CRB on Range Separation Estimation

In this section, the CRB for range separationΔ𝑘is derived using the two-point target data model from Section 4.4.1. Other bounds are determined as well including spatial separation and the target amplitudes. For a particular imaging scenario, the range separation CRB is shown in a figure across the possible range separations.

For multiple unknowns, the CRB is defined by the diagonals of the Fisher Information Matrix (FIM) inverse and provide a lower bound on the variance of any unbiased estimator

which is shown for a general, multiple unknown parameter case by [84]

var[ ˆ𝜃_𝑖(D) − 𝜃𝑖

] ≥[𝐽⁻¹]

𝑖𝑖 (6.1)

where “var” is the variance, ˆ𝜃𝑖(D) is the estimate of a particular unknown, non-random variable 𝜃𝑖, D is the observation space, and 𝐽 is the FIM. The elements of the FIM are the negative expected value of the double derivative log-likelihood function and provides a measure of the amount of information of an unknown parameter contained in the random process. Mathematically, the FIM is defined by [84]

𝐽𝑖𝑗 = −E[ ∂²ln 𝑃 (𝑑𝑘(𝑥, 𝑦) = 𝐷𝑘(𝑥, 𝑦) ∀𝑘, 𝑥, 𝑦)

∂𝜃_𝑖∂𝜃_𝑗

]

(6.2)

whereE is the expected value operation, “ln” is the natural log, and 𝑃 is the probability mass function (PMF) for all 3D FLASH LADAR observations with 𝑑𝑘(𝑥, 𝑦) as the real-izations of the observations. Assuming statistical independence of each volume element (voxel), the PMF for the data model is defined by

𝑃 [𝑑𝑘(𝑥, 𝑦) = 𝐷𝑘(𝑥, 𝑦) ∀𝑘, 𝑥, 𝑦] =

𝐾

∏

𝑘=1 𝑋

∏

𝑥=1 𝑌

∏

𝑦=1

𝑖𝑘(𝑥, 𝑦)^{𝑑𝑘(𝑥,𝑦)}exp {−𝑖^𝑘(𝑥, 𝑦)}

𝑑𝑘(𝑥, 𝑦)! (6.3)

where the assumed dominant noise source is photon (shot) noise described by the Poisson distribution. While lasers exhibit partial coherence meaning the negative binomial distri-bution should be used for the light statistics, this photon noise assumption is valid when the operating environment produces a large enough speckle parameter so that the nega-tive binomial distribution approaches the Poisson distribution [24]. Previous 3D FLASH LADAR work has shown the speckle parameter to be adequate to assume the Poisson dis-tribution [9]. Additionally, the Poisson disdis-tribution CRB provides a lower bound to the negative binomial CRB considering the higher negative binomial variance [60]. This fact creates a true lower bound (most pessimistic) with the Poisson distribution CRB regardless

After performing the required operations, the general solution for the FIM elements is determined to be [9], [39]

𝐽𝑖𝑗 =

Particular to this work, there are four non-random unknown variables in the data model, 𝜃 = [Δ𝑚, Δ𝑘, 𝐴𝑡, 𝐴𝑟], resulting in a 4x4 FIM with its elements determined to be

where

NOTE: The “×” in 𝐽¹²is a multiply operation. The FIM is inverted and the CRB for each of the unknowns is on the diagonal of the inverted FIM matrix with the range separation CRB at[𝐽⁻¹]₂₂. The purpose behind supplying the FIM element expressions was to provide enough information to enable the work to be reproduced. Although an example plot is given later in the section, the range separation CRB expression itself is not shown due to its length and complexity.

Besides the four random unknown parameters, the CRB also depends on non-random known parameters to include 𝜎𝑝𝑡, 𝜎ℎ, and 𝑡𝑠. In order to view a useful plot, all other unknown and known factors are held constant while theΔ𝑘 is stepped from the be-ginning to the end of the range extents. Following this procedure, Figure 6.1 shows the range separation bound for a specific scenario withΔ𝑚 = 1 pixel, 𝜎ℎ = 3 pixels, 𝜎𝑝𝑡 = 3 ns, 𝐴𝑡 = 0.5 × 10⁴ photons, 𝐴𝑟 = 2 × 10⁵ photons, 𝐵 (𝑥, 𝑦) ∼ N(750, 38) in units of photons, and range sampling 𝑡𝑠 = 1.876 ns. These values were chosen to represent a scenario where the 3D FLASH LADAR interrogates adjacent targets with different reflec-tivities while experiencing significant turbulence in the atmosphere. Furthermore, the bias definition is consistent with estimation results from experimental data.

−3 −2 −1 0 1 2 3 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10⁻³

Δ𝑘 (m) CRB(Δ𝑘)m2

Figure 6.1: This plot shows an example CRB with Δ𝑚 = 1 pixel, 𝜎ℎ = 3 pixels, 𝜎𝑝𝑡 = 3 ns, 𝐴𝑡 = 0.5 × 10⁴ photons, and 𝐴𝑟 = 2 × 10⁵ photons. The bound behaves appropriately considering the variance goes up as the sepa-ration becomes smaller corresponding to the notion that close-in targets are tougher to resolve. The peak of the bound occurs when the range and spa-tial coupling are at their maximum. Further, when the range separation near zero, the range coupling is diminished, but the bound doesn’t go to exactly zero because the spatial coupling is still present.

The shape of the curve in Figure 6.1 reflects the negative effects of the range and spatial blurring as the targets become closer. Although, the effects of range blurring are minimized when the targets are at nearly identical ranges and the bound primarily depends on the spatial blurring. Additionally, the increase in the bound past ±2 meters of range separation is due to the truncation of the pulse at those ranges. An assumption in the bound derivation is a fully contained pulse within the range extents. The impact is negligible considering the eventual application of the CRB towards range resolution. Targets with

±2 meters of range separation would be easily resolved. Changes in these values affect the bound in a predictive manner. For example, increasing 𝜎ℎ and 𝜎𝑝𝑡 doesn’t affect the general shape of the range separation CRB, but it does increase the bound’s magnitude due to increased spatial and range blurring hampering range separation estimation.

More specifically, Figure 6.2 shows several examples of how the range separation CRB is affected by changing parameters in the model including target amplitude, blurring severity, and spatial separation. Each individual figure holds all other parameters constant and plots the CRB while changing one parameter. For example, Figure6.2(a) changes the unknown target amplitude𝐴𝑡while keeping all other parameters constant. Unless otherwise noted, the standard values for the parameters are :𝜎_𝑝𝑡 = 3 ns, 𝛿_𝑚 = 1 pixel, 𝜎_ℎ = 3 pixels, 𝑡𝑠 = 1.876 ns, 𝐴𝑡 = 0.5 × 10⁴ photons, and𝐴𝑟 = 2 × 10⁵ photons. The next few para-graphs detail how the changing parameters effect the range separation CRB. The parameter changes affect only the bound values, but not the general shape of the bound.

Figure6.2(a) -𝐴𝑡effects. As the unknown target’s amplitude is increased, the bound decreases meaning that higher SNR values of the unknown target aids in range separation estimation efforts. (Inversely proportional to bound)

Figure6.2(b) -𝐴𝑟 effects. Changing the known target’s amplitude has the opposite effect of 𝐴_𝑡. As the 𝐴_𝑟 amplitude is increased, the bound also increases meaning that range separation estimation becomes more difficult due to the increased blurring between the targets. In other words, estimating the range separation becomes very difficult when

−3 −2 −1 0 1 2 3

Figure 6.2: Effects on CRB(Δ_𝑘) when changing several parameters in the model includ-ing target amplitude, blurrinclud-ing severity, and spatial separation.

(a)𝐴𝑡- inversely proportional to bound (b)𝐴_𝑟- proportional to bound

(c)𝜎ℎ - proportional to bound

(d)Δ𝑚 - inveresly proportional to bound

considering a very bright known target next to a much dimmer unknown target and vice versa. (Proportional to bound)

Figure6.2(c) -𝜎ℎeffects. As the blurring severity increases, the bound also increases meaning that more blurring (i.e. more coupling between the targets) hinders range separa-tion estimasepara-tion performance. (Inversely proporsepara-tional to bound)

Figure6.2(d) -Δ_𝑚 effects. Finally, as the spatial separation increases, there is a log-ical corresponding range separation estimation performance improvement due to decrease in coupling between the targets. (Inversely proportional to bound)