## Abstract

Imaging systems designed for point target detection and 3D reconstruction must be filtered in order to maximize signal-to-noise ratio and minimize position expectation error. An optimal whitening matched filter (WMF) is derived based on expected spatial target distribution and system colored noise. The expected noise is derived as a weighted combination of clutter aliasing, target aliasing, and detector noise. Further optimization of system performance is achieved by modification of the optical point spread function (PSF), so a sampling-balanced operation is achieved where all noise components are comparable. The improved performance of the optimized system is calculated and compared to the performance of other systems using other known linear postfilters with various optical PSF widths. It is shown that the WMF in a sampling-balanced system is a robust configuration that needs only minor modifications when scenario parameters are varied.

© 2010 Optical Society of America

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### Equations (21)

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(1)
$${i}_{r}(x)={i}_{c}(x)+{i}_{t}(x),$$
(2)
$${p}_{s}(x)={p}_{d}(x)\otimes {p}_{o}(x),$$
(3)
$${p}_{d}(x)\equiv \frac{r(x)}{\int r(x)\mathrm{d}x}=\frac{r(x)}{{S}_{d}}\mathrm{.}$$
(4)
$${i}_{pr}(x)={p}_{s}\otimes {i}_{r}(x)={i}_{pc}(x)+{i}_{pt}(x)\mathrm{.}$$
(5)
$${i}_{sr}(x)={S}_{d}\xb7{i}_{pr}(x)\xb7\text{comb}(x/{X}_{d})+{n}_{d}(x)\xb7\text{comb}(x/{X}_{d}),$$
(6)
$${n}_{s}(x)\equiv {i}_{sr}(x)-{i}_{pr}(x)\mathrm{.}$$
(7)
$${n}_{s}(x)={n}_{u}(x)+{n}_{ta}(x)\mathrm{.}$$
(8)
$${n}_{u}(x)\equiv \lfloor {S}_{d}\xb7{i}_{pc}(x)+{n}_{d}(x)\rfloor \xb7\text{comb}(x/{X}_{d})-{i}_{pc}(x),$$
(9)
$${n}_{ta}(x)\equiv {S}_{d}\xb7{i}_{pt}(x)\xb7\text{comb}(x/{X}_{d})-{i}_{pt}(x)\mathrm{.}$$
(10)
$${i}_{k}=k\otimes {i}_{sr}\equiv {i}_{kt}+{n}_{k},$$
(11)
$${N}_{u}(f)=\u3008|\mathrm{CFT}\{{n}_{u}\}{|}^{2}\u3009=\stackrel{\infty}{\underset{m\ne 0}{\sum _{m=-\infty}}}|{I}_{pc}(f-m\xb7{X}_{d}){|}^{2}+{N}_{0}=\stackrel{\infty}{\underset{m\ne 0}{\sum _{m=-\infty}}}|{P}_{s}(f-m\xb7{X}_{d}){|}^{2}\xb7|{I}_{c}(f-m\xb7{X}_{d}){|}^{2}+{N}_{0},$$
(12)
$${N}_{ta}(f)=\u3008|\mathrm{CFT}\{{n}_{ta}\}{|}^{2}\u3009=\stackrel{\infty}{\underset{m\ne 0}{\sum _{m=-\infty}}}|{I}_{pt}(f-m\xb7{X}_{d}){|}^{2}=\stackrel{\infty}{\underset{m\ne 0}{\sum _{m=-\infty}}}|{P}_{s}(f-m\xb7{X}_{d}){|}^{2}\xb7|{I}_{t}(f-m\xb7{X}_{d}){|}^{2},$$
(13)
$${N}_{sc}(f)={N}_{u}(f)+{C}_{p}\xb7{N}_{ta}(f),$$
(14)
$${C}_{p}\equiv \frac{\text{image area}}{\text{target erea}}=\frac{{X}_{d}^{2}\xb7M}{{S}_{W}({i}_{pt}{)}^{2}},$$
(15)
$${S}_{W}({i}_{pt})=\frac{\int |{i}_{pt}{|}^{2}\mathrm{d}x}{\mathrm{max}|{i}_{pt}{|}^{2}}\mathrm{.}$$
(16)
$$K(f)=\frac{{I}_{pt}(f)}{{N}_{sc}(f)}\mathrm{.}$$
(17)
$$k(x)=\mathrm{CIFT}\{K(f)\},$$
(18)
$${\mathrm{STC}}^{n}\equiv \frac{\mathrm{max}\{{i}_{pt}^{n}\}}{\mathrm{STD}\{{n}_{pc}^{n}\}},$$
(19)
$${\mathrm{SNR}}^{n}=\frac{\mathrm{max}\{{i}_{pt}^{n}\}}{\mathrm{STD}\{{n}_{d}\}}\mathrm{.}$$
(20)
$$\mathrm{SNR}=\frac{\mathrm{max}\{{i}_{kt}\}}{\mathrm{STD}\{{n}_{k}\}}\mathrm{.}$$
(21)
$${\sigma}_{\text{target}}=\frac{1}{2\pi \xb7\mathrm{SNR}\xb7{B}_{G}},$$