only down to a scale of ∼1.5 cm since, as is demonstrated in the discussion smaller scales are highly influenced by artifacts. The spectra were normalized to a sum of one as in the PSDs of the plant echoes. This analysis method enables an observation of the echo-structure relation independently of the cross section of the reflectors. Notice that while for the spectral analysis we calculated the PSDs of the original echoes, for the temporal analysis we did so for the envelope of the IR.
The uniform model creates a three-dimensional Poisson distribution of point reflectors (simulations were done in Matlab 7.0). It has two parameters: 1) d1 - the typical distance between the reflectors which can be easily translated into the λ parameter of the Poisson distribution, depicting the probability to find a reflector in a cubic volume with a side length of 0.5 cm (the basic unit of volume we used). 2) L – the side length of the entire cube ensonified by the signal. The first parameter was fit (see below) while the second was estimated from the time duration of the original echoes.
These models include more complex non-stationary distributions in which point reflectors are clustered in Poisson clouds. Due to the high complexity of the calculation of these models we only tested second-order models in which spherical Poisson clusters of reflectors were distributed around the centers found according to the uniform model. Higher-order models can be derived in a sequential fashion based on lower-order models by distributing spherical Poisson clusters of reflector around the reflectors of the lower level. Each order of the n-order models adds two parameters to the two initial ones of the uniform model: ri - the radius of the spheres of the i'th order and di- defining the typical distance between the reflectors in each cloud at level i (for the clusters i>1).
We tested models with different sets of parameters (d, r) spread along the entire range of physical plausible values (see below). 500 echoes were simulated for each set of parameters. The spectra of the echoes were computed in the same way as for the real echoes. We used the Kullback-Leibler (KL) divergence to choose the model that best fits the observed real data. Assuming that the different frequencies (scales) are independent we used the KL divergence to compare the probability distribution functions (PDF) of the model and the real data at each frequency. The PDFs of the frequencies of single spectra were highly non-gaussian in all frequencies (both for real data and models) and were therefore estimated by a histogram with 8 equally spaced bins. The similarity between model and measurement was then calculated according to:
d1 [cm] | 60 | 60 | 60 | 60 | 40 | 40 | 40 | 20 | 20 | 10 |
d2[cm] | 30 | 16 | 8 | 4 | 16 | 8 | 4 | 8 | 4 | 4 |
Combination of distance parameters (d1, d2) used for the second-order model. Each distance combination was tested with the following cluster radii (r2): 25, 50 or 75 cm.
To compare our results to the statistics of the echo after going through the biological filtering we applied the standard auditory system model (see
We would like to thank our three anonymous reviewers for their remarks.