Abstract

Simulations applied to hyperspectral imagery from the AVIRIS sensor are employed to quantitatively evaluate the performance of anomalous change detection algorithms. The evaluation methodology reflects the aim of these algorithms, which is to distinguish actual anomalous changes in a pair of images from the incidental differences that pervade the entire scene. By simulating both the anomalous changes and the pervasive differences, accurate and plentiful ground truth is made available, and statistical estimates of detection and false alarm rates can be made. Comparing the receiver operating characteristic (ROC) curves that encapsulate these rates provides a way to identify which algorithms work best under which conditions.

© 2008 Optical Society of America

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  1. R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
    [CrossRef]
  2. A. Schaum and A. Stocker, “Spectrally selective target detection,” in Proceedings of the International Symposium on Spectral Sensing Research (1997).
  3. A. Schaum and A. Stocker, “Long-interval chronochrome target detection,” Proceedings of the International Symposium on Spectral Sensing Research (1997).
  4. C. Clifton, “Change detection in overhead imagery using neural networks,” Appl. Intell. 18, 215-234 (2003).
    [CrossRef]
  5. A. Schaum and A. Stocker, “Linear chromodynamics models for hyperspectral target detection,” in Proceedings of the 2003 IEEE Aerospace Conference (IEEE, 2003), Vol. 4, pp. 1879-1885.
  6. A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
    [CrossRef]
  7. J. Theiler and S. Perkins, “Proposed framework for anomalous change detection,” in Proceedings of the ICML Workshop on Machine Learning Algorithms for Surveillance and Event Detection (29 June 2006, Pittsburgh, Pa.), pp. 7-14.
  8. J. Theiler and S. Perkins, “Resampling approach for anomalous change detection,” Proc. SPIE 6565, 65651U (2007).
    [CrossRef]
  9. Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), Jet Propulsion Laboratory (JPL), National Aeronautics and Space Administration (NASA), http://aviris.jpl.nasa.gov/.
  10. AVIRIS Free Standard Data Products, Jet Propulsion Laboratory (JPL), National Aeronautics and Space Administration (NASA), http://aviris.jpl.nasa.gov/html/aviris.freedata.html.
  11. I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760-1770(1990).
    [CrossRef]
  12. A. Schaum and A. Stocker, “Estimating hyperspectral target signature evolution with a background chromodynamics model,” in Proceedings of the International Symposium on Spectral Sensing Research (2003).
  13. A. Schaum and A. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
    [CrossRef]
  14. A. Schaum and E. Allman, “Advanced algorithms for autonomous hyperspectral change detection,” in the 33rd Applied Imagery Pattern Recognition Workshop (AIPR'04) (IEEE Computer Society, 2004), pp. 33-38.
    [CrossRef]
  15. J. Theiler, “Subpixel anomalous change detection in remote sensing imagery,” in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation (IEEE Computer Society, 2008), pp. 165-168.
    [CrossRef]
  16. T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Trans. Geosci. Remote Sens. 40, 1815-1823 (2002).
    [CrossRef]
  17. J. Theiler, “Sensitivity of anomalous change detection to small misregistration errors,” Proc. SPIE 6966, 69660X (2008).
    [CrossRef]

2008 (1)

J. Theiler, “Sensitivity of anomalous change detection to small misregistration errors,” Proc. SPIE 6966, 69660X (2008).
[CrossRef]

2007 (1)

J. Theiler and S. Perkins, “Resampling approach for anomalous change detection,” Proc. SPIE 6565, 65651U (2007).
[CrossRef]

2005 (1)

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

2004 (1)

A. Schaum and A. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

2003 (1)

C. Clifton, “Change detection in overhead imagery using neural networks,” Appl. Intell. 18, 215-234 (2003).
[CrossRef]

2002 (1)

T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Trans. Geosci. Remote Sens. 40, 1815-1823 (2002).
[CrossRef]

1998 (1)

A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
[CrossRef]

1990 (1)

I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760-1770(1990).
[CrossRef]

Al-Kofahi, O.

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

Allman, E.

A. Schaum and E. Allman, “Advanced algorithms for autonomous hyperspectral change detection,” in the 33rd Applied Imagery Pattern Recognition Workshop (AIPR'04) (IEEE Computer Society, 2004), pp. 33-38.
[CrossRef]

Andra, S.

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

Clifton, C.

C. Clifton, “Change detection in overhead imagery using neural networks,” Appl. Intell. 18, 215-234 (2003).
[CrossRef]

Conradsen, K.

A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
[CrossRef]

Kasetkasem, T.

T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Trans. Geosci. Remote Sens. 40, 1815-1823 (2002).
[CrossRef]

Nielsen, A. A.

A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
[CrossRef]

Perkins, S.

J. Theiler and S. Perkins, “Resampling approach for anomalous change detection,” Proc. SPIE 6565, 65651U (2007).
[CrossRef]

J. Theiler and S. Perkins, “Proposed framework for anomalous change detection,” in Proceedings of the ICML Workshop on Machine Learning Algorithms for Surveillance and Event Detection (29 June 2006, Pittsburgh, Pa.), pp. 7-14.

Radke, R. J.

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

Reed, I. S.

I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760-1770(1990).
[CrossRef]

Roysam, B.

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

Schaum, A.

A. Schaum and A. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

A. Schaum and A. Stocker, “Estimating hyperspectral target signature evolution with a background chromodynamics model,” in Proceedings of the International Symposium on Spectral Sensing Research (2003).

A. Schaum and E. Allman, “Advanced algorithms for autonomous hyperspectral change detection,” in the 33rd Applied Imagery Pattern Recognition Workshop (AIPR'04) (IEEE Computer Society, 2004), pp. 33-38.
[CrossRef]

A. Schaum and A. Stocker, “Spectrally selective target detection,” in Proceedings of the International Symposium on Spectral Sensing Research (1997).

A. Schaum and A. Stocker, “Long-interval chronochrome target detection,” Proceedings of the International Symposium on Spectral Sensing Research (1997).

A. Schaum and A. Stocker, “Linear chromodynamics models for hyperspectral target detection,” in Proceedings of the 2003 IEEE Aerospace Conference (IEEE, 2003), Vol. 4, pp. 1879-1885.

Simpson, J. J.

A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
[CrossRef]

Stocker, A.

A. Schaum and A. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

A. Schaum and A. Stocker, “Estimating hyperspectral target signature evolution with a background chromodynamics model,” in Proceedings of the International Symposium on Spectral Sensing Research (2003).

A. Schaum and A. Stocker, “Linear chromodynamics models for hyperspectral target detection,” in Proceedings of the 2003 IEEE Aerospace Conference (IEEE, 2003), Vol. 4, pp. 1879-1885.

A. Schaum and A. Stocker, “Long-interval chronochrome target detection,” Proceedings of the International Symposium on Spectral Sensing Research (1997).

A. Schaum and A. Stocker, “Spectrally selective target detection,” in Proceedings of the International Symposium on Spectral Sensing Research (1997).

Theiler, J.

J. Theiler, “Sensitivity of anomalous change detection to small misregistration errors,” Proc. SPIE 6966, 69660X (2008).
[CrossRef]

J. Theiler and S. Perkins, “Resampling approach for anomalous change detection,” Proc. SPIE 6565, 65651U (2007).
[CrossRef]

J. Theiler and S. Perkins, “Proposed framework for anomalous change detection,” in Proceedings of the ICML Workshop on Machine Learning Algorithms for Surveillance and Event Detection (29 June 2006, Pittsburgh, Pa.), pp. 7-14.

J. Theiler, “Subpixel anomalous change detection in remote sensing imagery,” in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation (IEEE Computer Society, 2008), pp. 165-168.
[CrossRef]

Varshney, P. K.

T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Trans. Geosci. Remote Sens. 40, 1815-1823 (2002).
[CrossRef]

Yu, X.

I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760-1770(1990).
[CrossRef]

Appl. Intell. (1)

C. Clifton, “Change detection in overhead imagery using neural networks,” Appl. Intell. 18, 215-234 (2003).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

I. S. Reed and X. Yu, “Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust. Speech Signal Process. 38, 1760-1770(1990).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

T. Kasetkasem and P. K. Varshney, “An image change detection algorithm based on Markov random field models,” IEEE Trans. Geosci. Remote Sens. 40, 1815-1823 (2002).
[CrossRef]

IEEE Trans. Image Process. (1)

R. J. Radke, S. Andra, O. Al-Kofahi, and B. Roysam, “Image change detection algorithms: a systematic survey,” IEEE Trans. Image Process. 14, 294-307 (2005).
[CrossRef]

Proc. SPIE (3)

J. Theiler and S. Perkins, “Resampling approach for anomalous change detection,” Proc. SPIE 6565, 65651U (2007).
[CrossRef]

J. Theiler, “Sensitivity of anomalous change detection to small misregistration errors,” Proc. SPIE 6966, 69660X (2008).
[CrossRef]

A. Schaum and A. Stocker, “Hyperspectral change detection and supervised matched filtering based on covariance equalization,” Proc. SPIE 5425, 77-90 (2004).
[CrossRef]

Remote Sens. Environ. (1)

A. A. Nielsen, K. Conradsen, and J. J. Simpson, “Multivariate alteration detection (MAD) and MAF postprocessing in multispectral, bitemporal image data: new approaches to change detection studies,” Remote Sens. Environ. 64, 1-19 (1998).
[CrossRef]

Other (9)

J. Theiler and S. Perkins, “Proposed framework for anomalous change detection,” in Proceedings of the ICML Workshop on Machine Learning Algorithms for Surveillance and Event Detection (29 June 2006, Pittsburgh, Pa.), pp. 7-14.

Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), Jet Propulsion Laboratory (JPL), National Aeronautics and Space Administration (NASA), http://aviris.jpl.nasa.gov/.

AVIRIS Free Standard Data Products, Jet Propulsion Laboratory (JPL), National Aeronautics and Space Administration (NASA), http://aviris.jpl.nasa.gov/html/aviris.freedata.html.

A. Schaum and A. Stocker, “Spectrally selective target detection,” in Proceedings of the International Symposium on Spectral Sensing Research (1997).

A. Schaum and A. Stocker, “Long-interval chronochrome target detection,” Proceedings of the International Symposium on Spectral Sensing Research (1997).

A. Schaum and A. Stocker, “Linear chromodynamics models for hyperspectral target detection,” in Proceedings of the 2003 IEEE Aerospace Conference (IEEE, 2003), Vol. 4, pp. 1879-1885.

A. Schaum and E. Allman, “Advanced algorithms for autonomous hyperspectral change detection,” in the 33rd Applied Imagery Pattern Recognition Workshop (AIPR'04) (IEEE Computer Society, 2004), pp. 33-38.
[CrossRef]

J. Theiler, “Subpixel anomalous change detection in remote sensing imagery,” in Proceedings of the IEEE Southwest Symposium on Image Analysis and Interpretation (IEEE Computer Society, 2008), pp. 165-168.
[CrossRef]

A. Schaum and A. Stocker, “Estimating hyperspectral target signature evolution with a background chromodynamics model,” in Proceedings of the International Symposium on Spectral Sensing Research (2003).

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Figures (7)

Fig. 1
Fig. 1

Broadband image from the 150 × 500 pixel chip that was used in this study. This is from the AVIRIS image labeled f960323t01p02_r04_sc01, which is in Florida, near the Kennedy Space Center. There are 224 spectral channels, spanning the visible to the near infrared.

Fig. 2
Fig. 2

ROC curves plotted for various anomaly detection algorithms: simple difference (SD), chronochrome (CC), covariance equalization (CE-I and CE-R), straight anomaly detection (RX), hyperbolic anomalous change detection (Hyper), and subpixel hyperbolic anomalous change (Subpix). These experiments used full-pixel anomalies generated according to method 1 from Subsection 3B. Four different pervasive changes were considered, as described in Subsection 3A: (a) smoothing, (b) noise, (c) spectral splitting, and (d) single-pixel misregistration.

Fig. 3
Fig. 3

Same as Fig. 2, but here the anomalous change was subpixel, as described in item 2 of the list in Subsection 3B.

Fig. 4
Fig. 4

Same as Fig. 2, but here the anomalous change was brightening or darkening of a pixel, as described in item 3 of the list in Subsection 3B.

Fig. 5
Fig. 5

Same as Fig. 2, but here the anomalous change was darkening or brightening of a pixel, as described in item 4 of the list in Subsection 3B.

Fig. 6
Fig. 6

Same as Fig. 2, but PCA was applied to the two images before seeking anomalous changes. Only the first five principal components were used.

Fig. 7
Fig. 7

Same as Fig. 2, but CCA was employed as a preprocessing stage, so that the dimension was reduced to d = 5 channels before change detection algorithms were applied. Note that, because of the substantially better performance achieved by this preprocessing, the vertical axis on these plots ranges only from 0.5 to 1.

Tables (1)

Tables Icon

Table 1 Coefficient Matrices a for the Quadratic Covariance-Based Anomalous Change Detectors

Equations (76)

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X = x x T ,
Y = y y T ,
C = y x T .
A ( x , y ) = [ x T y T ] Q [ x y ] ,
x ˜ = X 1 / 2 x ,
y ˜ = Y 1 / 2 y ,
C ˜ = Y 1 / 2 C X 1 / 2 ,
A ˜ ( x ˜ , y ˜ ) = A ( x , y ) = A ( X 1 / 2 x ˜ , Y 1 / 2 y ˜ ) = [ x ˜ T y ˜ T ] Q ˜ [ x ˜ y ˜ ] ,
Q ˜ = [ X 1 / 2 0 0 Y 1 / 2 ] Q [ X 1 / 2 0 0 Y 1 / 2 ] .
Q = [ X 1 / 2 0 0 Y 1 / 2 ] Q ˜ [ X 1 / 2 0 0 Y 1 / 2 ]
e = B y A x .
E = e e T = ( B y A x ) ( y T B T x T A T ) = B y y T B T B y x T A T A x y T B T + A x x T A T = B Y B T B C A T A C T B T + A X A T .
A ( x , y ) = e T e e T 1 e = e T E 1 e = ( B y A x ) T [ B Y B T B C A T A C T B T + A X A T ] 1 ( B y A x ) .
A ( x , y ) = [ x T y T ] Q [ x y ] ,
Q = [ A T B T ] [ B Y B T B C A T A C T B T + A X A T ] 1 [ A B ] .
e = y x ;
A SD ( x , y ) = ( y x ) T [ Y C C T + X ] 1 ( y x ) .
Q SD = [ I I ] [ Y C C T + X ] 1 [ I I ] .
e = y L x
A CC ( x , y ) = e T e e T 1 e = ( y C X 1 x ) T [ Y C X 1 C T ] 1 ( y C X 1 x ) .
Q CC = [ X 1 C T I y ] [ Y C X 1 C T ] 1 [ C X 1 I y ] .
Q ˜ CC = [ C ˜ T I y ] [ I y C ˜ C ˜ T ] 1 [ C ˜ I y ] .
[ I V T V I ] 1 = [ I + V T ( I V V T ) 1 V V T ( I V V T ) 1 ( I V V T ) 1 V ( I V V T ) 1 ]
= [ V T I ] [ I V V T ] 1 [ V I ] + [ I 0 0 0 ] .
Q ˜ CC = [ I x C ˜ T C ˜ I y ] 1 [ I x 0 0 0 ] .
A CC ( x , y ) = ( x C T Y 1 y ) T [ X C T Y 1 C ] 1 ( x C T Y 1 y ) ,
Q ˜ CC = [ I x C ˜ ] [ I x C ˜ T C ˜ ] 1 [ I x C ˜ T ]
= [ I x C ˜ T C ˜ I y ] 1 [ 0 0 0 I y ] .
e = Y 1 / 2 y X 1 / 2 x .
A CE-I ( x , y ) = ( Y 1 / 2 y X 1 / 2 x ) T [ 2 I Y 1 / 2 C X 1 / 2 X 1 / 2 C T Y 1 / 2 ] 1 ( Y 1 / 2 y X 1 / 2 x ) .
A CE-I ( x ˜ , y ˜ ) = ( y ˜ x ˜ ) T [ 2 I C ˜ C ˜ T ] 1 ( y ˜ x ˜ ) ,
Q ˜ CE-I = [ I I ] [ 2 I C ˜ C ˜ T ] 1 [ I I ] .
e = y Y 1 / 2 X 1 / 2 x .
A ( x , y ) = e T e e T 1 e = e T Y 1 / 2 Y 1 / 2 e e T Y 1 / 2 1 Y 1 / 2 e = e T Y 1 / 2 Y 1 / 2 e e T 1 Y 1 / 2 Y 1 / 2 e = e T e e T 1 e = A ( x , y ) .
e = Y 1 / 2 y R X 1 / 2 x = y ˜ R x ˜ .
C ˜ = U J V T ,
R = ( C ˜ C ˜ T ) 1 / 2 C ˜ .
A ˜ CE-R ( x ˜ , y ˜ ) = ( y ˜ R x ˜ ) T [ 2 I C ˜ R T R C ˜ T ] 1 ( y ˜ R x ˜ )
= ( y ˜ R x ˜ ) T [ 2 I 2 ( C ˜ C ˜ T ) 1 / 2 ] 1 ( y ˜ R x ˜ ) .
Q ˜ CE-R = 1 2 [ R I y ] [ I y ( C ˜ C ˜ T ) 1 / 2 ] 1 [ R I y ]
e = S Y 1 / 2 y R X 1 / 2 x = S y ˜ R x ˜ ,
Var ( e ) = e T e = trace ( e e T ) = trace ( ( S y ˜ R x ˜ ) ( y ˜ T S T x ˜ T R T ) )
= trace ( 2 I S C ˜ R T R C ˜ T S T )
= trace ( 2 I S C ˜ R T ( S C ˜ R T ) T )
= 2 d 2 trace ( S C ˜ R T )
trace ( S C ˜ R T ) = trace ( S U J V T R T ) trace ( J ) .
trace ( S C ˜ R T ) = trace ( S S T J R R T ) = trace ( J ) ,
Q ˜ CE-D = 1 2 [ R T S T ] [ I J ] 1 [ R S ] ,
trace ( S 1 C ˜ R 1 T ) = trace ( U S o C ˜ R o T U T ) = trace ( S o C ˜ R o T )
x = R X 1 / 2 x ,
y = S Y 1 / 2 y
x x T = I ,
y y T = I .
y x T = S Y 1 / 2 yx X 1 / 2 R T = S Y 1 / 2 C X 1 / 2 R T = S C ˜ R T = J .
e = [ x y ]
A ( x , y ) = e T e e T 1 e = [ x T y T ] [ X C T C Y ] 1 [ x y ] .
Q RX = [ X C T C Y ] 1
Q ˜ RX = [ I x C ˜ T C ˜ I y ] 1 .
P x ( x ) = P ( x , y ) d y ,
P y ( y ) = P ( x , y ) d x .
A ( x , y ) = log P x ( x ) + log P y ( y ) log P ( x , y ) .
P x ( x ) = ( 2 π ) d x / 2 | X | 1 / 2 exp ( 1 2 x T X 1 x ) ,
P y ( y ) = ( 2 π ) d y / 2 | Y | 1 / 2 exp ( 1 2 y T Y 1 y ) ,
P ( x , y ) = ( 2 π ) ( d x + d y ) / 2 | X C T C Y | 1 / 2 exp ( 1 2 [ x T y T ] [ X C T C Y ] 1 [ x y ] ) .
A ( x , y ) = constant 1 2 x T X 1 x 1 2 y T Y 1 y + 1 2 [ x T y T ] [ X C T C Y ] 1 [ x y ] ,
A ( x , y ) = [ x T y T ] Q Hyper [ x y ] ,
Q Hyper = [ X C T C Y ] 1 [ X 0 0 Y ] 1 ;
Q ˜ Hyper = [ I x C ˜ T C ˜ I y ] 1 [ I x 0 0 I y ] .
Q ˜ θ = [ I x C ˜ T C ˜ I y ] 1 [ I x θ C ˜ T θ C I y ] 1
Q ˜ subpix = [ I x C ˜ T C ˜ I y ] 1 [ 0 C ˜ T C ˜ 0 ] [ I x C ˜ T C ˜ I y ] 1 .
[ I x C ˜ T C ˜ I y ] 1 = [ I x + C ˜ T ( I y C ˜ C ˜ T ) 1 C ˜ C ˜ T ( I y C ˜ C ˜ T ) 1 ( I y C ˜ C ˜ T ) 1 C ˜ ( I y C ˜ C ˜ T ) 1 ] .
P 2 ( y ) = y ( 1 + ϵ η ) ,
P 1 ( y ) = [ I k 0 0 0 ] y ,
P 2 ( y ) = [ 0 0 0 I n k ] y .
( x , y ) ( x , f ( y ) ) .
e = [ x y ]

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