Abstract

This paper introduces a computationally efficient algorithm for synthesis of a distortion tolerant correlation filter and associated threshold, denoted collectively as the enhanced matched filter (EMF). Application areas of EMF include imagery based automatic target detection and recognition and biometrics. The EMF is synthesized from a set of training images characterizing the target of interest within the expected distortion range. A distinguishing feature of EMF is the ascribed threshold, which is a byproduct of the filter computation process and does not rely on nontarget trainers. The EMF performance is compared to that of the synthetic discriminant function using realistic test scenarios.

© 2013 Optical Society of America

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References

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  1. G. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
    [CrossRef]
  2. A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
    [CrossRef]
  3. L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
    [CrossRef]
  4. A. Papoulis, Signal Analysis (McGraw-Hill, 1977).
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).
  6. A. VanderLugt, Optical Signal Processing (Wiley, 1992).
  7. A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
    [CrossRef]
  8. D. Hao, S. M. Seitz, and N. Snavely, “The dimensionality of scene appearance,” in Proceedings of the 12th International Conference on Computer Vision (IEEE, 2009), pp. 1917–1924.
  9. T. M. Caelli and Z. Q. Liu, “On the minimum number of templates required for shift, rotation and size invariant pattern recognition,” Pattern Recogn. 21, 205–216 (1988).
    [CrossRef]
  10. H. J. Caulfield and W. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef]
  11. H. J. Caulfield and M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
    [CrossRef]
  12. K. Heidary and H. J. Caulfield, “Application of supergeneralized matched filters to target classification,” Appl. Opt. 44, 47–54 (2005).
    [CrossRef]
  13. R. B. Johnson and K. Heidary, “A unified approach for database analysis and application to ATR performance metrics,” Proc. SPIE 7696, 76960Z (2011).
    [CrossRef]
  14. K. Heidary and H. J. Caulfield, “Needles in a haystack: fast spatial search for targets in similar-looking backgrounds,” J. Franklin Inst. 349, 2935–2955 (2012).
    [CrossRef]
  15. D. Casasent and D. Psaltis, “Scale invariant correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
    [CrossRef]
  16. T. D. Wilkinson, D. C. O’Brien, and R. J. Mears, “Scale-invariant binary phase-only matched filter using a ferroelectric-liquid-crystal spatial light modulator,” Appl. Opt. 33, 4452–4453 (1994).
    [CrossRef]
  17. D. Mendlovic, E. Marom, and N. Konforti, “Improved rotation or scale invariant matched filter,” Appl. Opt. 28, 3814–3819 (1989).
    [CrossRef]
  18. J. Rosen and J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef]
  19. C. F. Hester and D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef]
  20. A. Mahalanobis, B. V. K. Kumar, and D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef]
  21. Z. Bahri and B. V. K. Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. 5, 562–571(1988).
    [CrossRef]
  22. A. Mahalanobis, B. V. K. Kumar, S. Song, S. R. Sims, and J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
    [CrossRef]
  23. A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
    [CrossRef]
  24. S. H. Hong and B. Javidi, “Optimum nonlinear composite filter for distortion-tolerant pattern recognition,” Appl. Opt. 41, 2172–2178 (2002).
    [CrossRef]
  25. S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).
  26. J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
    [CrossRef]
  27. http://staff.science.uva.nl/~aloi/ .

2012 (1)

K. Heidary and H. J. Caulfield, “Needles in a haystack: fast spatial search for targets in similar-looking backgrounds,” J. Franklin Inst. 349, 2935–2955 (2012).
[CrossRef]

2011 (1)

R. B. Johnson and K. Heidary, “A unified approach for database analysis and application to ATR performance metrics,” Proc. SPIE 7696, 76960Z (2011).
[CrossRef]

2007 (1)

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

2005 (2)

J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
[CrossRef]

K. Heidary and H. J. Caulfield, “Application of supergeneralized matched filters to target classification,” Appl. Opt. 44, 47–54 (2005).
[CrossRef]

2002 (1)

2001 (1)

A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
[CrossRef]

1994 (4)

L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
[CrossRef]

T. D. Wilkinson, D. C. O’Brien, and R. J. Mears, “Scale-invariant binary phase-only matched filter using a ferroelectric-liquid-crystal spatial light modulator,” Appl. Opt. 33, 4452–4453 (1994).
[CrossRef]

A. Mahalanobis, B. V. K. Kumar, S. Song, S. R. Sims, and J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef]

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

1989 (2)

1988 (2)

Z. Bahri and B. V. K. Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. 5, 562–571(1988).
[CrossRef]

T. M. Caelli and Z. Q. Liu, “On the minimum number of templates required for shift, rotation and size invariant pattern recognition,” Pattern Recogn. 21, 205–216 (1988).
[CrossRef]

1987 (1)

1982 (1)

1980 (1)

1976 (1)

D. Casasent and D. Psaltis, “Scale invariant correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

1969 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

1960 (1)

G. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

Bahri, Z.

Z. Bahri and B. V. K. Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. 5, 562–571(1988).
[CrossRef]

Belhumeur, P. N.

A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
[CrossRef]

Bone, P.

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

Bower, M.

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

Burghouts, G. J.

J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
[CrossRef]

Caelli, T. M.

T. M. Caelli and Z. Q. Liu, “On the minimum number of templates required for shift, rotation and size invariant pattern recognition,” Pattern Recogn. 21, 205–216 (1988).
[CrossRef]

Casasent, D.

Caulfield, H. J.

Chatwin, C.

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

Cherry, R.

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

Day, N.

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

Epperson, J. F.

Forman, A. V.

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

Georghiades, A. S.

A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
[CrossRef]

Geusebroek, J. M.

J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).

Hao, D.

D. Hao, S. M. Seitz, and N. Snavely, “The dimensionality of scene appearance,” in Proceedings of the 12th International Conference on Computer Vision (IEEE, 2009), pp. 1917–1924.

Heidary, K.

K. Heidary and H. J. Caulfield, “Needles in a haystack: fast spatial search for targets in similar-looking backgrounds,” J. Franklin Inst. 349, 2935–2955 (2012).
[CrossRef]

R. B. Johnson and K. Heidary, “A unified approach for database analysis and application to ATR performance metrics,” Proc. SPIE 7696, 76960Z (2011).
[CrossRef]

K. Heidary and H. J. Caulfield, “Application of supergeneralized matched filters to target classification,” Appl. Opt. 44, 47–54 (2005).
[CrossRef]

Hester, C. F.

Hong, S. H.

Javidi, B.

Johnson, R. B.

R. B. Johnson and K. Heidary, “A unified approach for database analysis and application to ATR performance metrics,” Proc. SPIE 7696, 76960Z (2011).
[CrossRef]

Konforti, N.

Kriegman, D. J.

A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
[CrossRef]

Kumar, B. V. K.

Liu, Z. Q.

T. M. Caelli and Z. Q. Liu, “On the minimum number of templates required for shift, rotation and size invariant pattern recognition,” Pattern Recogn. 21, 205–216 (1988).
[CrossRef]

Mahalanobis, A.

Maloney, W.

Marom, E.

Mears, R. J.

Mendlovic, D.

Netishen, C.

L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
[CrossRef]

Novak, L. M.

L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
[CrossRef]

O’Brien, D. C.

Owirka, G.

L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
[CrossRef]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

Psaltis, D.

D. Casasent and D. Psaltis, “Scale invariant correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Rehman, S.

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

Rosen, J.

Seitz, S. M.

D. Hao, S. M. Seitz, and N. Snavely, “The dimensionality of scene appearance,” in Proceedings of the 12th International Conference on Computer Vision (IEEE, 2009), pp. 1917–1924.

Shamir, J.

Sims, S. R.

Smeulders, A. W. M.

J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
[CrossRef]

Snavely, N.

D. Hao, S. M. Seitz, and N. Snavely, “The dimensionality of scene appearance,” in Proceedings of the 12th International Conference on Computer Vision (IEEE, 2009), pp. 1917–1924.

Song, S.

Turin, G.

G. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

A. VanderLugt, Optical Signal Processing (Wiley, 1992).

Weinberg, M. H.

Wilkinson, T. D.

Young, R.

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

Appl. Opt. (10)

H. J. Caulfield and W. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef]

H. J. Caulfield and M. H. Weinberg, “Computer recognition of 2-D patterns using generalized matched filters,” Appl. Opt. 21, 1699–1704 (1982).
[CrossRef]

K. Heidary and H. J. Caulfield, “Application of supergeneralized matched filters to target classification,” Appl. Opt. 44, 47–54 (2005).
[CrossRef]

T. D. Wilkinson, D. C. O’Brien, and R. J. Mears, “Scale-invariant binary phase-only matched filter using a ferroelectric-liquid-crystal spatial light modulator,” Appl. Opt. 33, 4452–4453 (1994).
[CrossRef]

D. Mendlovic, E. Marom, and N. Konforti, “Improved rotation or scale invariant matched filter,” Appl. Opt. 28, 3814–3819 (1989).
[CrossRef]

J. Rosen and J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
[CrossRef]

C. F. Hester and D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef]

A. Mahalanobis, B. V. K. Kumar, and D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef]

A. Mahalanobis, B. V. K. Kumar, S. Song, S. R. Sims, and J. F. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef]

S. H. Hong and B. Javidi, “Optimum nonlinear composite filter for distortion-tolerant pattern recognition,” Appl. Opt. 41, 2172–2178 (2002).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. S. Georghiades, P. N. Belhumeur, and D. J. Kriegman, “From few to many: illumination cone models for face recognition under variable lighting and pose,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 643–660 (2001).
[CrossRef]

Int. J. Comput. Vis. (1)

J. M. Geusebroek, G. J. Burghouts, and A. W. M. Smeulders, “The Amsterdam library of object images,” Int. J. Comput. Vis. 61, 103–112 (2005).
[CrossRef]

IRE Trans. Inf. Theory (1)

G. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

J. Comput. Inf. Sci. (1)

S. Rehman, P. Bone, R. Young, and C. Chatwin, “Object detection and recognition in cluttered scenes using fully scale and in-plane invariant synthetic discriminant function filters,” J. Comput. Inf. Sci. 1, 15–18 (2007).

J. Franklin Inst. (1)

K. Heidary and H. J. Caulfield, “Needles in a haystack: fast spatial search for targets in similar-looking backgrounds,” J. Franklin Inst. 349, 2935–2955 (2012).
[CrossRef]

J. Opt. Soc. Am. (1)

Z. Bahri and B. V. K. Kumar, “Generalized synthetic discriminant functions,” J. Opt. Soc. Am. 5, 562–571(1988).
[CrossRef]

Opt. Commun. (1)

D. Casasent and D. Psaltis, “Scale invariant correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Pattern Recogn. (3)

A. Mahalanobis, A. V. Forman, N. Day, M. Bower, and R. Cherry, “Multi-class SAR ATR using shift-invariant correlation filters,” Pattern Recogn. 27, 619–626 (1994).
[CrossRef]

T. M. Caelli and Z. Q. Liu, “On the minimum number of templates required for shift, rotation and size invariant pattern recognition,” Pattern Recogn. 21, 205–216 (1988).
[CrossRef]

L. M. Novak, G. Owirka, and C. Netishen, “Radar target identification using spatial matched filters,” Pattern Recogn. 27, 607–617 (1994).
[CrossRef]

Proc. SPIE (1)

R. B. Johnson and K. Heidary, “A unified approach for database analysis and application to ATR performance metrics,” Proc. SPIE 7696, 76960Z (2011).
[CrossRef]

Other (5)

A. Papoulis, Signal Analysis (McGraw-Hill, 1977).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, 1996).

A. VanderLugt, Optical Signal Processing (Wiley, 1992).

D. Hao, S. M. Seitz, and N. Snavely, “The dimensionality of scene appearance,” in Proceedings of the 12th International Conference on Computer Vision (IEEE, 2009), pp. 1917–1924.

http://staff.science.uva.nl/~aloi/ .

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Figures (8)

Fig. 1.
Fig. 1.

Top and bottom rows show five samples of objects-138 and 160, respectively, from ALOI. All images are 144×192 pixels.

Fig. 2.
Fig. 2.

Top row and the left three images in the bottom row show the trainer set for a particular instantiation of the experiment. The rightmost image in the bottom row is the EMF correlation filter. The numbers at the top of trainers denote image indices from ALOI, and the number at the top of EMF is the nominal threshold.

Fig. 3.
Fig. 3.

Performance comparison of EMF and SDF.

Fig. 4.
Fig. 4.

Effect of noise on performance of SDF and EMF filters. The numbers in the legend denote image SNR.

Fig. 5.
Fig. 5.

Effect of noise on performance of SDF and EMF filters. The numbers in the legend denote image SNR.

Fig. 6.
Fig. 6.

Top row and left image in the bottom row show the trainer set for a particular instantiation of the experiment. The right image in the bottom row is the EMF correlation filter. The numbers at the top of trainers denote image indices from ALOI, and the number at the top of EMF is the nominal threshold.

Fig. 7.
Fig. 7.

Performance comparisons of EMF and SDF at different noise levels. The numbers in the legend denote image SNR.

Fig. 8.
Fig. 8.

Effect of number of trainers on the EMF performance. TCIU comprises the first 14 images of object-160 in ALOI. Test images consist of 71 images of ALOI object-138 and the nontrained-on target TCIU elements. The numbers in the legend denote number of trainers that are randomly selected from TCIU. Performance results are averaged across 100 trials of the experiment.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

{t˜q(m,n):1qQ,0mM1,0nN1},
tq(m,n)=tq(m,n)1MNn=0N1m=0M1t˜q(m,n)n=0N1m=0M1[t˜q(m,n)1MNn=0N1m=0M1t˜q(m,n)]2,
Λ=[λp,q]Q×Q,λp,p=1,λp,q=λq,p,
λp,q=maxm,ncp,q(m,n),
cp,q(m,n)=1MNv=0N1u=0M1Cp,q(u,v)ej2π(muM+nvN);0m,uM1,0n,vN1,
Cp,q(u,v)=Tp(u,v)Tq*(u,v),
Tp(u,v)=n=0N1m=0M1tp(m,n)ej2π(muM+nvN),
γq=1p=1QλpqQ+ϵ,
k:γkγqq1qQ,A(m,n)=Tk(m,n),
q(mq,nq):dq(mq,nq)dq(m,n),
dq(m,n)=1MNv=0N1u=0M1A(u,v)Tq*(u,v)ej2π(umM+vnN),
T^q(u,v)=ej2π(umqM+vnqN)Tq(u,v),
F(u,v)=q=1QγqT^q(u,v)1MNv=0N1u=0M1|q=1QγqT^q(u,v)|2,
f(m,n)=1MNv=0N1u=0M1F(u,v)ej2π(muM+nuN),
ρ=min1qQ{maxm,n[1MNv=0N1u=0M1F(u,v)Tq*(u,v)ej2π(muM+nvN)]},
w(m,n)={1;0mM1,0nN10;MmM¯1,NnN¯1.
Z(u,v)=S(u,v)F˜*(u,v),
A(u,v)=S(u,v)W*(u,v),
B(u,v)=S¯¯(u,v)W*(u,v),
S¯¯(u,v)=n=0N1m=0M1s2(m,n)ej2π(muM¯+nvN¯),
F˜(u,v)=n=0N1m=0M1f(m,n)ej2π(muM¯+nvN¯),
E(u,v)=B(u,v)1MNA(u,v).
c(m,n)=z(m,n)e(m,n),
z(m,n)=1MN¯v=0N¯1u=0M¯1Z(u,v)ej2π(muM¯+nvN¯),
e(m,n)=1MN¯v=0N¯1u=0M¯1E(u,v)ej2π(muM¯+nvN¯).
X=[τ1τ2τQ],
φ=Σ1X(XΣ1X)1δ,
φ=X(XX)1δ.

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