Abstract

In image recognition applications, complex decision regions in the image space are needed. Linear filtering forms the decision regions by hyperplanes in the image space. We determine the decision region formed by Fourier-plane nonlinear filtering. In the case in which power law nonlinearity is applied in the Fourier plane, the decision region turns out to be approximately an n-dimensional parabola that opens toward the direction of the reference vector. That is, the intersection of the decision region with any plane (two-dimensional vector space) not containing any vector parallel to the reference vector is a bounded convex region enclosed by a closed curve. The size of the convex region depends on the filter nonlinearity, which determines the distortion robustness and discrimination capability of the filter. It can be adjusted by choosing different Fourier-plane nonlinearities and/or different threshold values at the output plane. These types of regions are desirable and well suited in image recognition. Analytical and numerical solutions are provided.

© 1999 Optical Society of America

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References

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  2. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
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    [CrossRef]
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    [CrossRef] [PubMed]
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1998

1996

1995

1994

1993

1992

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

1990

P. Refregier, “Filter design for optical image recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef] [PubMed]

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

1989

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 1511, 77 (1989).

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

1984

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

1982

1969

April, G.

Arsenault, H. H.

Casasent, D.

D. Weber, D. Casasent, “Quadratic filters for object classification and detection,” in Optical Pattern Recognition, D. Casasent, T. Chao, eds., Proc. SPIE3073, 2–13 (1997).
[CrossRef]

Caufield, H. J.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Fazlolahi, A.

Fazlollahi, A. H.

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 1511, 77 (1989).

Gianino, P. D.

Gonzalez, R. C.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).

Goudaul, F.

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Horner, J.

Horner, J. L.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 1511, 77 (1989).

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

Hsu, Y.-N.

Javidi, B.

Laude, V.

Li, H.-Y.

Li, J.

Mahalonobis, A.

A. Mahalonobis, “Review of correlation filters and their application scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. CR65 of Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 240–260.

Maloney, W. T.

Nadler, M.

M. Nadler, E. P. Smith, Pattern Recognition Engineering (Wiley, New York, 1993).

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Painchaud, D.

Psaltis, D.

H.-Y. Li, Y. Qiao, D. Psaltis, “Optical network for real-time face recognition,” Appl. Opt. 32, 5026–5035 (1993).
[CrossRef] [PubMed]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Qiao, Y.

Refregier, P.

Réfrégier, Ph.

Smith, E. P.

M. Nadler, E. P. Smith, Pattern Recognition Engineering (Wiley, New York, 1993).

Tang, Q.

Tou, J. T.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974).

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Wang, J.

Weber, D.

D. Weber, D. Casasent, “Quadratic filters for object classification and detection,” in Optical Pattern Recognition, D. Casasent, T. Chao, eds., Proc. SPIE3073, 2–13 (1997).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical image correlation with binary spatial light modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Opt. Lett.

Proc. IEEE

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 1511, 77 (1989).

Other

A. Mahalonobis, “Review of correlation filters and their application scene matching,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Vol. CR65 of Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 240–260.

D. Weber, D. Casasent, “Quadratic filters for object classification and detection,” in Optical Pattern Recognition, D. Casasent, T. Chao, eds., Proc. SPIE3073, 2–13 (1997).
[CrossRef]

M. Nadler, E. P. Smith, Pattern Recognition Engineering (Wiley, New York, 1993).

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1967).

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Figures (13)

Fig. 1
Fig. 1

Fourier-plane binary nonlinear filtering forms an elliptic cone-shaped decision region. The intersection of this region with any hyperplane that is parallel to two arbitrary axes is a real ellipse.

Fig. 2
Fig. 2

Regular parabola with vertex vector r supported by hyperplane L.

Fig. 3
Fig. 3

Skewed parabola with vertex vector r supported by hyperplane L.

Fig. 4
Fig. 4

Image of a tank used as a reference signal.

Fig. 5
Fig. 5

Output values of the Fourier-plane nonlinear filter [Eq. (13), with k=0]. Contour lines were formed by thresholding the output of the nonlinear filter with use of threshold values for T ranging from 0.75 to 0.95 in increments of 0.5 [Eq. (15)]. The x and y axes are the values of the input image at two different pixels. The values of the input image are the same at all other pixels. Contour lines are the output values obtained with Eq. (13).

Fig. 6
Fig. 6

Output values of the Fourier-plane nonlinear filter [Eq. (13), with k=0.3]. Contour lines were formed by thresholding the output of the nonlinear filter with use of threshold values for T ranging from 0.75 to 0.95 in increments of 0.5 [Eq. (15)]. The x and y axes are the values of the input image at two different pixels. The values of the input image are the same at all other pixels. Contour lines are the output values obtained with Eq. (13).

Fig. 7
Fig. 7

Cross sections of the output values for different values of k ranging from 0 to 1 and T=0.97. Contour lines indicate an output value of 0.97 obtained by using Eq. (13) for different values of k. The x and y axes are values of the input image at two different pixels. The values of the input image are the same at all other pixels.

Fig. 8
Fig. 8

Decision region of nonlinear filtering intersected by a series of planes that are parallel to an arbitrary xy plane. The decision region is obtained by thresholding the output of the Fourier-plane nonlinear filter [Eq. (15), with nonlinearity index k=0] at threshold value T=0.95. The x, y, and z coordinates are the values of the input signals at three different pixels. The intersecting planes are given by equations z=2.5, z=5, , z=60.

Fig. 9
Fig. 9

Comparison of the decision region of nonlinear filtering intersected by a series of arbitrary planes parallel to the xy plane, with use of two different threshold values. The decision region in (a) was obtained by thresholding the output of the Fourier-plane nonlinear filter [Eq. (15), with nonlinearity index k=0] at threshold value T=0.98. The decision region in (b) was obtained by using a threshold value of T=0.95. The x, y, and z coordinates are the values of the input signals at three different pixels. The intersecting planes are given by the equations z=2.5, z=5, , z=60.

Fig. 10
Fig. 10

Image of a Mig jet used as a reference, or target, image.

Fig. 11
Fig. 11

Output values of the Fourier-plane nonlinear filter [Eq. (13), with k=0]. Contour lines were formed by thresholding the output of the nonlinear filter with use of threshold values for T ranging from 0.75 to 0.95 in increments of 0.5 [Eq. (15)]. The x and y axes are the values of the input image at two different pixels. The values of the input image are the same at all other pixels. Contour lines are the output values obtained with Eq. (13).

Fig. 12
Fig. 12

Output values of the nonlinear filter [Eq. (13), with nonlinearity index k=0.3]. Contour lines were formed by thresholding the output of the nonlinear filter with use of threshold values T=0.85, 0.9, and 0.95 [Eq. (15)]. The x and y axes are the values of the input image at two different pixels. The values of the input image are the same at all other pixels. Contour lines are the output values obtained with Eq. (13).

Fig. 13
Fig. 13

Cross sections of the output values for different values of k ranging from 0 to 1 and T=0.99. Contour lines indicate an output value of 0.97 obtained by using Eq. (13) for different values of k. The x and y axes are values of the input image at two different pixels. The values of the input image are the same at all other pixels.

Equations (39)

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R={s:Y(s)>T},
s·r(s)(r)>T,0<T<1,
s·r=s1r1+s2r2,
v=(v12v12+v22v22)1/2.
R=s:s·r(s)(r)>T.
R={s:s·v>0}{s:s·w<0},
v=1-T2T(r1, r2)+(-r2, r1),
w=1-T2T(r1, r2)+(r2,-r1).
g=sgn(r·s-T),
r·s=Torj=0n-1ri sj=T.
Rj=l=0n-1r0 exp2πiljn,j=0, 1, , n-1.
S·R=real(S)·real(R)+imag(S)·imag(R).
Gj=|RjSj|k-1Rj·Sj,j=0, 1, , n-1.
Y(s)=1nj=0n-1Gj=1nj=0n-1|RjSj|k-1Rj·Sj.
R=s:1nj=0n-1|RjSj|k-1Rj·Sj>T.
R=s:1nj=0n-1|RjSj|k-1Rj·Sj=T.
S·R=Real(S)·Real(R)+Imag(S)·Imag(R).
Gj=|RjSj|k-1Rj·Sj,j=0, 1, , n-1.
Y(s)=1nj=0n-1Gj=1nj=0n-1|RjSj|k-1Rj·Sj.
R=s:1nj=0n-1|RjSj|k-1Rj·Sj>T.
R=s:1nj=0n-1|RjSj|k-1Rj·Sj=T.
L={m+v:mM0}.
R=s:Y(s)=1nj=0n-1[cos(ϕRj)cos(ϕSj)+sin(ϕRj)sin(ϕSj)]>T,
R=s:Y(s)=1nj=0n-1[cos(ϕRj)cos(ϕSj)+sin(ϕRj)sin(ϕSj)]=T,
wj=exp(iθj)ej.
1nj=0n-1[cos(ϕRj)cos(ϕwj)+sin(ϕRj)sin(ϕwj)]=0.
L={F-1(w):wLw},
Y(s)=1nj=0n-1[cos(ϕRj)cos(ϕSj)+sin(ϕRj)sin(ϕSj)]=0,foranysL.
F(Δ)=Y(pr+Δ),
F(qΔ)=(1/n)j=0n-1 Rj·(Rj+qdj)|Rj||Rj+qdj|,
Aj(q)=Rj·(Rj+qdj)|Rj||Rj+qdj|.
Y(s)=1nj=0n-1(|Rj|-1|Rj+qdj|-1(|Rj|2+Rj·qdj).
I=1nj=0n-1(|Rj||Rj+qdj|-1,
II=qnj=0n-1|Rj|-1|Rj+qdj|-1(Rj·dj).
Y(s)>T,when|q|issufficientlysmall.
q|Rj+qdj|-11|dj|1-|Rj|2+Rj·2qdj|2qdj|2.
1nj=0n-1|Rj|-1|dj|-1(Rj·dj)=Y(Δ)(sinceΔM0)=0.
Y(r+qΔ)<TforanyvectorΔM0,providedthat|q|islarge.
Y(pr+qΔ)<TforanyvectorΔM0,providedthat|q|islarge.

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