Abstract

Fukunaga–Koontz transform (FKT), stemming from principal component analysis (PCA), is used in many pattern recognition and image-processing fields. It cannot capture the higher-order statistical property of natural images, so its detection performance is not satisfying. PCA has been extended into kernel PCA in order to capture the higher-order statistics. However, thus far there have been no researchers who have definitely proposed kernel FKT (KFKT) and researched its detection performance. For accurately detecting potential small targets from infrared images, we first extend FKT into KFKT to capture the higher-order statistical properties of images. Then a framework based on Kalman prediction and KFKT, which can automatically detect and track small targets, is developed. Results of experiments show that KFKT outperforms FKT and the proposed framework is competent to automatically detect and track infrared point targets.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. B. Scholkopf, A. Smola, and K. R. Muller, "Nonlinear component analysis as a kernel eigenvalue problem," Neural Comput. 10, 1299-1319 (1998).
    [CrossRef]
  14. A. Bal and M. S. Alam, "Quadratic correlation filter based target tracking in FLIR image sequences," Proc. SPIE 5909, 590906 (2005).
    [CrossRef] [PubMed]
  15. J. Yang, A. Frangi, J. Yang, D. Zhang, and Z. Jin, "KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition," IEEE Trans. Pattern Anal. Mach. Intell. 27, 230-244 (2005).
    [CrossRef] [PubMed]
  16. S. Kim and B. Oommen, "On utilizing search methods to select subspace dimensions for kernel-based nonlinear subspace classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 27, 136-141 (2005).
    [CrossRef] [PubMed]
  17. J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
    [CrossRef] [PubMed]
  18. D. S. K. Chan, D. A. Langan, and D. A. Staver, "Spatial-processing techniques for the detection of small targets in IR clutter," Proc. SPIE 1305, 53-62 (1990).
    [CrossRef] [PubMed]
  19. K. L. Anderson and R. A. Iltis, "A tracking algorithm for infrared images based on reduced sufficient statistics," IEEE Trans. Aerosp. Electron. Syst. 33, 464-472 (1997).
    [CrossRef]
  20. P. S. Maybeck, R. L. Jensen, and D. A. Harnly, "Adaptive extended Kalman filter for target image tracking," IEEE Trans. Aerosp. Electron. Syst. AES-17, 173-180 (1981).
    [CrossRef]

2006

2005

A. Bal and M. S. Alam, "Automatic target tracking in FLIR image sequences using intensity variation function and template modeling," IEEE Trans. Instrum. Meas. 54, 1846-1852 (2005).
[CrossRef] [PubMed]

Zh. Liu, Ch. Chen, X. Shen, and X. Zou, "Detection of small objects in image data based on the nonlinear principal component analysis neural network," Opt. Eng. 44, 093604 (2005).
[CrossRef] [PubMed]

K. Kim, M. Franz, and B. Scholkopf, "Iterative kernel principal component analysis for image modeling," IEEE Trans. Pattern Anal. Mach. Intell. 27, 1351-1366 (2005).
[CrossRef] [PubMed]

A. Bal and M. S. Alam, "Quadratic correlation filter based target tracking in FLIR image sequences," Proc. SPIE 5909, 590906 (2005).
[CrossRef] [PubMed]

J. Yang, A. Frangi, J. Yang, D. Zhang, and Z. Jin, "KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition," IEEE Trans. Pattern Anal. Mach. Intell. 27, 230-244 (2005).
[CrossRef] [PubMed]

S. Kim and B. Oommen, "On utilizing search methods to select subspace dimensions for kernel-based nonlinear subspace classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 27, 136-141 (2005).
[CrossRef] [PubMed]

2004

S. S. Young, H. Kwon, S. Z. Der, and N. M. Nasrabadi, "Adaptive target detection in forward-looking infrared imagery using the eigenspace separation transform and principal component analysis," Opt. Eng. 43, 1767-1776 (2004).
[CrossRef]

S. R. F. Sims and A. Mahalanobis, "Performance evaluation of quadratic correlation filters for target detection and discrimination in infrared imagery," Opt. Eng. 43, 1705-1711 (2004).
[CrossRef]

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef] [PubMed]

L. Yang, J. Yang, and K. Yang, "Adaptive detection for infrared small target under sea-sky complex background," Electron. Lett. 40, 1083-1085 (2004).
[CrossRef]

2002

M. H. Yang, D. J. Kriegman, and N. Ahuja, "Detecting faces in images: a survey," IEEE Trans. Pattern Anal. Mach. Intell. 24, 35-58 (2002).

2000

A. P. Tzannes and D. H. Brooks, "Point target detection in IR image sequences: a hypothesis-testing approach based on target and clutter temporal profile modeling," Opt. Eng. 39, 2270-2278 (2000).
[CrossRef]

1998

B. Scholkopf, A. Smola, and K. R. Muller, "Nonlinear component analysis as a kernel eigenvalue problem," Neural Comput. 10, 1299-1319 (1998).
[CrossRef]

1997

K. L. Anderson and R. A. Iltis, "A tracking algorithm for infrared images based on reduced sufficient statistics," IEEE Trans. Aerosp. Electron. Syst. 33, 464-472 (1997).
[CrossRef]

1993

T. Soni, J. R. Zeidler, and W. H. Ku, "Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data," IEEE Trans. Image Process. 2, 327-340 (1993).
[CrossRef] [PubMed]

1990

D. S. K. Chan, D. A. Langan, and D. A. Staver, "Spatial-processing techniques for the detection of small targets in IR clutter," Proc. SPIE 1305, 53-62 (1990).
[CrossRef] [PubMed]

1981

P. S. Maybeck, R. L. Jensen, and D. A. Harnly, "Adaptive extended Kalman filter for target image tracking," IEEE Trans. Aerosp. Electron. Syst. AES-17, 173-180 (1981).
[CrossRef]

1970

K. Fukunaga and W. Koontz, "Representation of random processes using the finite Karhunen-Loeve expansion," Inf. Control 16, 85-101 (1970).
[CrossRef] [PubMed]

Appl. Opt.

Electron. Lett.

L. Yang, J. Yang, and K. Yang, "Adaptive detection for infrared small target under sea-sky complex background," Electron. Lett. 40, 1083-1085 (2004).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef] [PubMed]

K. L. Anderson and R. A. Iltis, "A tracking algorithm for infrared images based on reduced sufficient statistics," IEEE Trans. Aerosp. Electron. Syst. 33, 464-472 (1997).
[CrossRef]

P. S. Maybeck, R. L. Jensen, and D. A. Harnly, "Adaptive extended Kalman filter for target image tracking," IEEE Trans. Aerosp. Electron. Syst. AES-17, 173-180 (1981).
[CrossRef]

IEEE Trans. Image Process.

T. Soni, J. R. Zeidler, and W. H. Ku, "Performance evaluation of 2-D adaptive prediction filters for detection of small objects in image data," IEEE Trans. Image Process. 2, 327-340 (1993).
[CrossRef] [PubMed]

IEEE Trans. Instrum. Meas.

A. Bal and M. S. Alam, "Automatic target tracking in FLIR image sequences using intensity variation function and template modeling," IEEE Trans. Instrum. Meas. 54, 1846-1852 (2005).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

J. Yang, A. Frangi, J. Yang, D. Zhang, and Z. Jin, "KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition," IEEE Trans. Pattern Anal. Mach. Intell. 27, 230-244 (2005).
[CrossRef] [PubMed]

S. Kim and B. Oommen, "On utilizing search methods to select subspace dimensions for kernel-based nonlinear subspace classifiers," IEEE Trans. Pattern Anal. Mach. Intell. 27, 136-141 (2005).
[CrossRef] [PubMed]

M. H. Yang, D. J. Kriegman, and N. Ahuja, "Detecting faces in images: a survey," IEEE Trans. Pattern Anal. Mach. Intell. 24, 35-58 (2002).

K. Kim, M. Franz, and B. Scholkopf, "Iterative kernel principal component analysis for image modeling," IEEE Trans. Pattern Anal. Mach. Intell. 27, 1351-1366 (2005).
[CrossRef] [PubMed]

Inf. Control

K. Fukunaga and W. Koontz, "Representation of random processes using the finite Karhunen-Loeve expansion," Inf. Control 16, 85-101 (1970).
[CrossRef] [PubMed]

Neural Comput.

B. Scholkopf, A. Smola, and K. R. Muller, "Nonlinear component analysis as a kernel eigenvalue problem," Neural Comput. 10, 1299-1319 (1998).
[CrossRef]

Opt. Eng.

S. S. Young, H. Kwon, S. Z. Der, and N. M. Nasrabadi, "Adaptive target detection in forward-looking infrared imagery using the eigenspace separation transform and principal component analysis," Opt. Eng. 43, 1767-1776 (2004).
[CrossRef]

Zh. Liu, Ch. Chen, X. Shen, and X. Zou, "Detection of small objects in image data based on the nonlinear principal component analysis neural network," Opt. Eng. 44, 093604 (2005).
[CrossRef] [PubMed]

A. P. Tzannes and D. H. Brooks, "Point target detection in IR image sequences: a hypothesis-testing approach based on target and clutter temporal profile modeling," Opt. Eng. 39, 2270-2278 (2000).
[CrossRef]

S. R. F. Sims and A. Mahalanobis, "Performance evaluation of quadratic correlation filters for target detection and discrimination in infrared imagery," Opt. Eng. 43, 1705-1711 (2004).
[CrossRef]

Proc. SPIE

A. Bal and M. S. Alam, "Quadratic correlation filter based target tracking in FLIR image sequences," Proc. SPIE 5909, 590906 (2005).
[CrossRef] [PubMed]

D. S. K. Chan, D. A. Langan, and D. A. Staver, "Spatial-processing techniques for the detection of small targets in IR clutter," Proc. SPIE 1305, 53-62 (1990).
[CrossRef] [PubMed]

Other

J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis (Cambridge U. Press, 2004).
[CrossRef] [PubMed]

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Figures (15)

Fig. 1
Fig. 1

Three infrared images from three different image sequences. (a) The first frame of sequence 1 (Seq. 1 for short) with an airplane in the background of the sky. The size is 286 × 246 . (b) The first frame of sequence 2 (Seq. 2 for short) with a yacht in the background of the sea. The size is 288 × 236 . (c) The first frame of sequence 3 (Seq. 3 for short) with a tank on the ground in the background. The size is 125 × 125 .

Fig. 2
Fig. 2

Target training images generated by the Gaussian intensity function.

Fig. 3
Fig. 3

Cropping the background training images (subimages) from the three images of Fig. 1. All cropping windows have the same size: 11 × 11 .

Fig. 4
Fig. 4

Diagram of target detection with KFKT.

Fig. 5
Fig. 5

(Color online) Eigenvalues obtained by training KFKT for detecting potential targets from the three images of Fig. 1. Horizontal axis represents the eigenvector number. (a) Eigenvalues for detecting targets from Fig. 1(a). (b) Eigenvalues for detecting targets from Fig. 1(b). (c) Eigenvalues for detecting targets from Fig. 1(c).

Fig. 6
Fig. 6

(Color online) Detection results and their plots of FKT and KFKT for the image of Fig. 1(a). (a1), (a2), and (a3) are the detection results of FKT for Fig. 1(a), its plot, and its target local plot, respectively. (a1′), (a2′), and (a3′) are the detection result of KFKT for Fig. 1(a), its plot, and its local plot, respectively.

Fig. 7
Fig. 7

(Color online) Detection results and their plots of FKT and KFKT for the image of Fig. 1(b). (a1), (a2), and (a3) are the detection results of FKT for Fig. 1(b), its plot, and its target local plot, respectively. (a1′), (a2′), and (a3′) are the detection results of KFKT for Fig. 1(b), its plot, and its local plot, respectively.

Fig. 8
Fig. 8

(Color online) Detection results and their plots of FKT and KFKT for the image of Fig. 1(c). (a1), (a2), and (a3) are the detection results of FKT for Fig. 1(c), its plot, and its target local plot, respectively. (a1′), (a2′), and (a3′) are the detection results of KFKT for Fig. 1(c), its plot, and its local plot, respectively.

Fig. 9
Fig. 9

Proposed framework for automatically detecting and tracking small targets.

Fig. 10
Fig. 10

(Color online) Tracking process and results of KP_KFKT for Seq. 1.

Fig. 11
Fig. 11

(Color online) Tracking process and results of KP_KFKT for Seq. 2.

Fig. 12
Fig. 12

(Color online) Tracking process and results of KP_KFKT for Seq. 3.

Fig. 13
Fig. 13

(Color online) Tracking error plot of KP, KP_KFT, and KP_KFKT for Seq. 1. The horizontal axis is the frame number and the vertical axis is the error.

Fig. 14
Fig. 14

(Color online) Tracking error plot of KP, KP_KFT, and KP_KFKT for Seq. 2. The horizontal axis is the frame number and the vertical axis is the error.

Fig. 15
Fig. 15

(Color online) Tracking error plot of KP, KP_KFT, and KP_KFKT for Seq. 3. The horizontal axis is the frame number and the vertical axis is the error.

Tables (2)

Tables Icon

Table 1 Performance Comparison of Detection Methods

Tables Icon

Table 2 Average Error of Three Methods for Three Sequences

Equations (52)

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Φ :   n F ,
x Φ ( x ) .
X ˜ = [ x ˜ 1 x ˜ 2 x ˜ n 1 ] ,
= [ Φ ( x 1 ) Φ ( x 2 ) Φ ( x n 1 ) ] ,
Y ˜ = [ y ˜ 1 y ˜ 2 y ˜ n 2 ] ,
= [ Φ ( y 1 ) Φ ( y 2 ) Φ ( y n 2 ) ] .
T 0 = X ˜ X ˜ T ,
T 0 ( i , j ) = x ˜ i x ˜ j T = ( Φ ( x i ) Φ ( x j ) ) = K ( x i , x j ) ,
B 0 = Y ˜ Y ˜ T ,
B 0 ( i , j ) = y ˜ i y ˜ j T = ( Φ ( y i ) Φ ( y j ) ) = K ( y i , y j ) .
Σ = T 0 + B 0 = Ψ Λ Ψ T ,
P = Ψ Λ 1 / 2 .
P T ( T 0 + B 0 ) P = I .
X ˜ ^ = P T X ˜ = [ P T Φ ( x 1 ) P T Φ ( x 2 ) P T Φ ( x n 1 ) ] ,
Y ˜ ^ = P T Y ˜ = [ P T Φ ( y 1 ) P T Φ ( y 2 ) P T Φ ( y n 2 ) ] .
T ^ = X ˜ ^ X ˜ ^ T = P T X ˜ X ˜ T P = P T T 0 P ,
B ^ = Y ˜ ^ Y ˜ ^ T = P T Y ˜ Y ˜ T P = P T B 0 P .
P T ( T 0 + B 0 ) P = T ^ + B ^ = I .
T ^ θ i = λ i θ i ,
T ^ θ i = ( I B ^ ) θ i = λ i θ i ,
B ^ θ i = ( 1 λ i ) θ i .
M = F D ( z ) .
Ξ 1 = [ θ 1 θ 2 θ t 1 ] ,
Ξ 2 = [ θ t θ t 1 θ t t 2 + 1 ] .
W 1 = Ξ 1 T z ˜ ^ ,
W 2 = Ξ 2 T z ˜ ^ .
M = W 1 T W 1 W 2 T W 2 = z ˜ ^ T ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) z ˜ ^ = ( P T Φ ( z ) ) T ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) ( P T Φ ( z ) ) = Φ ( z ) T [ P ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) P T ] Φ ( z ) = Φ ( z ) T Q Φ ( z ) ,
Q = P ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) P T = ( Ψ Λ 1 / 2 ) ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) ( Ψ Λ 1 / 2 ) T .
T = T 0 I n 1 T 0 T 0 I n 1 + I n 1 T 0 I n 1 .
f j = 1 β j γ j T [ k ( x 1 , z ) , k ( x 2 , z ) , … ,  k ( x n 1 , z ) ] ,
j = 1  , … ,  m 1 .
f = [ f 1 f 2 f m 1 ] = 1 ( z ) .
f ^ = P T f .
W 1 = Ξ 1 T f ^ .
W 2 = Ξ 2 T f ^ .
M = W 1 T W 1 W 2 T W 2 = ( f ^ T Ξ 1 Ξ 1 T f ^ ) ( f ^ T Ξ 2 Ξ 2 T f ^ ) = ( P T f ) T ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) ( P T f ) = f T [ P ( Ξ 1 Ξ 1 T Ξ 2 Ξ 2 T ) P T ] f = f T Q f = 1 ( z ) T Q 1 ( z ) .
F D ( z ) = 1 ( z ) T Q 1 ( z ) .
I ( x , y ) = I max exp ( 1 2 [ ( x x 0 ) 2 σ x 2 + ( y y 0 ) 2 σ y 2 ] ) .
SCR = S C ,
SCRG = SCR out SCR in ,
BSF = C in C out .
X k + 1 = F X k + ω k .
z k = H X k + v k .
E [ ω k ] = E [ v k ] = 0 ,
E [ ( ω k v k ) ( ω k T v k T ) ] = [ Q k 0 0 R k ] .
F = [ 1 T 0 0 0 1 0 0 0 0 1 T 0 0 0 1 ] ,
H = [ 1 0 1 0 ] .
X ^ k + 1 k = F X ^ k k .
P k + 1 k = F P k k F T + Q k .
P k + 1 k + 1 1 = P k + 1 k 1 + H T R k + 1 1 H .
K k + 1 = P k + 1 k + 1 H T R k + 1 1 .
X ^ k + 1 k + 1 = X ^ k + 1 k + K k + 1 ( z k + 1 H X ^ k + 1 k ) .

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