Two Point Target Range and Spatial Separation Estimator

With FOliage PENetration (FOPEN) applications, this section develops a range sep-aration estimator by using a least squares approach adapted from previous work that only considered two targets within a single pixel in a non-blurry environment [5]. While no noise source is specified in the subsequent development, estimator results in a shot-noise limited environment are given in Section6.2.

4.4.1 Two Point Target Data Model. The mean of the observations in units of photons of a two point target scene interrogated by a 3D FLASH LADAR are defined by a convolution between the object and the system point-spread-function (PSF) added to a pixel bias or [25], [37]

𝑖𝑘(𝑥, 𝑦) =

𝑀

∑

𝑚=1 𝑁

∑

𝑛=1

𝑜𝑘(𝑚, 𝑛) ℎ ((𝑥 − 𝑚, 𝑦 − 𝑛) + 𝐵 (𝑥, 𝑦) (4.18)

where(𝑥, 𝑦) are the pixel plane coordinates with 𝑥 ∈ [1, 𝑋] and 𝑦 ∈ [1, 𝑌 ], 𝑘 is the range dimension coordinate, and (𝑚, 𝑛) are the object plane coordinates with 𝑚 ∈ [1, 𝑀] and 𝑛 ∈ [1, 𝑁]. The integer range dimension variable 𝑘 ∈ [0, K − 1] corresponds to a range distance𝑟_𝑘in units of meters according to

𝑟𝑘 = 𝐾𝑠+( 𝑘 ⋅ 𝑡𝑠⋅ 𝑐 2

)

(4.19)

with𝐾𝑠 being the initial/starting range of the data cube, 𝑡𝑠 as the range sampling interval in seconds, and𝑐 being the speed of light in meters per second.

2 4 6 8 10 12 14 16 18 20

Figure 4.1: (a) For illustrative purposes, this figure is a range image of the truth data where the reference target is in the center of the array at 1000 meters with the unknown target placed atΔ𝑚 = 2 pixels and Δ𝑘 = 1.7 meters.

(b) Defined by Equation (4.22), this shows the ideal waveforms of the un-known𝑝 (𝑟𝑘− 𝐾^𝑡) and reference target 𝑝 (𝑟𝑘− 𝐾^𝑟) from Figure4.1(a) with a pulse-width standard deviation𝜎𝑝𝑡= 0.88 ns.

Considering both range and spatial dimensions, the two point target scene consists of one target at a known position and one target at an unknown position. The targets are constructed this way since the paper’s focus is on range separation between the targets and not absolute range. This assumption keeps the parameter of interest (range separation) in-tact while simplifying the data model by preventing an additional unknown parameter. The targets are considered point targets spatially, but do provide a returned waveform. Consid-ering the two point target scene illustrated by Figures4.1(a) and (b) , the object𝑜 (𝑥, 𝑦) is

defined by

𝑜_𝑘(𝑚, 𝑛) = 𝐴_𝑡𝑝 (𝑟_𝑘− (𝐾𝑟− Δ𝑘)) 𝛿 (𝑚 − Δ𝑚, 𝑛) + 𝐴_𝑟𝑝 (𝑟_𝑘− 𝐾𝑟) 𝛿 (𝑚, 𝑛) . (4.20)

where𝐴𝑡and 𝐴𝑟 are the point target amplitudes,𝑝 (𝑟𝑘− (𝐾^𝑟− Δ^𝑘)) and 𝑝 (𝑟𝑘− 𝐾^𝑟) are the received pulse shapes with𝐾𝑟 as the known reference target andΔ𝑘 as the range sep-aration between the known and unknown target (𝐾𝑡) orΔ𝑘 = 𝐾𝑟− 𝐾^𝑡. While the range sampling capability 𝑟𝑘 of the LADAR is fixed by the receiver electronics, the unknown target 𝐾𝑡could occur anywhere within the range gate to include ranges between samples.

Also, the spatial point targets are defined by Kronecker delta functions𝛿 (𝑚 − Δ^𝑚, 𝑛) and 𝛿 (𝑚, 𝑛) and ℎ (𝑥, 𝑦) is the known system PSF. The final term is the pixel bias 𝐵 (𝑥, 𝑦) and is intended to account for any ambient light, dark current, electron noise, and pixel-to-pixel impulse response variations. This bias is assumed known and to be governed by the Poisson distribution due to the discrete, random nature of these noise sources. Concerning the validity of the assuming a known pixel bias, it is target independent and can be sepa-rately determined during LADAR operation by a calibration step where the data is collected without activating the laser.

Performing the convolution in Equation (4.18) results in the simplified form

𝑖𝑘(𝑥, 𝑦) = 𝐴𝑡𝑝 (𝑟𝑘− (𝐾^𝑟− Δ^𝑘)) ℎ (𝑥 − Δ^𝑚, 𝑦) + 𝐴𝑟𝑝 (𝑟𝑘− 𝐾^𝑟) ℎ (𝑥, 𝑦) + 𝐵 (𝑥, 𝑦) (4.21) where the received pulse shapes are assumed symmetric Gaussian and defined by

𝑝 (𝑟𝑘) = 1

√2𝜋𝜎𝑝𝑑

exp

{−(𝑟^𝑘)² 2𝜎²_𝑝𝑑

}

(4.22)

with 𝜎𝑝𝑑 as the pulse-width standard deviation in units of meters and defined as 𝜎𝑝𝑑 = 𝑐𝜎𝑝𝑡/2 where 𝜎𝑝𝑡 is the pulse-width standard deviation in units of seconds. Gaussian-shaped pulses are a valid approximation for the pulse shapes transmitted from 3D FLASH

definition is changed such that there are two pulse-shape standard deviations concerning Equation (4.22): one for the leading edge (pre-target) and another for the trailing edge (post-target). Although the effects of asymmetrical pulses on the CRB and range sepa-ration estimation is a source of additional research, the symmetry or lack thereof in the received pulses does not change the conclusion that an optimal pulse exists given the range resolution metric. Symmetrical pulse-shapes are assumed for simplicity and are simply a subset of asymmetrical pulse-shapes.

Furthermore, a spatially, invariant 2D Gaussian PSF is chosen because its differenti-ation is straight-forward while still providing a function to sufficiently blur a target scene.

This type of impulse response has been used previously to describe blurring due to atmo-spheric turbulence [37]. The PSF is defined as

ℎ (𝑥, 𝑦) = 1

2𝜋𝜎_ℎ² exp{ −(𝑥²+ 𝑦²) 2𝜎²_ℎ

}

(4.23)

where𝜎ℎ > 0 is the PSF standard deviation (measured in units of pixels) and is affected by light diffraction effects, receiver optic’s quality, and atmospheric turbulence.

4.4.2 Estimator Derivation. Given the variable definitions from Equations (4.18)-(4.23), the sum squared error term is the sum of the square of the difference between the observed data and the estimate or

𝐸 (Δ𝑘) =∑

𝑘

∑

𝑥

∑

𝑦

(𝑑𝑘(𝑥, 𝑦) − 𝑖^𝑘(𝑥, 𝑦))² (4.24)

where the dependence on Δ𝑘 will subsequently be dropped for conciseness. There are four unknowns including the two amplitudes, range separation, and spatial separation. The procedure to find the range and spatial separation estimates is to iteratively step through each possible combination of range and spatial separation values (these values are known a priori) and then determine the amplitudes that minimize the error. After an exhaustive search of combinations of range and spatial separations, the combination that results in the least sum square error is chosen as the estimates.

For a particular amplitude𝐴𝑖, the approach is to take the derivative of the error (Equa-tion (4.24)) with respect to that amplitude, set the result equal to zero,∂𝐸/∂𝐴_𝑖 = 0, and solve for the amplitude term. This method gives the amplitude value that minimizes the error term due to the positivity of the second derivative. Since there are two amplitudes, a well-posed system of equations is set up by performing∂𝐸/∂𝐴𝑡 = 0 and ∂𝐸/∂𝐴𝑟 = 0 and solving for𝐴_𝑡and𝐴_𝑟respectively resulting in two equations and two unknowns shown by

𝐶11𝐴𝑡+ 𝐶12𝐴𝑟 = 𝐷1

The amplitudes are then determined by solving the system of equations.

The following provides the estimation steps:

1. Select a range separation,Δ𝑘

2. Select a spatial separation,Δ𝑚

3. Determine the estimates for amplitudes, 𝐴𝑡 and 𝐴𝑟, via the system of equations in (4.25)

6. Select range and spatial separation corresponding to the smallest error