Abstract

A method for designing and implementing quadratic correlation filters (QCFs) for shift-invariant target detection in imagery is presented. The QCFs are quadratic classifiers that operate directly on the image data without feature extraction or segmentation. In this sense the QCFs retain the main advantages of conventional linear correlation filters while offering significant improvements in other respects. Not only is more processing required for detection of peaks in the outputs of multiple linear filters but choosing the most suitable among them is an error-prone task. All channels in a QCF work together to optimize the same performance metric and to produce a combined output that leads to considerable simplification of the postprocessing scheme. The QCFs that are developed involve hard constraints on the output of the filter. Inasmuch as this design methodology is indicative of the synthetic discriminant function (SDF) approach for linear filters, the filters that we develop here are referred to as quadratic SDFs (QSDFs). Two methods for designing QSDFs are presented, an efficient architecture for achieving them is discussed, and results from the Moving and Stationary Target Acquisition and Recognition synthetic aperture radar data set are presented.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  6. D. Torreiri, “A linear transform that simplifies and improves neural network classifiers,” presented at the International Conference on Neural Networks, Washington, D.C., 1996.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. Three Class Public MSTAR data set, CD version, released for the Defense Advanced Research Projects Agency by Veda, Inc., the Dayton Group, 1997.
  15. A. Mahalanobis, L. Ortiz, B. V. K. Vijaya Kumar, “Performance of the MACH/DCCF algorithms on the 10-class public release MSTAR data set,” in Algorithms for Synthetic Aperture Radar Imagery VI, E. G. Zelnio, eds., Proc. SPIE3721, 285–291 (1999).
    [CrossRef]

1999 (1)

1997 (1)

D. Casasent, R. Shenoy, “Feature space trajectory for distorted object classification and pose estimation in SAR,” Opt. Eng. 36, 2719–2728 (1997).
[CrossRef]

1994 (1)

1992 (1)

1986 (1)

1980 (1)

1964 (1)

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

Ahn, J.

B. Bhanu, J. Ahn, “A system for model-based recognition of articulated objects,” in International Conference on Pattern Recognition (IEEE Computer Society, Washington, D.C., 1998), pp. 1812–1815, 1998).

Bhanu, B.

B. Bhanu, J. Ahn, “A system for model-based recognition of articulated objects,” in International Conference on Pattern Recognition (IEEE Computer Society, Washington, D.C., 1998), pp. 1812–1815, 1998).

Casasent, D.

D. Casasent, R. Shenoy, “Feature space trajectory for distorted object classification and pose estimation in SAR,” Opt. Eng. 36, 2719–2728 (1997).
[CrossRef]

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

Chan, L. A.

L. A. Chan, S. Z. Der, N. M. Nasrabadi, “Neural based target detectors for multi-band infrared imagery,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 1–36.

Chellappa, R.

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

Der, S. Z.

L. A. Chan, S. Z. Der, N. M. Nasrabadi, “Neural based target detectors for multi-band infrared imagery,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 1–36.

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

Epperson, J.

Goudail, F.

Hester, C. F.

Mahalanobis, A.

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

A. Mahalanobis, L. Ortiz, B. V. K. Vijaya Kumar, “Performance of the MACH/DCCF algorithms on the 10-class public release MSTAR data set,” in Algorithms for Synthetic Aperture Radar Imagery VI, E. G. Zelnio, eds., Proc. SPIE3721, 285–291 (1999).
[CrossRef]

A. Mahalanobis, R. Muise, S. R. Stanfill, A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Eng. (to be published).

Mahmoud, Hesham

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

Muise, R.

A. Mahalanobis, R. Muise, S. R. Stanfill, A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Eng. (to be published).

Nasrabadi, N. M.

L. A. Chan, S. Z. Der, N. M. Nasrabadi, “Neural based target detectors for multi-band infrared imagery,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 1–36.

Ortiz, L.

A. Mahalanobis, L. Ortiz, B. V. K. Vijaya Kumar, “Performance of the MACH/DCCF algorithms on the 10-class public release MSTAR data set,” in Algorithms for Synthetic Aperture Radar Imagery VI, E. G. Zelnio, eds., Proc. SPIE3721, 285–291 (1999).
[CrossRef]

Page, V.

Redman, B.

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

Réfrégier, P.

Sharma, R.

J. Starch, R. Sharma, S. Shaw, “A unified approach to feature extraction for model based ATR,” in Algorithms for Synthetic Aperture Radar Imagery III, E. G. Zelnio, R. J. Douglass, eds., Proc. SPIE2757, 294–305 (1996).
[CrossRef]

Shaw, S.

J. Starch, R. Sharma, S. Shaw, “A unified approach to feature extraction for model based ATR,” in Algorithms for Synthetic Aperture Radar Imagery III, E. G. Zelnio, R. J. Douglass, eds., Proc. SPIE2757, 294–305 (1996).
[CrossRef]

Shenoy, R.

D. Casasent, R. Shenoy, “Feature space trajectory for distorted object classification and pose estimation in SAR,” Opt. Eng. 36, 2719–2728 (1997).
[CrossRef]

Sims, S. R. F.

Stanfill, S. R.

A. Mahalanobis, R. Muise, S. R. Stanfill, A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Eng. (to be published).

Starch, J.

J. Starch, R. Sharma, S. Shaw, “A unified approach to feature extraction for model based ATR,” in Algorithms for Synthetic Aperture Radar Imagery III, E. G. Zelnio, R. J. Douglass, eds., Proc. SPIE2757, 294–305 (1996).
[CrossRef]

Torreiri, D.

D. Torreiri, “A linear transform that simplifies and improves neural network classifiers,” presented at the International Conference on Neural Networks, Washington, D.C., 1996.

Van Nevel, A.

A. Mahalanobis, R. Muise, S. R. Stanfill, A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Eng. (to be published).

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Zheng, Q.

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

Appl. Opt. (3)

IEEE Trans. Inf. Theory (1)

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

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

Opt. Eng. (1)

D. Casasent, R. Shenoy, “Feature space trajectory for distorted object classification and pose estimation in SAR,” Opt. Eng. 36, 2719–2728 (1997).
[CrossRef]

Opt. Lett. (1)

Other (8)

J. Starch, R. Sharma, S. Shaw, “A unified approach to feature extraction for model based ATR,” in Algorithms for Synthetic Aperture Radar Imagery III, E. G. Zelnio, R. J. Douglass, eds., Proc. SPIE2757, 294–305 (1996).
[CrossRef]

B. Bhanu, J. Ahn, “A system for model-based recognition of articulated objects,” in International Conference on Pattern Recognition (IEEE Computer Society, Washington, D.C., 1998), pp. 1812–1815, 1998).

S. Z. Der, Q. Zheng, R. Chellappa, B. Redman, Hesham Mahmoud, “View based recognition of military vehicles in LADAR imagery using CAD model matching,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 151–187.

L. A. Chan, S. Z. Der, N. M. Nasrabadi, “Neural based target detectors for multi-band infrared imagery,” in Image Recognition and Classification, Algorithms, Systems and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), pp. 1–36.

D. Torreiri, “A linear transform that simplifies and improves neural network classifiers,” presented at the International Conference on Neural Networks, Washington, D.C., 1996.

A. Mahalanobis, R. Muise, S. R. Stanfill, A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Eng. (to be published).

Three Class Public MSTAR data set, CD version, released for the Defense Advanced Research Projects Agency by Veda, Inc., the Dayton Group, 1997.

A. Mahalanobis, L. Ortiz, B. V. K. Vijaya Kumar, “Performance of the MACH/DCCF algorithms on the 10-class public release MSTAR data set,” in Algorithms for Synthetic Aperture Radar Imagery VI, E. G. Zelnio, eds., Proc. SPIE3721, 285–291 (1999).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Efficient architecture for implementing the quadratic filter correlates the input image with a bank of linear filters to yield partial results that are squared and added to produce the desired quadratic output for every point in the input image.

Fig. 2
Fig. 2

FKQCF output: solid curve, target class; dotted curve, clutter class.

Fig. 3
Fig. 3

Left, example of a target image used for Figs. 4, 7, 10, and 14 below. Right, example of a clutter image used for Figs. 4, 7, 10, and 14.

Fig. 4
Fig. 4

Left, example FKQCF target correlation surface. Right, example FKQCF clutter correlation surface.

Fig. 5
Fig. 5

ROC curve for a FKQCF for the MSTAR data test set.

Fig. 6
Fig. 6

Output responses of test data for a RQQCF. Solid curve, target; dashed curve, clutter.

Fig. 7
Fig. 7

Left, example RQQCF target correlation surface. Right, example RQQCF clutter correlation surface.

Fig. 8
Fig. 8

ROC curve for a RQQCF.

Fig. 9
Fig. 9

Left, ̂T (QSDF) response to test data; right, T (full-rank QSDF) response. Solid curves, targets; dashed curves, clutter.

Fig. 10
Fig. 10

Left, example QSDF target correlation surface; right, example QSDF clutter correlation surface.

Fig. 11
Fig. 11

ROC curve for a QSDF algorithm.

Fig. 12
Fig. 12

Singular values of B T B. Horizontal line, threshold value for reduced-rank inverse.

Fig. 13
Fig. 13

Top left, clutter and target output responses from a SSQSDF for the training data. Top right, clutter and target output responses from a SSQSDF for the test data. Bottom left, clutter and target output responses from the full-rank SSQSDF for the training data. Bottom left, clutter and target output responses from a full-rank SSQSDF for the test data.

Fig. 14
Fig. 14

Left, example SSQSDF target correlation surface; right, example SSQSDF clutter correlation surface.

Fig. 15
Fig. 15

ROC curve for a SSQSDF design.

Equations (34)

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

xTh=αx is a target<αx is not a target.
X=x1 x2  xNx,Y=y1 y2  yNy.
XYTh=c= [ 1 1 1 N x 0 0 0 N y ]T ,
h=X  Ya
h=X  YX  YTX  Y-1c.
EyTh2=EhTyyTh=hTEyyThhTYYTh,
h=YYT-1X  YX  YTYYT-1X  Y-1c.
xTTx=αx is a target<αx is not a target.
X  YTTX  Y=C, X  YTTX  TY=C, XTTXXTTYYTTXYTTY=C.
C=I00-I,
T=XXT+YYT-1XXT-YYTXXT+YYT-1.
T=RX+RY-1RX-RYRX+RY-1.
XC=x1-mX x2-mX  xNx-mX,YC=y1-mY y2-mY  yNy-mY,
X XC Y YCTTX XC Y YC=C, X XC Y YCTTX TXC TY TYC=C, XTTXXTTXCXTTYXTTYCXCTTXXCTTXCXCTTYXCTTYCYTTXYTTXCYTTYYTTYCYCTTXYCTTXCYCTTYYCTTYC=C.
C=I000000000-I00000.
T=RX+CX+RY+CY-1RX-RY×RX+CX+RY+CY-1,
X=x1 x2  xNx,Y=y1 y2  yNy,Z=z1 z2  zN.
T=i=1N βiziziT
EXzTTz=1,EYzTTz=-1.
xiTTxi-EXzTTz=0  i=1,, NX,yiTTyi-EYzTTz=0  i=1,, NY.
EXzTTz=EXzTi=1N βiziziTz=EXi=1N βizTziziTz=i=1N βiziTEXzzTzii=1N βiziTRxzi=1,EYzTTz=EYzTi=1N βiziziTz=EYi=1N βizTziziTz=i=1N βiziTEYzzTzii=1N βiziTRYzi=-1,
xiTTxi-EXzTTzj=1N βjzjTxixiT-RXzj=0  i=1,, NX,yiTTyi-EYzTTzj=1N βjzjTyiyiT-RYzj=0  i=1,, NY.
p=z1TRXz1 z2TRXz2  zNTRXzNT,q=z1TRYz1 z2TRYz2  zNTRYzNT,pi=z1TxixiT-RXz1 z2TxixiT-RXz2  zNTxixiT-RXzNT i,qi=z1TyiyiT-RYz1 z2TyiyiT-RYz2  zNTyiyiT-RYzNT i,c=1 -1 0  0T,β=β1  βNT.
Bβ=c,
B=p q p1  pNX q1  qNYT.
β=BTB-1BTc.
T=ZΔZT.
y=zTTz=zTFFTz=vTv.
vim, n=xm, nfim, n,  1iN,
ym, n=i=1N |vim, n|2=i=1N |xm, nfim, n|2.
T=FFT-GGT.
ym, n=i=1N2 |xm, nfim, n|2-i=1N1 |xm, ngim, n|2,
T=UΛUT.
T=u1 u2  umλ100λmu1 u2  umT-ud-n ud-n+1  ud-λd-n00-λdud-n ud-n+1  udT.

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