Development of the local maximum variety of ranked-order filters

John Bart Wilburn

doi:10.1364/JOSAA.19.001994

Journal of the Optical Society of America A
Vol. 19,
Issue 10,
pp. 1994-2004
(2002)
•https://doi.org/10.1364/JOSAA.19.001994

Development of the local maximum variety of ranked-order filters

John Bart Wilburn

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John Bart Wilburn, "Development of the local maximum variety of ranked-order filters," J. Opt. Soc. Am. A 19, 1994-2004 (2002)

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Abstract

A form of ranked-order filters is introduced as the local maximum filter. The construction of the local maximum filter is described, followed by a discussion of its function and some of its more important properties, and an example application of a two-dimensional local maximum filter is provided to illustrate the detection of single-pixel targets against a cloud clutter background. The closing discussion provides a mathematical development of the filter.

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SNR	Local Maximum Filter
	$ϑ_{+} (17) * I$		$ϑ_{oct} (33) * I$		$ϑ_{oct} (33) * L (I)$
	$P_{d}$	$P_{fa}$	$P_{d}$	$P_{fa}$	$P_{d}$	$P_{fa}$
0	0	$4.00 \times 10^{- 2}$	0	$1.73 \times 10^{- 2}$	0	$2.30 \times 10^{- 2}$
1.5	0.08	$3.77 \times 10^{- 2}$	0.04	$1.68 \times 10^{- 2}$	0.20	$1.76 \times 10^{- 2}$
2.0	0.42	$2.29 \times 10^{- 2}$	0.31	$1.18 \times 10^{- 2}$	0.54	$7.85 \times 10^{- 3}$
2.5	0.62	$1.24 \times 10^{- 2}$	0.55	$6.40 \times 10^{- 3}$	0.66	$5.89 \times 10^{- 3}$
3.0	0.70	$9.87 \times 10^{- 3}$	0.62	$5.25 \times 10^{- 3}$	0.79	$3.83 \times 10^{- 3}$
3.5	0.78	$7.19 \times 10^{- 3}$	0.71	$4.11 \times 10^{- 3}$	0.85	$2.26 \times 10^{- 3}$
4.0	0.86	$4.50 \times 10^{- 3}$	0.83	$2.78 \times 10^{- 3}$	0.87	$1.37 \times 10^{- 3}$
4.5	0.91	$3.45 \times 10^{- 3}$	0.86	$2.01 \times 10^{- 3}$	0.90	$9.78 \times 10^{- 4}$
5.0	0.92	$2.97 \times 10^{- 3}$	0.88	$1.72 \times 10^{- 3}$	0.92	$7.81 \times 10^{- 4}$
5.5	0.94	$2.21 \times 10^{- 3}$	0.91	$1.36 \times 10^{- 3}$	0.93	$4.86 \times 10^{- 4}$
6.0	0.96	$1.34 \times 10^{- 3}$	0.93	$6.65 \times 10^{- 4}$	0.94	$4.87 \times 10^{- 4}$
6.5	0.97	$1.05 \times 10^{- 3}$	0.94	$5.71 \times 10^{- 4}$	0.94	$3.89 \times 10^{- 4}$
7.0	0.98	$8.63 \times 10^{- 4}$	0.96	$4.77 \times 10^{- 4}$	0.95	$2.91 \times 10^{- 4}$
7.5	0.99	$5.76 \times 10^{- 4}$	0.98	$3.82 \times 10^{- 4}$	0.96	$1.93 \times 10^{- 4}$

SNR	Local Maximum Filter
	$ϑ_{+} (17) * I$		$ϑ_{oct} (33) * I$		$ϑ_{oct} (33) * L (I)$
	$P_{d}$	$P_{fa}$	$P_{d}$	$P_{fa}$	$P_{d}$	$P_{fa}$
0	0	$4.00 \times 10^{- 2}$	0	$1.73 \times 10^{- 2}$	0	$2.30 \times 10^{- 2}$
1.5	0.08	$3.77 \times 10^{- 2}$	0.04	$1.68 \times 10^{- 2}$	0.20	$1.76 \times 10^{- 2}$
2.0	0.42	$2.29 \times 10^{- 2}$	0.31	$1.18 \times 10^{- 2}$	0.54	$7.85 \times 10^{- 3}$
2.5	0.62	$1.24 \times 10^{- 2}$	0.55	$6.40 \times 10^{- 3}$	0.66	$5.89 \times 10^{- 3}$
3.0	0.70	$9.87 \times 10^{- 3}$	0.62	$5.25 \times 10^{- 3}$	0.79	$3.83 \times 10^{- 3}$
3.5	0.78	$7.19 \times 10^{- 3}$	0.71	$4.11 \times 10^{- 3}$	0.85	$2.26 \times 10^{- 3}$
4.0	0.86	$4.50 \times 10^{- 3}$	0.83	$2.78 \times 10^{- 3}$	0.87	$1.37 \times 10^{- 3}$
4.5	0.91	$3.45 \times 10^{- 3}$	0.86	$2.01 \times 10^{- 3}$	0.90	$9.78 \times 10^{- 4}$
5.0	0.92	$2.97 \times 10^{- 3}$	0.88	$1.72 \times 10^{- 3}$	0.92	$7.81 \times 10^{- 4}$
5.5	0.94	$2.21 \times 10^{- 3}$	0.91	$1.36 \times 10^{- 3}$	0.93	$4.86 \times 10^{- 4}$
6.0	0.96	$1.34 \times 10^{- 3}$	0.93	$6.65 \times 10^{- 4}$	0.94	$4.87 \times 10^{- 4}$
6.5	0.97	$1.05 \times 10^{- 3}$	0.94	$5.71 \times 10^{- 4}$	0.94	$3.89 \times 10^{- 4}$
7.0	0.98	$8.63 \times 10^{- 4}$	0.96	$4.77 \times 10^{- 4}$	0.95	$2.91 \times 10^{- 4}$
7.5	0.99	$5.76 \times 10^{- 4}$	0.98	$3.82 \times 10^{- 4}$	0.96	$1.93 \times 10^{- 4}$

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Journal of the Optical Society of America A