We begin by considering two populations: one comprised of white Gaussian noise, 𝒩 0, 𝜎O , with variance equivalent to unity; and a second population consisting of
a bounded rank perturbation with additive white Gaussian noise, in which the measurement noise is assumed statistically similar to the noise population. To fully develop the depth-based ranking algorithm, initially the standard binary hypothesis test is considered
𝐻"∶= 𝑛 𝑡 (4.13a)
𝐻#∶= 𝑠 𝑡 + 𝑛 𝑡 (4.13b)
However, Equations (4.13a) and (4.13b) do not properly account for the iterative testing that is to be accomplished via the ranking algorithm; specifically, the binary hypothesis test is more properly written as
𝐻"e≔ 𝑛 𝑡 (4.14a)
𝐻#e∶= 𝑠e 𝑡 + 𝑛 𝑡 (4.14b)
where the superscript 𝑗 denotes the rank value of the time-reversal operator under test, and is chosen from the sequence of real numbers formed by, 𝑗 = 1,2,3, … , 𝑛 , and is assumed for all further instances of the superscript 𝑗 in this chapter. Traditionally, the selection of the null or alternative hypothesis is chosen based upon a pre-determined, or adaptive, threshold that transforms Equation (4.14a) and (4.14b) into
𝛿Thresholde ⋚
×Øy ×Ny
(4.15)
which requires a threshold. Previously we showed this to be a non-parametrically derived quantity represented by a volume in n-dimensional space. This quantity may be determined with training data or empirically as is the case with this work. This quantity is then used as the approximate of the noise bounds on the time- reversal operator. Each scattering center manifests as a bounded perturbation on the target covariance function, and owing to its localized phenomenology, is isolated to a single and unique singular value; this stands in stark contrast to thermal/environmental noise that is always full-rank, indicating a diffuse statistical process; and clutter that is never tightly bound, but partial or full-rank, indicative of a simple or complex statistical process, respectively. This bounded phenomenology is the principal reason detection and ranking is possible. Rather than attempting to determine which selection procedure could choose the correct partial-to-full rank of a clutter or noise process, the selection algorithm only has to
contend with one or more finite bounded perturbations isolated to a single singular value. Now, admittedly our object model is a collection of finite-sized target scattering centers, but even complex extended targets are modeled as collections of scattering centers; and more importantly phenomenologically appear as discrete or diffuse collection of point-scatterers (point-scatterer clouds).
Differential dispersion values are determined from the difference of the volume defined by the contour, 𝐷?y
, and that of the volume of a second contour, 𝐷™. Previously, 𝛽, was defined as unity, and that definition holds true here, where
we are attempting to ascertain the effective rank of the target time-reversal operator function non-parametrically. The value of unity for contour, 𝐷?y,
encompasses all population values for an assumed rank of our target time-reversal operator, in our depth functional. The value of contour, 𝐷™, is between the values
of 𝑝 ∈ 0,1 , and constitutes the inner radius of the annulus that is comprised of our annular volumetric parameter used for our recursive binary hypothesis test. For this case, whilst the value of 𝑝 has previously been defined as, 𝑝 ∈
0.5,0.75,0.5 , we initially fix the value at 𝑝 = 0.5; thus, maximizing the possible annular volume of our depth functional, and ensures the collapse of this annular volume is due to the absence of a signal; as the singular values for our corruptive noise process is, 𝒩 0, 𝜎O , with 𝜎O = 1; thereby ensuring a clustering of noise
singular values that are far removed from signal singular values.
As our example consists of a finite number of dielectric scattering centers of the singly-spread target-or rather, the range-extended target, as there is no assumed motion smearing the resultant echoed received signal-the ranking algorithm does not need to run any longer than the computational time required to find the effective rank of the corruptive noise process; defined as the rank value at which point the binary hypothesis test choose 𝐻". For each step in our recursive algorithm, the contours, 𝐷?y and 𝐷™ are defined. Each contour is effectively the
boundary for a value for our depth functional 𝐷?y
∶= 𝑉?y
and 𝐷™∶= 𝑉™. Having
defined the depth function values, the differential dispersion is calculated from
𝑉eF = 𝑉?Fy− 𝑉™F (4.16)
The threshold value is derived from signal-free training data comprised of only the underlying corruptive noise process. Similarly, we calculate the annular volume of the corruptive noise process by choosing 𝛽 = 1 and 𝑝 = 0.5.
where the normalized threshold statistic is found from 𝛿Thresholde = PThreshold
y
PThresholdy (4.18)
In this manner, regardless of the actual measurement noise, the threshold for determination of the binary hypothesis test is non-parametrically derived. Before we can jump into ranking, a method of false alarm control is required. As was the case for depth-based detection, the Chebyshev Inequality is applied to our nonparametric threshold annular region, in an effort to minimize Type II errors. A number of 𝑉Thresholde values are calculated based on target-free training data, and the mean and standard deviation of the vectorized quantities are determined. The Chebyshev Inequality is valid for any population, provided a mean and standard deviation exist. We choose the false alarm rate by varying a single parameter, 𝑘 in the expression
𝑉Thresholde = 𝜇e+ 𝑘𝜎e (4.19)
where 𝜇e and 𝜎e were previously derived from training data. 𝑘 is the effective
number of standard deviations that are required to encompass a minimum population mean of 1 −õ#ü . In our case, for a desired false alarm rate of 6%, 𝑘 is required to be set at 𝑘 = 4; whereas for a desired false alarm rate of 4% or 3%, 𝑘 is 5 and 6 respectively. The threshold for the binary hypothesis test is equivalent to 𝛿Thresholde = PThreshold; lyamny y P Threshold; lyamny y ⇒ 1 (4.20)
which is to say, unity. This is a nonparametrically derived quantity that ensures the applicability of the binary hypothesis test over a wide-range of corruptive noise processes, without compromising the integrity of the depth-based detector.
Determination of the confidence interval is based on the estimate of the annular volume between the 𝑉? and 𝑉™; as the annulus region contracts with each
recursive contraction of contour, 𝑉?y
, there is a point at which the annulus of the signal-free is volumetrically equivalent to the annulus of the time-reversal operator under test. It is important to note that whilst noise-thermal, environmental or otherwise-is full-rank, there is a bounded perturbation resulting from the presence of scattering center reflection(s) that is manifest as a localized singular value(s) in the time-reversal operator; this offset has previously been described as arising from Newton's Third Law, but is more colloquially referred to as eigenvalue
repulsion. Now, one the rank of the echoing scattering center(s) is exceeded, the singular values of the time-reversal operator under test collapses to values that are essentially equivalent to the signal-free time-reversal operator, derived from signal- free training data. Since the values under test differ only in statistical variability, the Chebyshev inequality derived threshold value, 𝛿Threshold should eclipse the volumetric bounds of noise-on-noise annular volumes, and result in a null hypothesis being chosen for the binary hypothesis test, see Equation (4.13a) and (4.13b). In this manner, the differential volume serves as the confidence bounds for our depth-based ranking algorithm. To conclude the depth-based ranking algorithm, the final binary hypothesis test is then
𝛿e ⋚ ×Øy ×Ny 𝛿
Threshold
e (4.21)
where 𝑗 has previously been defined as the rank value under test, and is a sequence of real numbers formed by, 𝑗 = 1,2,3, … , 𝑛 . The value of 𝑗, or the effective rank of the time-reversal response matrix under test, for which the test last hypothesis determines alternative hypothesis, 𝐻#, is true also determines the final rank value of the matrix; thus, the first value at which 𝐻" is determined true, terminates the recursive depth-based ranking algorithm, and also indicates the point at which the annular region of the differential volume collapses, and only contains noisy singular values.
In this section, a bespoke depth-based ranking algorithm was introduced, along with a method for controlling the Type II error of the nonparametric binary hypothesis test. For the succeeding section, a series of examples are presented, focused on a common scenario, and the results of the depth-based ranking algorithm are determined based upon a receiver-operating characteristic curve (ROC).