Depth-Based Detection Examples

In this example, there are three isotropic scattering centers comprising an over- resolved (ie-range-extended) target that are within the scene under illumination by the distributed sensor network. Since the Born approximation was utilized to develop the time-reversal operator, the reflectivity of the target scattering centers are fixed, with an associated volume of zero. The surrounding background medium is assumed to be that of a standard Gaussian noise type. We further approximate the form of the noise to be white Gaussian noise, 𝒩 0, 𝜎O _{, with the variance fixed}

at a value of unity. The signal-to-noise ratio is varied from −3𝑑𝐵 to 10 𝑑𝐵 in order to capture a broad range of conditions resulting from the sensor network pre- detection fusion process. The resulting signal subspace rank is then three, with all other rank components being associated with the random noise process. As the noise is increased, the eigenvalue repulsion between the noise and signal weakens; this weakening results in missed detections and the potential for false alarms. This later statement implies a fundamental limit of efficacy for the depth- based detection algorithm for a given range of signal-to-noise ratio values, that is addressed and overcome in Chapter 6.

Fundamentally, there is no set constraint on the underlying statistical distribution for either the signal or the noise; white Gaussian noise was chosen so that the results of this novel depth-based detection method are readily comparable against traditional monostatic receiver operating characteristics-ie single pulse and envelope detectors.

Figure 15: Data Depth Functional Annular Volume Comparison for Noise Only vs Signal+Noise

4.3.1 Depth-Based Detector Performance: Non-Parametric Case

In order to develop an adequate probability of detection curve, a Monte Carlo simulation was performed at each signal-to-noise ratio (SNR) sample. The number of Monte Carlo runs per SNR increment was large enough to ensure a representative sample point was captured that would be free from spurious statistical anomalies arising from the random noise process. At present, the depth- based detector is not optimized to run on a parallel processing computing architecture, so is single-threaded, and computationally inefficient. As computational efficiency was not an initial goal of this effort, this was an acceptable trade for initial demonstration of efficacy of the depth-based methods for detection

and ranking. Thus, the number of Monte Carlo simulations were kept to a statistically significant value, but not increased to a number that would effectively smooth the generated receiver operating characteristic figures. For the remainder of this section, oscillations and/or jitters present in the receiver operating characteristic figures should be attributed to smaller number of effective Monte Carlo simulations run for each example. A follow-on effort is underway to investigate computationally efficient implementations for depth-based methods, to make them more relevant for real-time sensing systems. A detection was recorded if, and only if, the annular volume increased sufficiently to exceed the detection threshold. For each 𝑝h€ _{= 0.5 contour, the probability of detection was recorded}

for false alarm rates of 6% through 0.01%. The role of any detection algorithm is to ensure a sufficient balance is struck between suppressing Type I and Type II statistical errors within the detection hypothesis test to warrant their use; the role of the associated tracking algorithm-should one be utilized-is to drive the false alarm rate as low as possible.

Figure 16: Probability of Detection vs SNR for Depth-Based Detection Statistic

From Figure 16 the receiver operating characteristic (ROC) curves only appear for false alarm values of, 𝑃Q- = 6% to 𝑃Q- = 0.01%, owing to the restrictive form of the

Type I error control used in the depth-based detector. For comparison, if we assume a standard form of a radar detector, as found in [39], a performance comparison is possible for a single-pulse detector (no integration within the

receiver) for noise corruption described by a normal Gaussian distribution. The form of the detector is shown below

𝑃¬ =#_O− Φ 𝑥 − 2×SNR (4.11)

where x is the detection threshold (ie-3 times the noise power) and 𝜙 is the error function. The performance of the depth-based detector does indeed outperform the classical single-pulse detector, as shown in Table 4, particularly as the probability of false alarm decreases.

Table 4: Probability of Detection Comparison with Chebyshev Inequality

PFA x Required SNR for Classical Detector Required SNR for Depth- Based Detector Improvement Factor of Depth-Based Detector 6% 1.88 3.2 dB 3.6 dB -0.4 dB 4% 2.05 4.2 dB 3.7 dB +0.7 dB 3% 2.17 4.5 dB 3.8 dB +1.3 dB 1% 2.33 5.1 dB 4.5 dB +0.6 dB 0.1% 3.08 7.7 dB 4.6 dB +3.1 dB 0.01% 3.62 10 dB 6.0 dB +4.0 dB

4.3.2 Knowledge-Aided Depth-Based Detector

The Chebyshev Inequality is a more restrictive detection criterion; which was deliberately chosen to ensure the depth-based detection algorithm is general for any given class of distribution-both known and unknown. We would expect the detection probabilities are less than that of Equation (4.11), in which the function is derived from a normal Gaussian noise process assumption. In our example, the corruptive noise distribution is also assumed to be normal Gaussian; so, if we choose to re-run the same simulation by assuming our depth-function has complete a priori knowledge of the corruptive noise distribution, then the following false-alarm rates in Table 5 are more appropriate, and are found from the error function. We note, that owing to the conservative nature of the Chebyshev Inequality, the equivalent false-alarm rate-assuming Gaussian noise-for values greater than 𝑃Q- ≥ 10=O are significantly better than those shown in Table 5, but

are purposefully kept to 𝑃_Q- = 10=d _{and 𝑥 = 6.23 for more relative}

Table 5: Probability of False Alarm for Knowledge-Aided Depth-Based Detector vs Classical Detector

Chebyshev’s Inequality, PFA 𝒌𝝈 x Equivalent PFA for Gaussian Noise 6% 4𝜎 3.62 10-4 4% 5𝜎 4.75 10-6 3% 6𝜎 5.61 10-8 1% 9𝜎 𝑥 ≫ 6.23 𝑃Q-≪ 10=d 0.1% 10𝜎 𝑥 ≫ 6.23 𝑃_Q-≪ 10=d 0.01% 32𝜎 𝑥 ≫ 6.23 𝑃Q-≪ 10=d

For reliable detection performance, the probability of detection was set to 𝑃_¬ = 0.9. The Knowledge-Aided Depth-Based (KA-DB) detector in comparison with the classical single-pulse detector of Equation (4.11) is given in Table 6; which does show significantly better performance; especially when the false alarm rate is low- i.e. 𝑃Q- ≤ 10=D.

Table 6: Probability of Detection Comparison for Knowledge-Aided Depth-Based Detector vs Classical Detector 𝒌𝝈 Required SNR for Classical Detector Required SNR for Depth-Based Detector Improvement Factor of Depth- Based Detector 𝟑𝝈 10 dB 3.6 dB +6.4 dB 𝟒𝝈 15.6 dB 3.7 dB +11.9 dB 𝟔𝝈 20.8 dB 3.8 dB +17.0 dB 𝟗𝝈 𝑆𝑁𝑅 ≫ 25 𝑑𝐵 4.5 dB <+20.5 dB 𝟏𝟎𝝈 𝑆𝑁𝑅 ≫ 25 𝑑𝐵 4.6 dB <+20.4 dB 𝟑𝟐𝝈 𝑆𝑁𝑅 ≫ 25 𝑑𝐵 6.0 dB <+19.0 dB

4.3.3 Depth-Based and Knowledge-Aided Depth-Based Detector Performance vs. Envelop Detector

More realistically, a radar sensor utilizes the envelope of the received signal to perform a binary hypothesis test. If we assume a general form of the envelope detector [39]

SNR= A + 0.12AB + 1.7B (4.12)

with A= ln ".hO₅

+i and B = ln

56

#=56 , then the depth-based and knowledge-aided

depth-based detector performance is found in table Table 7 and 8, respectively. When knowledge of the underlying noise distribution was assumed, the depth-

based methods were shown to improve on the classical single-pulse magnitude threshold detector and classical envelope detector by a factor of up to > +20𝑑𝐵 for the case of a 𝑃_Q- of less than 1%.

Table 7: Probability of Detection Comparison for Depth-Based Detector vs Envelope Detector

PFA Required SNR for Envelope Detector Required SNR for Depth-Based Detector Improvement Factor of Depth- Based Detector 6% 6.7 dB 3.6 dB +3.1 dB 4% 7.2 dB 3.7 dB +3.5 dB 3% 7.6 dB 3.8 dB +3.8 dB 1% 9.0 dB 4.5 dB +4.5 dB 0.1% 11.9 dB 4.6 dB +7.3 dB 0.01% 14.8 dB 6.0 dB +8.8 dB

Table 8: Probability of Detection Comparison for Knowledge-Aided Depth-Based Detector vs Envelope Detector 𝒌𝝈 Required SNR for Envelope Detector Required SNR for Depth-Based Detector Improvement Factor of Depth-Based Detector 𝟒𝝈 12.5 dB 3.6 dB +8.9 dB 𝟓𝝈 17.9 dB 3.7 dB +14.2 dB 𝟔𝝈 23.3 dB 3.8 dB +19.5 dB 𝟗𝝈 𝑆𝑁𝑅 ≫ 26.0 𝑑𝐵 4.5 dB >+21.5 dB 𝟏𝟎𝝈 𝑆𝑁𝑅 ≫ 26.0 𝑑𝐵 4.6 dB >+21.4 dB 𝟑𝟐𝝈 𝑆𝑁𝑅 ≫ 26.0 𝑑𝐵 6.0 dB >+20.0 dB

One benefit of the depth-based detector is the basis of its nonparametric formulation, which does not rely on the underlying corruptive noise process to conform to a univariate or bivariate distribution; in fact, the depth-based detector should be more optimal for cases in which the underlying noise process is multivariate and not adequately described by a second-order moment method.