Extending Sensor Capability Using Concurrent

Extending Sensor Capability Using Concurrent Random Sampling Thanh Dang, Nirupama Bulusu, Wu-chi Feng Department of Computer Science Portland State University dangtx,nbulusu,[email protected] Wen Hu CSIRO ICT Centre Queensland Centre for Advanced technologies (QCAT) [email protected]

Abstract Conventionally, in order to reconstruct a signal, sensors need to sample the signal at the Nyquist rate, which is twice the frequency of the signal. However, embedded sensors are often resource-impoverished devices, which are unable to sense high frequency signals such as acoustic and vibration signals due to limited resources such as energy, memory and bandwidth requirements. This paper proposes and evaluates a novel concurrent random sampling (CRS) algorithm which allows distributed sensors to independently, concurrently and randomly sample a signal at a much lower frequency than the Nyquist rate. This algorithm is based on recent breakthrough achievements in compressive sampling theory that exploit the fact that most natural signals are sparse. CRS is elegant and completely distributed. Our experimental results show that sensors can sample at a rate lower than the signal frequency by a factor of 10 with no overhead. As a result, sensors now can not only capture high frequency signals but also potentially conserve energy, memory and bandwidth.

1. Introduction In this paper, we address the problem of enabling highfrequency signal capture with a low-powered sensor network. we need to come with a more precise problem statement. Wireless sensor network technology has witnessed great advances in the last decade, in devices, algorithms and software. Networks of low power, small form factor embedded sensors such as the motes have been deployed in many applications to monitor buildings and bridges, vineyards, turtles and toads. However, this technology has not yet crossed

over to many scientific and medical applications where low cost, dense sampling of the environment is desirable. The common feature amongst these applications is the need to capture high frequency signals such as the human voice, cane-toad song, seismic signals or vibrations. The few systems developed for applications requiring high frequency signal capture have been highly application specific. They employ hybrid or tiered architectures, or specialized DSP hardware. It is challenging to capture high frequency signals with resource-impoverished sensor devices such as the motes. Low-powered sensors cannot sample fast enough to capture high frequency signals (acoustic) because classically sensors need to sample twice as fast as the signal frequency in order to reconstruct the signals. Memory limited sensors cannot store the large volume of data generated in a short period of time. Bandwidth-limited sensors cannot transmit a large amount of data in real time. 1 ? The classical approach to high-frequency sampling, multi-rate signal processing [22] uses multiple sensors in parallel to sample a source concurrently. The drawback with this approach is that capturing high frequency signals requires very precise time synchronization across sensors. Moreover, the cumulative sampling rate across sensors still has to meet or exceed the Nyquist rate. Thus, this approach does not really overcome the fundamental bandwidth and 1 POLISH this: For example, it is impossible for MicaZ to sample at 20Khz. Because it will require 400 instructions per sample, which are not enough for interrupt handling, hardware operation and signal processing. Signal processing is needed because MicaZ has only 4K memory. Have to log to flash memory, hence need more instruction. Assuming one instruction takes one CPU cycle. If one instruction take more than one CPU cycles, the situation is even worse. If one sample is 2 bytes length, the amount of data needed to be transferred equals to 20,000*8 = 320k bit per second. The effective bandwidth is about 125kbit/s for tmote , mica2? Micaz

memory limitations of resource-impoverished sensors. Our proposed solution, which is based on compressive sampling theory, overcomes these limitations by exploiting the observation that most natural signals are sparse in some domain, and that low-cost, low-power sensors can be densely deployed. We make the following explicit assumptions: • Our first assumption forms the basis for compressive sampling theory. We assume that signals are sparse in some domain. A signal can be a combination of a small number of basis functions in a domain. Fortunately, this assumption is strongly held in practice because most natural signals are sparse in some domain eg. fourier, wavelet, or curvelet. In this paper, the signal is assumed to be sparse in the frequency domain. • Second, we assume that the network is dense, so that the signal propagations are similar from a source to different sensors. As in multi-rate signal processing, in our approach sensors sample concurrently. However, each sensor starts sampling randomly at a much lower rate with random jitter. The original signal can be reconstructed at a back end server using Dantzig selector [12], a robust optimization method that has been proved to be well suited for the reconstruction. There has been limited previous work applying compressive sensing to sensor networks [24, 16, 3]. The key difference between our approach and prior work on compressive sensing is that we do not calculate random projections in the network, which must be performed either at the hardware level using specialized hardware [27], or by exchanging samples in the network [24] In contrast to prior work, we sample the signal directly and randomly in time. Therefore, we do not need to use random projection. The random samples are themselves random projections from the frequency domain to the time domain (explained in section 3.2). The advantage of our approach is that we can use existing hardware for compressive sensing, without incurring any additional communication overhead for exchanging samples among sensors. While the reconstruction phase is also present in previous work, we use a newer technique developed by Candes. The main contributions of this paper are as follows:

• An evaluation of an animal-classification algorithm on an acoustic signal reconstructed from concurrent random sampling (CRS) to show that it is viable to use concurrent random sampling in real world applications. The rest of this paper is organized as follows. We will briefly introduce the theory behind compressive sensing in Section 3. We will present the CRS algorithm and an architecture that employs CRS in Section 3. We will discuss experimental results in Section 4. We will review related work in Section 5 and conclude in Section 6.

2. Compressive Sensing Overview Our algorithm builds on compressive sensing and we provide a brief overview in this section. Compressive sensing is a new field that is based on an underlying sampling theory that allows us to reconstruct a signal from a few measurements. For an excellent overview, the reader is refered to lectures by Baraniuk [2] or Candes [4]. The key idea is that if a signal is sparse — the signal can be expressed in a small number of linear combinations of basis functions in some domain — we can reconstruct the signal from a few measurements obtained from random projection. We briefly present two main techniques in obtaining the measurements: random projection and random sampling. Random projection: Let f is the original signal of length n. The m measurements can be obtained by projecting f to a projection matrix P of m by n random entries. This technique is used in a single pixel camera [27] and work in [24, 16]. Random sampling: Let f is the original signal of length n. Random sampling takes random measurements as random entries in f . Hence, random sampling does not explicitly do a random projection on the original signal. This technique is used in [5, 20]. For the purpose of this paper, we discuss theorem 2.1 [7] that presents the theory of compressive sensing for time and frequency domains.

• A concurrent random sampling algorithm (CRS) in sensor networks that extends the state-of-the-art in compressive sensing can lower the sampling rate at each sensor by a factor of 10 or more depending on the signals and network density.

Theorem 2.1. Suppose that f ∈ Rn and is supported on a set |T |. If we sample at randomly selected m frequency locations ω1 , ..., ωm

• An evaluation on both simulated and real data to show that sensors now can not only capture high frequency signals but also potentially conserve energy, memory and bandwidth.

Solving the optimization problem

m ≥ C ∗ |T | ∗ log n

min g

n−1 X t=0

|g(t)|,

Notation n t T w Ω f C g(t) gˆ fˆ gˆ(ω) M m N k

Meaning signal length Time unit Support set, indicating which time position f is non-zero Frequency unit A Set of all frequencies in f Original signal of length n A constant factor An estimate of f in time domain Fourier representation of g and f Fourier coefficient of g at frequency ω down sampling factor total number of measurements number of sensors jitter scaling factor Table 1. Notation table

where the variables are in the notation table 2. This theorem states that if we have a signal that is sparse in the time domain such as a spike train, we do not need to sample the signal at the Nyquist rate. Instead, we need to collect only a small number of random samples in frequency domain. This number is proportional to the log of the signal length and the number of non zero coefficients in the time domain. This is an important result. In the past, in order to capture spike train signals, we needed to sample at a very high frequency and collect a large number of samples in the time domain. This theorem tells us that a small number of samples in the frequency domain is enough to reconstruct the signal. Several optimization techniques [8, 12, 9, 15] have been developed to solve 2.1. In most practical applications, there is noise in the measurements. Fortunately, another result of this theory states that the reconstruction error is bounded and proportional to the noise level in most cases [11]. A complete discussion of this theory can be found in [10, 11, 9, 12, 15, 13, 7, 2, 4, 8].

3. Concurrent Random Sampling

number of random samples in the frequency domain. Similarly, if the signal is sparse in frequency domain, we can reconstruct the signal exactly using a small number of random samples in the time domain. A straightforward corollary to theorem 2.1 is as follows. Corollary 3.1. Suppose that f ∈ Ω and is supported on a set |T | If we sample at m randomly selected time steps, t1 , ..., tm where m ≥ C ∗ |T | ∗ log n Solving the optimization problem min g

n−1 X

|g(w)|,

w=0

such that g(ti ) = f (ti ) for i = 1..n will give f exactly with overwhelming probability. This result is useful because in practice we have many signals that are sparse in the frequency domain. For example, vibration signals from a machine are often concentrated on some frequencies specific to that machine, acoustic signals from an animal are often concentrated on some frequencies specific to that animal. Figure 2 shows electroencephalogram (EEG) signal, which is nearly sparse in time domain but is not sparse in frequency domain. Figure 1 shows signals that are sparse in frequency domain (cane toad song, human voice, and piano music). In order to capture such a signal, we now do not need to sample at the signal frequency. We can sample at a much lower rate depending on how sparse the signal is. Canetoad song in frequency domain 10 5 0 0 Signal amplitude

such that gˆ(ω) = fˆ(ω) for all ω ∈ Ω will give f exactly with overwhelming probability.

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Having reviewed compressive sensing, we discuss the intuition behind our Concurrent Random Sampling algorithm and describe the algorithm in detail.

3.1

Intuition

Theorem 2.1 tells us that if a signal is sparse in the time domain, we can reconstruct the signal exactly using a small

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Figure 1. Signals that are sparse in frequency domain

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Figure 2. EEG signal that is near sparse in time domain but not sparse in frequency domain (Source: Ben Greenstein, UCLA CENS)

We can further exploit one characteristic of randomness. The summation of k independent random variables is likely to be random. Hence, each sensor can now independently and randomly collect samples in the time domain and transmit these samples back to a sink. Each sensor uses a random seed to generate the random sampling time. The sink only needs to know the random seeds that the sensors use to determine the time of each sample. The combination of all collected samples is basically a sequence of random samples drawn from a the signal.

3.2

Concurrent Random Sampling Algorithm

Figure 3 illustrates how CRS works. Sensor 1, 2, and 3 randomly sample a source and transmit the samples back to a sink. The last graph shows the summation of all the samples in time. The summation is actually a random sequence of samples in time. Based on the theory of compressive sensing, we can reconstruct the signal using these samples. The sampling time is determined in the following equation. sample − time = M ∗ randsample(n/M, m/N ) (1) +k ∗ round(randn(m/N, 1)) (2) where • n is the signal length. • M is the down sampling factor. M controls the minimum time interval between each sample.

probability distribution of sampling

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Figure 3. Concurrent Random Sampling: The dots represent sampling points in time. The bell curve represents the probability distribution of a sampling point. Each sensor independently and randomly samples a source at a low sampling rate with random jitter. The samples collected at the sink are basically random samples drawn from the source.

• m is the total number of measurements. • N is the number of sensors. • Function randsample(a, b) picks b arbitrary numbers with independent identical distribution (i.i.d) from 1 to a. Function randn(a, b) generates an a-by-b matrix with random entries, chosen from a normal distribution with zero mean and variance one. • k is the scaling factor that control how large the jitter, which we will discuss about shortly, is. We use N sensors to collect m samples. Hence, each sensor needs to collect only m/N samples. If a sensor collects m/N samples randomly, there will be a chance that the sensor collects two samples that are close together in time. In that case, the sensor still needs to be able to sample at Nyquist rate. We overcome this limitation by choosing sampling time randomly in a short period and dilate it to the full time scale. The sampling time sequence is determined by M ∗ randsample(n/M, m/N ), which means we select m/N sample time randomly from the time scale [1,n/M ] and dilate the sample time to the time scale [0, n] by multiplying the sampling time by M . The sampling time is now random from 1 to n and are at least M time units apart. This ensures that we not only collect a sub number of samples but also sample at a rate lower by a factor of M compared

needs to preempt the current execution, switch the context and does the sampling. Hence, there is a certain delay depending on the state of the processor. However, these two noise sources can be consider as one and the Dantzig selector can reconstruct the signal with bounded error. The nature of CRS makes it robust to transmission loss. Since the data is collected randomly; if some samples are lost during the transmission, we can still reconstruct the signal. In addition, sensors do not need to be synchronized at all in time. When there is a triggering event, sensors collect data independently. After transmitting all the data to the sink, we can synchronize the sensors and calculate the sampling time at each sensor. This is the key difference compared to tranditional multi-rate data acquisition techniques [22] where sensors need to be very well synchronized with each other at sampling time.

4. Experiments and Results Figure 4. Co-located sensors In this section, we conduct experiments to evaluate: to the Nyquist rate. In order to increase the randomness of the sampling time. We introduce time jitter, which has a normal distribution at the sampling time. We generate jitter using k ∗ round(randn(m/N, 1)). The signal can be reconstructed using the optimization techniques in [8, 12, 9, 15]. Specifically, we use the Dantzig selector [12], which has been shown to be robust to noise. We will not describe this technique here. Readers may refer to the references for a complete discussion. The intuition behind this approach is that dilation is a linear projection from the time scale [0, n/M ] to the time scale [1, n], adding random jitter is a linear operation. The sample time is a linear operation on random variables. Hence, the sample time preserves some randomness as Gaussian noises transformed through linear operations are still Gaussian noises. We are aware that in (1) the sample time follows a Bernuoulli distribution and the sample time is not completely random. Surprisingly, the reconstruction works very well even when the sample time is not completely random. In order to make this algorithm work, we assume that the sensor network is dense and the signal attenuation from the source to all sensors in a co-located group is the same as shown in Figure 4. This assumption is reasonable because we can have several sensors stacked together as an array to record a signal. We believe that even with different signal attenuation from a source to sensors, this algorithm may still work because only the energy of the signal changes, the signal sparsity or signal frequencies do not change. There are several sources of noise in the measurements. The first noise source is due to the conversion from an analog to a digital signal. The second noise source is due to the inaccurate sampling time. When a timer fires, the processor

• How well can CRS capture a high frequency signal? • How does the CRS reconstruction error affect the classification results?

4.1

Experimental Setup

We use a microphone on a laptop to randomly sample signal sources. We consider three types of signal sources: cane-toad songs, the human voice, and the music tones. We reconstruct the signal using the Dantzig selector [12], which is available in l1magic [6], and cvx matlab packages [21]. We use the reconstructed signals of cane-toad songs to classify cane-toad types. We will compare the classification results on data collected with CRS to those on data collected with Nyquist sampling to show that CRS does not greatly degrade the performance of the classification system without retraining the system. We believe that if we retrain the classification system, we can achieve even better results.

4.2

Experimental Results

Figure 5 shows the a portion of an original signal and the sampling points of 3 sensors. Clearly, each sensor now can sample at a much lower rate than the signal frequency. Figure 6 and Figure 7 shows the reconstructed signals collected using CRS the signals captured at the Nyquist rates. We sample a human voice and a piano music song. We supper impose the two signals to show that they are very closed together. These figures can be viewed best in color mode. Figure 10 and Figure 11 show the reconstructed signal collected using CRS and the original signal of cane toad’s

Reconstructed Signal Collected Using CRS 2 Sensors Downsampling at a Factor of 10

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Figure 7. CRS reconstructed signal of human voice ”Hello, How are you!” (best view in color mode)

call. Not surprisingly, when sampling at 1 kHz, Figure 10 shows that the reconstructed signals are closer to the original signal with the increasing number of sensors sampling simultaneously. In general, although sensors sample a rate lower by a factor of 10, the reconstructed signals capture most features of the original signal. Figure 11 reconfirms the above observations in frequency domain. Figure 8 shows the relative error of the reconstructed signal using CRS with one sensor sampling at different rates and the signal sampled at the Nyquist rate. Using only sensor and sampling at a rate lower by a factor of 3, the relative error is about 25%. Figure 9 shows the relative error — the ratio between error energy and the signal sampled at the Nyquist rate energy — between the reconstructed signal using CRS with different number of sensors and the signal sampled at the Nyquist rate. We do not consider the mean square error because there is a phase shift between the reconstructed signal and the original signal. The phase shift cause a huge error although they are relatively similar. The sensors sample the signal a rate lower by a factor of 10. Obviously, the more number of sensors randomly sample a signal is, the more smaller relative error is. With 5 sensors, the relative error is about 15%. Table 2 shows the detection rates using signal collected using CRS with one sensor sampling at 3.3Khz, 2Khz, and 1Khz. When random sampling at 1 kHz (a factor of 10 reduction from the Nyquist rate), the number of cane toad detections is 8 or 45% smaller than sampling at 10 kHz. When the sampling rates are greater than 2 kHz (a factor of 5 reduction from the Nyquist rate), the number of cane toad

Sampling Rate Original Signal (10 Khz) 1 Khz 2 Khz 3.3 Khz

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Table 2. Classification performance using signal collected using CRS. One sensor samples at 1 kHz, 2 kHz, and 3.3 kHz. The classifier makes 10, 15, and 14 cane toad detections respectively.

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Relative Error of CRS Reconstructed Signal (Sampling Rate 1Khz) 80 70 Relative error (100%)

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detections is less than 23% from the original 18 detections. Recalled that Figure 9 shows that the related error rate is about 25% when sampling at 3.3 kHz. Table 3 shows the detection rates using signal collected using CRS with different number of sensors sampling at 1 Khz. It shows cane-toad classification accuracy appears to be peak when the number of sensors are more than 2 sensors. The peak values are less than 17% from original results sampling at 10 kHz (Figure 9 shows that the related error rate is around 15% with 5 sensors). The maximum number of detections (18), which is equal to that of original results sampling at 10 kHz, is achieved when 4 sensors are sampling simultaneously. Counter-intuitively, the number of detections, when 5 sensors are sampling simultaneously, is 15, which 3 less than that when 4 sensors are sampling simultaneously. Figure 9 shows that the relative error rates decrease exponentially as the increasing of the sensors. We believe that the exceptional performance of classifier with 4 sensors is a random behavior of peak accuracy. Number of Sensors

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Figure 9. Relative error in signal energy of CRS reconstructed signal of different number of sensors sampling at a rate lower by a factor of 10 compared to the Nyquist rate for the cane toad song

Table 3. Classification performance using signal collected using CRS. Different number of sensors sample at 1 Khz.

5. Related Work In this section, we would like to review and contrast our work from previous work in the field. There are two main

Figure 10. CRS reconstructed signals in time domain (cane toad). A) 2 sensors sampling at 1 kHz, B) 3 sensors sampling at 1 kHz, C) 4 sensors sampling at 1 kHz, D) 5 sensors sampling at 1 kHz, E) Original signal sampling at the Nyquist rate.

Figure 11. Spectrogram of reconstructed signal collected using CRS (cane toad). A) 2 sensors sampling at 1 kHz, B) 3 sensors sampling at 1 kHz, C) 4 sensors sampling at 1 kHz, D) 5 sensors sampling at 1 kHz, E) Original signal sampling at the Nyquist rate.

approaches to processing data in the network. The first approach, which we refer to as post-facto data compression, is to collect all the data, compress the data and transmit the data back to the sink. The second approach, which we refer to as compressive sensing, is most closely related to our work. The key idea is to collect data while compressing it at the same time, and transmit the data back to the sink. In some sense, it collects compressive data. Also relevant to our work are system optimizations for enabling highfrequency sampling, which we briefly comment upon.

5.1. Post-facto Data Compression The main goal of post-facto data compression in sensor networks is to reduce the amount of data stored or transmitted in the sensor network in order to conserve memory, bandwidth and energy. Several attempts have either tried to exploit the spatial redundancy [14] or temporal redundancy [26] of sensor data. One advantage of this approach, in contrast to compressive sensing [26], is that it is possible to do lossless compression (eg. [26]). While on-board data compression can conserve bandwidth and long-term memory, it can be computationally expensive. Moreover, the key limitation is that sensors have to collect all the data first and process it later. Therefore, sensors can not capture high frequency phenomena beyond their fundamental sampling limitations.

5.2. Compressive Sensing Compressive sensing differs from post-facto data compression in that the collected data is already small (or compressed) due to sparse sampling. Therefore, compressive sensing is fast and allows sensors to capture most high frequency phenomena. There are different approaches to sample and reconstruct the data. Rabbat et al. [24] propose a system that simultaneously computes random projections of the sensor data and disseminates them throughout the network using a gossiping algorithm. This approach implies that sensors still need to exchange all the data amongst themselves in order to compute the random projection in the network. Rabbat’s technique is intended to facilitate the extraction of spatial characteristics of the environment (for example, estimation of the temperature field in a room, or network health diagnostics) from any small subset of nodes in the sensor network. Thus, the scheme incurs the overhead of data exchange among sensor nodes for computing the random projection. Moreover, random projections are used to reduce the storage or communication requirements of samples collected at the Nyquist rate, or with special hardware, as opposed to random sampling to facilitate signal sampling at a rate considerably lower than the Nyquist rate. Therefore, this algorithm cannot be used to enable

resource-limited sensors to capture high frequency phenomena beyond their sampling capabilities. The work of Duarte et al. [16] is a direct application of their earlier work in distributed compressed sensing [3]. In this work, each sensor independently collects random Gaussian measurements and transmits the data to the sink. It is important to note that Duarte’s work relies on a random projection of the signal source. This random projection can be generated by using specialized hardware. For example, Baraniuk’s research group generates random projection of an image by building a special camara that randomly rotating the mirror in a CCD array and collecting the reflected light using a single photo sensor [27]. In the absence of such special hardware, it is unclear how to random projection can be done for signals such as accoustic signal without actually first sampling the signal at the Nyquist rate. While Duarte et al show that compressive sensing can be accomplished in a distributed manner, it does not show how random projection can be implemented. It assumes a hypothetical implementation of distributed compressive sensing for the signals. In addition, in order to collect random Gaussian measurements, they either have to collect all the data and then do the Gaussian projections or they need new hardware that can do Gaussian projections automatically. Our approach focuses on a solution that might be implementable on existing hardware (eg. motes) and does not collect Gaussian measurements. Instead, we collect measurements randomly in the time domain, which is actually better than using a random basis [7]. A minor difference in our work is that we use the Dantzig selector [12] which has been shown to be more robust and accurate than previous reconstruction techniques [12]. We find that our work is closely related to the work of Ragheb et al. [25, 20]. Ragheb et al. have developed a hardware platform that enables random sampling of a signal source. A similar work in [5] also considers sampling a signal at a sub Nyquist rate but non-uniformly in time. These two works focus either on extending the capability of analog-to-digital converters or on allowing a direct conversion from analog signals to information. Theese techniques are not intended for a distributed environment. Therefore, we introduce concurrency to extend the random sampling approach to densley distributed sensor networks.

5.3

System Optimization Frequency Sampling

for

High-

Greenstein et al. [17] have developed a system for high-frequency data collection with resource-limited sensors. While their motivating objective is the same as our work, the approach is completely different. Greenstein tries to optimize the software system to reduce the processing time for a higher data acquistion rate. In particular, Green-

stein modifies the code for accessing memory to only generate an interrupt after a RAM buffer is filled with data instead of after each sample conversion. This approach reduces the number of hardware interrupts in data acquisition. However, it still samples data at the Nyquist rate. Therefore, there is a theoretical limit regardless of how well the system is optimized.

6. Conclusion and Future Work We have proposed CRS, a Concurrent Random Sampling algorithm that enables us to sample high frequency signals at a much lower rate than the Nyquist rate. We have extended the state-of-the-art in compressive sensing, by applying compressive sampling theory to enable sparse sampling, without requiring either special hardware, or for sensors to coordinate with each other in order to perform random projections in the network. Our algorithm can be applied to signals that are sparse in the frequency domain, such as acoustic and vibration signals. Our evaluation results on real-world data sets show that CRS can reduce the sampling rate to one-tenth of the Nyquist rate. The energy of the error between the reconstructed signal with CRS and the original signal is within 10-15% of the original signal. Finally, to understand how the error in the reconstructed signal impacts an application, we considered a real-world application, cane-toad monitoring. Our preliminary results show that it is possible to detect cane-toads by running a classification algorithm on the reconstructed signal, even without retraining the algorithm over the reconstructed signals. In principle, CRS could be implemented on any hardware platform. However, we are currently implementing CRS on the Crossbow MicaZ platform because our objective is to facilitate sparse sampling of high-frequency signals with a low-powered sensor network. We also plan to investigate improved metrics for characterizing the error in the reconstructed signal. Over the next few months, we plan to evaluate CRS over a wider range of data sets. We would also like to work with members of the sensor network community, to see whether CRS-enabled sparse sampling can benefit their applications. Our hope is that CRS will facilitate high-frequency signal capture for a wide range of applications, without the complexity of application-specific innetwork processing, or the packaging and deployment costs of expensive sensors.

7. Acknowlegements We would like to acknowledge Ben Greenstein, Nithya Ramanathan, and Lewis Girod for providing us with some of the data sets studied in this paper. The research described

in this project was funded by NSF grant 0424602, via a subcontract to Portland State University.

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