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.

© 2002 Optical Society of America

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References

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  1. J. B. Wilburn, “Developments in generalized ranked-order filters,” J. Opt. Soc. Am. A 15, 1054–1099 (1998).
    [CrossRef]
  2. J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 233–332.
  3. P. R. Halmos, Naïve Set Theory (Van Nostrand, Princeton, N.J., 1960), Chaps. 7, 8.
  4. J. R. Schoenfeld, Mathematical Logic (Addison-Wesley, Reading, Mass., 1967), Chaps. 6, 7.
  5. A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
    [CrossRef]
  6. D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
    [CrossRef]
  7. H. G. Longbotham, “Theory of order statistic filters and their relationship to FIR filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 275–287 (1989).
    [CrossRef]
  8. J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 255–281.

1998 (1)

J. B. Wilburn, “Developments in generalized ranked-order filters,” J. Opt. Soc. Am. A 15, 1054–1099 (1998).
[CrossRef]

1991 (1)

D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
[CrossRef]

1989 (1)

H. G. Longbotham, “Theory of order statistic filters and their relationship to FIR filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 275–287 (1989).
[CrossRef]

1983 (1)

A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
[CrossRef]

Aragon, J.

D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
[CrossRef]

Bovik, A.

A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
[CrossRef]

Eberly, D.

D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
[CrossRef]

Halmos, P. R.

P. R. Halmos, Naïve Set Theory (Van Nostrand, Princeton, N.J., 1960), Chaps. 7, 8.

Huang, T. S.

A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
[CrossRef]

Longbotham, H. G.

D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
[CrossRef]

H. G. Longbotham, “Theory of order statistic filters and their relationship to FIR filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 275–287 (1989).
[CrossRef]

Muson, D.

A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
[CrossRef]

Schoenfeld, J. R.

J. R. Schoenfeld, Mathematical Logic (Addison-Wesley, Reading, Mass., 1967), Chaps. 6, 7.

Wilburn, J. B.

J. B. Wilburn, “Developments in generalized ranked-order filters,” J. Opt. Soc. Am. A 15, 1054–1099 (1998).
[CrossRef]

J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 233–332.

J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 255–281.

IEEE Trans. Acoust. Speech Signal Process. (3)

A. Bovik, T. S. Huang, D. Muson, “A generalization of median filtering using linear combinations of order statistics,” IEEE Trans. Acoust. Speech Signal Process. 31, 1342–1350 (1983).
[CrossRef]

D. Eberly, H. G. Longbotham, J. Aragon, “Complete classification of roots to 1-dimensional median and ranked-order filters,” IEEE Trans. Acoust. Speech Signal Process. 39, 197–200 (1991).
[CrossRef]

H. G. Longbotham, “Theory of order statistic filters and their relationship to FIR filters,” IEEE Trans. Acoust. Speech Signal Process. 37, 275–287 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. B. Wilburn, “Developments in generalized ranked-order filters,” J. Opt. Soc. Am. A 15, 1054–1099 (1998).
[CrossRef]

Other (4)

J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 233–332.

P. R. Halmos, Naïve Set Theory (Van Nostrand, Princeton, N.J., 1960), Chaps. 7, 8.

J. R. Schoenfeld, Mathematical Logic (Addison-Wesley, Reading, Mass., 1967), Chaps. 6, 7.

J. B. Wilburn, Theory of Ranked-Order Filters with Applications to Feature Extraction and Interpretive Transforms, Vol. 112 in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic, San Diego, Calif., 2000), pp. 255–281.

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Figures (8)

Fig. 1
Fig. 1

Cross filter.

Fig. 2
Fig. 2

X filter.

Fig. 3
Fig. 3

Octagonal filter.

Fig. 4
Fig. 4

Hexagonal filter.

Fig. 5
Fig. 5

Example applications of the octagonal local maximum and threshold filters.

Fig. 6
Fig. 6

Dynamic scene with moving target: First sequential image pair. (a) Simulation of view of Earth at nadir from an orbital sensor. (b) Filtered image.

Fig. 7
Fig. 7

Dynamic scene with moving target: Second sequential image pair; (a) and (b) as in Fig. 6.

Fig. 8
Fig. 8

Dynamic scene with moving target: third sequential image pair; (a) and (b) as in Fig. 6.

Tables (2)

Tables Icon

Table 1 Pd and Pfa of the Local Maximum Filter at SNR = 0 – 7.5

Equations (47)

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S1=000453568437463000,
000[40513253648546]37463000.
1st application:000[850641512533404465326]37463000.
Cond.A:j=06,
Cond.B:l=0andj=k,k=3.
2nd application,1st increment: 0004[840631502523454315366]7463000.
2nd increment:00045[830761622513444305356]463000.
ϕ(N) * S1=000008000000,
45[30516283848576]463,
45[830841852763624515306]463,
Sn=010132453568437463576452101021000343010,
Σn=000000000008000000070000000020000040000.
Σn=000000000008000000070000000000000040000.
Dn=000000450568407460576450000000000040000.
u=(u1)ν1, (u2)ν2,, (un)νn.
ui; (ui)=((ui), (ui+1),, (ui+2k)), vi=νi, νi+1,, νi+2k.
β((ui), j)=(ui)j,
β(vi, j)=vji,
(ui)j=((ui)0,, (ui+2k)2k),
vji=ν0i,, ν2ki+2k,
ui=(ui)jvji.
u=i=1n-2k ui.
KR(a)=0ifR(a)=1if¬R(a),
(a)˜R(a).
(a1)˜R(a1),(a2)˜R(a2),, (aq)˜R(aq)
I(u1, u1)=(u1, u1)iff R(u1)
I(un-2k, un-2k)=(un-2k, un-2k)iff R(un-2k),
I(u, u)=i=1n-2k I(ui, ui).
(ui, ui)=μx[H(G(ui, x), vi)].
πm2k(xα0, xβ1,, xq2k)=xjm
O(x)=(x)αηα0, (x)βηβ1,, (x)qηq2k,
(x)α(x)β  (x)q
O(ui)=(ui+0)jηj0,, (ui+2k)jηj2k.
G(ui, (ui)jηjm)=[πm2k(O(ui))].
H(G(ui, (ui)kνki), vi)=μxx<N[x=k & (ui)xνxi=(ui)jηjm  (ui)xνxi=(0)xνxi].
(ui, ui)=(0)0ν0i,, (0)k-1νk-1i, (ui)jηjm,(0)k+1νk+1i,, (0)2kν2ki.
I(u, u)=(u1, u1)·K¬R(u1)(un-2k, un-2k)·K¬R(un-2k),k=(N-1)/2,
I(u, u)=i=1n-2k (ui, ui)·K¬R(ui).
Cond.A:ui istrueRA(Mx)(ui):ui=(ui)jvji,j=02k,N=2k+1.
Cond.B(ordinarymaximum):(ui)kνki=(ui)jηj0RB(OMx)(ui):ηjm,m=0,j=02k,
Cond.B(localmaximum):(ui)kνki=(ui)kηk0RB(LMx)(ui):ηjm,m=0,j=k.
Cond.C(maximum):RC(OMx)(ui):(ui)jηj0Thresh.
Cond.A:ui istrueRA(Mx)(ui):ui=(ui)jvji,j=02k,N=2k+1,
Cond.B(ordinarymedian):(ui)kνki=(ui)jηjkRB(OMd)(ui):ηjm,m=k,j=02k,
Cond.B(localmedian):(ui)kνki=(ui)kηkkRB(LMd)(ui):ηjm,m=k,j=k.
X×Y=[ζ|δX, γY, ζδ, γ].

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