Abstract

We propose a novel low-complexity recursive filter to efficiently recover quantitative phase from a series of noisy intensity images taken through focus. We first transform the wave propagation equation and nonlinear observation model (intensity measurement) into a complex augmented state space model. From the state space model, we derive a sparse augmented complex extended Kalman filter (ACEKF) to infer the complex optical field (amplitude and phase), and find that it converges under mild conditions. Our proposed method has a computational complexity of NzN logN and storage requirement of 𝒪(N), compared with the original ACEKF method, which has a computational complexity of 𝒪(NzN3) and storage requirement of 𝒪(N2), where Nz is the number of images and N is the number of pixels in each image. Thus, it is efficient, robust and recursive, and may be feasible for real-time phase recovery applications with high resolution images.

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt.46, 7978–7981 (2007).
    [CrossRef] [PubMed]
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  13. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  14. D. H. Dini and D. P. Mandic, “Class of widely linear complex Kalman filters,” IEEE Trans. Neural Netw. Learn. Syst.23, 775–786 (2012).
    [CrossRef]
  15. R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of basic Engineering82, 35–45 (1960).
    [CrossRef]
  16. K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
    [CrossRef]
  17. T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
    [CrossRef]
  18. A. J. Krener, “The convergence of the extended Kalman filter,” in “Directions in mathematical systems theory and optimization,” (Springer, 2003), pp. 173–182.
    [CrossRef]
  19. D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A16, 1005–1015 (1999).
    [CrossRef]

2012

Z. Jingshan, J. Dauwels, M. A. Vázquez, and L. Waller, “Efficient Gaussian inference algorithms for phase imaging,” Proc. IEEE ICASSP2012 pp. 25–30.

D. H. Dini and D. P. Mandic, “Class of widely linear complex Kalman filters,” IEEE Trans. Neural Netw. Learn. Syst.23, 775–786 (2012).
[CrossRef]

S. S. Gorthi and E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett.37, 707–709 (2012).
[CrossRef] [PubMed]

2011

2010

2007

2005

2004

T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
[CrossRef]

1999

D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A16, 1005–1015 (1999).
[CrossRef]

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

1992

1988

1984

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

1982

1972

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik35, 237–246 (1972).

1960

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of basic Engineering82, 35–45 (1960).
[CrossRef]

Acosta, E.

Barbastathis, G.

Bruyninckx, H.

T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
[CrossRef]

Dauwels, J.

Z. Jingshan, J. Dauwels, M. A. Vázquez, and L. Waller, “Efficient Gaussian inference algorithms for phase imaging,” Proc. IEEE ICASSP2012 pp. 25–30.

Dini, D. H.

D. H. Dini and D. P. Mandic, “Class of widely linear complex Kalman filters,” IEEE Trans. Neural Netw. Learn. Syst.23, 775–786 (2012).
[CrossRef]

Fienup, J.

Gerchberg, R.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik35, 237–246 (1972).

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gorthi, S. S.

Gunther, S.

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

Jingshan, Z.

Z. Jingshan, J. Dauwels, M. A. Vázquez, and L. Waller, “Efficient Gaussian inference algorithms for phase imaging,” Proc. IEEE ICASSP2012 pp. 25–30.

Kalman, R. E.

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of basic Engineering82, 35–45 (1960).
[CrossRef]

Krener, A. J.

A. J. Krener, “The convergence of the extended Kalman filter,” in “Directions in mathematical systems theory and optimization,” (Springer, 2003), pp. 173–182.
[CrossRef]

Lee, D. J.

Lefebvre, T.

T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
[CrossRef]

Luo, Y.

Mandic, D. P.

D. H. Dini and D. P. Mandic, “Class of widely linear complex Kalman filters,” IEEE Trans. Neural Netw. Learn. Syst.23, 775–786 (2012).
[CrossRef]

Osten, W.

Paxman, R.

Pedrini, G.

Ponda, S.

Reif, K.

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

Roggemann, M. C.

Saxton, W.

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik35, 237–246 (1972).

Schonbrun, E.

Schulz, T.

Schutter, J. D.

T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
[CrossRef]

Soto, M.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

Tian, L.

Tsang, M.

Unbehauen, R.

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

Vázquez, M. A.

Z. Jingshan, J. Dauwels, M. A. Vázquez, and L. Waller, “Efficient Gaussian inference algorithms for phase imaging,” Proc. IEEE ICASSP2012 pp. 25–30.

Waller, L.

Welsh, B. M.

Yang, S.

Yang, S. Y.

Yaz, E.

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

Zhang, Y.

Appl. Opt.

IEEE Trans. Autom. Control

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Trans. Autom. Control44, 714–728 (1999).
[CrossRef]

IEEE Trans. Neural Netw. Learn. Syst.

D. H. Dini and D. P. Mandic, “Class of widely linear complex Kalman filters,” IEEE Trans. Neural Netw. Learn. Syst.23, 775–786 (2012).
[CrossRef]

Internat. J. Control

T. Lefebvre, H. Bruyninckx, and J. D. Schutter, “Kalman filters for nonlinear systems: a comparison of performance,” Internat. J. Control77, 639–653 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Journal of basic Engineering

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of basic Engineering82, 35–45 (1960).
[CrossRef]

Opt. Commun.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik35, 237–246 (1972).

Proc. IEEE ICASSP

Z. Jingshan, J. Dauwels, M. A. Vázquez, and L. Waller, “Efficient Gaussian inference algorithms for phase imaging,” Proc. IEEE ICASSP2012 pp. 25–30.

Other

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

A. J. Krener, “The convergence of the extended Kalman filter,” in “Directions in mathematical systems theory and optimization,” (Springer, 2003), pp. 173–182.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Minimum singular value of observability matrix with 30 different initializations of the covariance matrix Q0 and P0. The matrix Q0 is initialized as qU, with q ranging from 70 to 7000, and P0 is initialized as pU, with p ranging from 50 to 5000.

Fig. 2
Fig. 2

Maximum singular value of covariance matrix with 30 different initializations of p and q in the covariance matrix.

Fig. 3
Fig. 3

Minimum singular value of covariance matrix with 30 different initializations of p and q in the covariance matrix.

Fig. 4
Fig. 4

(a) Data Set 1: synthetic images with strong noise (100 × 100 pixels with size of 1μm×1μm). (b) Data Set 2: experimental data acquired by a microscope (150×150 pixels with size of 2μm × 2μm). (c) Data Set 3: experimental data of large size (492 × 656 pixels with size of 0.74μm × 0.74μm) obtained by a microscope.

Fig. 5
Fig. 5

Recovered intensity and phase [radian] image from synthetic Data Set 1 by ACEKF, diagonalized CEKF, and the proposed sparse ACEKF.

Fig. 6
Fig. 6

Estimated intensity and phase [radians] of Data Set 2 by ACEKF, diagonalized CEKF, and the proposed sparse ACEKF.

Fig. 7
Fig. 7

(a) Estimated height [nm] for Data Set 3 by ACEKF, diagonalized CEKF, and the proposed sparse ACEKF. (b) Depth along the black line in (a).

Tables (2)

Tables Icon

Table 1 Sparse augmented complex extended Kalman filter for estimating a wave field.

Tables Icon

Table 2 Comparison of different methods. Each image is divided into 4 blocks of size 50 × 50 for ACEKF, while the proposed sparse ACEKF and diagonalized CEKF processes the images without separating into blocks.

Equations (59)

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A ( x , y , z ) z = i λ 4 π 2 A ( x , y , z ) ,
p [ I ( x , y , z ) | A ( x , y , z ) ] = e γ | A ( x , y , z ) | 2 ( γ | A ( x , y , z ) | 2 ) I ( x , y , z ) I ( x , y , z ) ! ,
H = diag ( exp [ i λ π ( u 1 2 L x 2 + v 1 2 L y 2 ) Δ z ] , , exp [ i λ π ( u M 2 L x 2 + v N 2 L y 2 ) Δ z ] ) ,
b n = H b n 1 .
I n = γ | a n | 2 + v ,
I n = γ | a ^ n | 2 + γ diag ( a ^ n * ) ( a n a ^ n ) + γ diag ( a ^ n ) ( a n * a ^ n * ) + v ,
state : [ b n b n * ] = [ H 0 0 H * ] [ b n 1 b n 1 * ]
observation : I n = [ J n J n * ] [ b n b n * ] γ | a ^ n | 2 + v , with v ~ ( 0 , R ) ,
R = γ diag ( a ^ n * ) diag ( a ^ n ) and J n = γ diag ( a ^ n * ) K H .
S n = [ S n Q S n P ( S n P ) * ( S n Q ) * ]
S ^ n Q = H S n 1 Q H H ,
S ^ n P = H S n 1 P H .
S n Q = S ^ n Q ( S ^ n Q J n H + S ^ n P J n T ) ( J n S ^ n Q J n H + J n S ^ n P J n T + J n * ( S ^ n Q ) * J n T + J n * ( S ^ n P ) * J n H + R ) 1 ( J n S ^ n Q + J n * ( S ^ n P ) * ) ,
S n P = S ^ n P ( S ^ n Q J n H + S ^ n P J n T ) ( J n S ^ n Q J n H + J n S ^ n P J n T + J n * ( S ^ n Q ) * J n T + J n * ( S ^ n P ) * J n H + R ) 1 ( J n S ^ n P + J n * ( S ^ n Q ) * ) ,
G n = ( S n Q J n H + S n P J n T ) R 1 ,
b n = b ^ n + G n ( I n γ | a ^ n | 2 ) .
S n 1 Q = Q n 1
S n 1 P = P n 1 E ,
Q ^ n = Q n 1
P ^ n = H P n 1 H
Q n = Q ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( Q ^ n + ( P ^ n ) * )
P n = P ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( P ^ n + ( Q ^ n ) * )
S n Q = Q n
S n P = P n E ,
G n = ( S n Q J n H + S n P J n T ) R 1 = ( Q n + P n ) ( J n ) 1 q 1 .
b n = b ^ n + G n ( I n γ | a ^ n | 2 ) .
b ^ n = H b n 1
Q ^ n = Q n 1
P ^ n = H P n 1 H
a ^ n = K H b ^ n
Q n = Q ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( Q ^ n + ( P ^ n ) * )
P n = P ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( P ^ n + ( Q ^ n ) * )
b n = b ^ n + ( Q n + P n ) ( J n ) 1 q 1 ( I n γ | a ^ n | 2 ) .
M = 1 2 [ U U j U j U ] ,
[ Re ( b n ) Im ( b n ) ] = M [ b n b n * ] ,
state : [ Re ( b n ) Im ( b n ) ] = A n [ Re ( b n 1 ) Im ( b n 1 ) ] ,
observation : I n = C n [ Re ( b n ) Im ( b n ) ] γ | a ^ n | 2 + v ,
A n = M [ H 0 0 H * ] M 1 ,
C n = [ J n J n * ] M 1 .
A n a ¯ ,
C n c ¯ ,
s _ U S n s ¯ U ,
v _ U v .
P ^ n = H P n 1 H
Q ^ n = H Q n 1 H H = Q n 1 .
S ^ n Q = H S n 1 Q H H + H Q n 1 H H = Q ^ n
S ^ n P = HS n 1 P H = HP n 1 EH = HP n 1 HE = P ^ n E .
S n Q = S ^ n Q ( S ^ n Q J n H + S ^ n P J n T ) ( J n S ^ n Q J n H + J n S ^ n P J n T + J n * ( S ^ n Q ) * J n T + J n * ( S ^ n P ) * J n H + R ) 1 ( J n S ^ n Q + J n * ( S ^ n P ) * ) .
S n Q = S ^ n Q ( S ^ n Q K D H + S ^ n P K * D T ) ( D K H S ^ n Q K D H + D K H S ^ n P K * D T + D * K T ( S ^ n Q ) * K * D T + D * K T ( S ^ n P ) * K D H + q D K H K D H ) 1 ( D K H S ^ n Q + D * K T ( S ^ n P ) * ) ,
S n Q = S ^ n Q ( S ^ n Q K + S ^ n P K * D ( D * ) 1 ) ( K H S ^ n Q K + K H S ^ n P K * D ( D * ) 1 + D 1 D * K T ( S ^ n Q ) * K * D ( D * ) 1 + D 1 D * K T ( S ^ n P ) * K + q K H K ) 1 ( K H S ^ n Q + D 1 D * K T ( S ^ n P ) * ) .
S n Q = S ^ n Q ( S ^ n Q K + S ^ n P K * ) ( K H S ^ n Q K + K H S ^ n P K * + K T ( S ^ n Q ) * K * + K T ( S ^ n P ) * K + q K H K ) 1 ( K H S ^ n Q + K T ( S ^ n P ) * ) .
S n Q = S ^ n Q ( S ^ n Q K + S ^ n P K * ) ( K H ( S ^ n Q + S ^ n P K * K H + K K T ( S ^ n Q ) * K * K H + K K T ( S ^ n P ) * + q I ) K ) 1 ( K H S ^ n Q + K T ( S ^ n P ) * ) .
S n Q = S ^ n Q ( S ^ n Q + S ^ n P K * K H ) ( S ^ n Q + S ^ n P K * K H + K K T ( S ^ n Q ) * K * K H + K K T ( S ^ n P ) * + q I ) 1 ( S ^ n Q + K K T ( S ^ n P ) * ) .
S n P = S ^ n P ( S ^ n Q + S ^ n P K * K H ) ( S ^ n Q + S ^ n P K * K H + K K T ( S ^ n Q ) * K * K H + K K T ( S ^ n P ) * + q I ) 1 ( S ^ n P + K K T ( S ^ n Q ) * ) .
S n Q = Q ^ n ( Q ^ n P ^ n EE ) ( Q ^ n + P ^ n EE + E ( Q ^ n ) * E + E ( P ^ n ) * E + q I ) 1 ( Q ^ n + E ( P ^ n ) * E )
S n P = P ^ n E ( Q ^ n + P ^ n EE ) ( Q ^ n + P ^ n EE + E ( Q ^ n ) * E + E ( P ^ n ) * E + q I ) 1 ( P ^ n E + E ( Q ^ n ) * EE ) .
S n Q = Q ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( Q ^ n + ( P ^ n ) * )
S n P = P ^ n E ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( P ^ n E + ( Q ^ n ) * E ) = ( P ^ n ( Q ^ n + P ^ n ) ( Q ^ n + P ^ n + ( Q ^ n ) * + ( P ^ n ) * + q I ) 1 ( P ^ n + ( Q ^ n ) * ) ) E .
G n = ( S n Q J n H + S n P J n T ) R 1 = ( S n Q J n H + S n P J n T ) ( J n H ) 1 ( J n ) 1 q 1 = ( S n Q + S n P E ) ( J n ) 1 q 1 = ( Q n + P n ) ( J n ) 1 q 1 .

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