Abstract

We recently presented a new asynchronous demodulation method for phase-sampling interferometry. The method is based in the principal component analysis (PCA) technique. In the former work, the PCA method was derived heuristically. In this work, we present an in-depth analysis of the PCA demodulation method.

© 2011 Optical Society of America

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References

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  1. J. Vargas, J. A. Quiroga, and T. Belenguer, Opt. Lett. 36, 1326 (2011).
    [CrossRef] [PubMed]
  2. http://en.wikipedia.org/wiki/Principal_component_analysis.
  3. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2007).
  4. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
    [CrossRef]
  5. G. H. Golum and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).
  6. Z. Y. Wang and B. T. Han, Opt. Lett. 29, 1671 (2004).
    [CrossRef] [PubMed]

2011 (1)

2007 (1)

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2007).

2004 (2)

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Z. Y. Wang and B. T. Han, Opt. Lett. 29, 1671 (2004).
[CrossRef] [PubMed]

1996 (1)

G. H. Golum and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Belenguer, T.

Golum, G. H.

G. H. Golum and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2007).

Han, B. T.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Quiroga, J. A.

Van Loan, C. F.

G. H. Golum and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Vargas, J.

Wang, Z. Y.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2007).

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Opt. Lett. (2)

Other (4)

http://en.wikipedia.org/wiki/Principal_component_analysis.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2007).

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

G. H. Golum and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

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Equations (22)

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x = [ x 1 , x 2 , , x N ] T ,
C = ( x m x ) ( x m ) T .
D = U C U T ,
y = U ( x m x ) .
I n ( x , y ) = S ( x , y ) + B ( x , y ) cos ( Φ ( x , y ) + δ n ) ,
I n , k = S k + B k cos ( Φ k + δ n ) ,
x = ( S 1 + B 1 cos ( Φ 1 + δ 1 ) S 2 + B 2 cos ( Φ 2 + δ 1 ) S M + B M cos ( Φ M + δ 1 ) S 1 + B 1 cos ( Φ 1 + δ N ) S 2 + B 2 cos ( Φ 2 + δ N ) S M + B M cos ( Φ M + δ N ) ) ,
{ m x } k = 1 N n I n , k S k .
x ˜ = ( B 1 cos ( Φ 1 + δ 1 ) B 2 cos ( Φ 2 + δ 1 ) B M cos ( Φ M + δ 1 ) B 1 cos ( Φ 1 + δ N ) B 2 cos ( Φ 2 + δ N ) B M cos ( Φ M + δ N ) ) .
C i j = k = 1 M ( a i u k + b i v k ) ( a j u k + b j v k ) .
C i j = k = 1 M a i a j u k u k + 2 a i b j u k v k + b i b j v k v k .
C i j = k = 1 M A i j u k u k + E i j u k v k + F i j v k v k .
k = 1 M u k v k = k = 1 M B k 2 cos Φ k sin Φ k k = 1 M u k u k = k = 1 M B k 2 cos 2 Φ k , k = 1 M u k v k = k = 1 M B k 2 cos Φ k sin Φ k k = 1 M v k v k = k = 1 M B k 2 sin 2 Φ k .
C = α A + β F ,
A = [ cos δ 1 cos δ N ] T [ cos δ 1 cos δ N ] , F = [ sin δ 1 sin δ N ] T [ sin δ 1 sin δ N ] ,
λ A = i = 1 N cos 2 δ i , λ F = i = 1 N sin 2 δ i ,
w A = [ cos δ 1 , , cos δ N ] λ A , w F = [ sin δ 1 , , sin δ N ] λ F .
n = 1 N cos δ n sin δ n 0 ,
A w F = 0 , F w A = 0.
y 1 , k = n = 1 N B k cos ( Φ k + δ n ) cos δ n , y 2 , k = n = 1 N B k cos ( Φ k + δ n ) sin δ n .
y 1 , k B k cos Φ k , y 2 , k B k sin Φ k .
Φ k = arctan ( y 2 , k y 1 , k ) .

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