Abstract

What we believe to be a novel three-dimensional (3D) phase unwrapping algorithm is proposed to unwrap 3D wrapped-phase volumes. It depends on a quality map to unwrap the most reliable voxels first and the least reliable voxels last. The technique follows a discrete unwrapping path to perform the unwrapping process. The performance of this technique was tested on both simulated and real wrapped-phase maps. And it is found to be robust and fast compared with other 3D phase unwrapping algorithms.

© 2007 Optical Society of America

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  2. M. D. Pritt and J. S. Shipman, "Least-square two-dimensional phase unwrapping using FFTs," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
    [CrossRef]
  3. D. C. Ghiglia and L. A. Romero, "Minimum Lp-norm two-dimensional phase unwrapping," J. Opt. Soc. Am. A 13, 1999-2013 (1996).
    [CrossRef]
  4. R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
    [CrossRef]
  5. S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Two-dimensional phase unwrapping using a hybrid genetic algorithm," Appl. Opt. 46, 730-743 (2007).
    [CrossRef] [PubMed]
  6. M. A. Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, "Robust, simple, and fast algorithm for phase unwrapping," Appl. Opt. 35, 5847-5852 (1996).
    [CrossRef] [PubMed]
  7. W. Xu and I. Cumming, "A region-growing algorithm for InSAR phase unwrapping," IEEE Trans. Geosci. Remote Sens. 34, 2044-2046 (1996).
  8. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Appl. Opt. 41, 7437-7444 (2002).
    [CrossRef] [PubMed]
  9. F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
    [CrossRef]
  10. M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
    [CrossRef]
  11. R. Cusack and N. Papadakis, "New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2001).
    [CrossRef]
  12. X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
    [CrossRef]
  13. J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  14. M. Jenkinson, "Fast, automated, N-dimensional phase unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
    [CrossRef] [PubMed]
  15. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
    [CrossRef]
  16. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

2007 (1)

2006 (1)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

2005 (1)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

2003 (1)

M. Jenkinson, "Fast, automated, N-dimensional phase unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

2001 (3)

R. Cusack and N. Papadakis, "New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2001).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

2000 (1)

F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

1998 (1)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

1996 (3)

1995 (1)

R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
[CrossRef]

1994 (1)

M. D. Pritt and J. S. Shipman, "Least-square two-dimensional phase unwrapping using FFTs," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Abdul-Rahman, H. S.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

Burton, D. R.

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Two-dimensional phase unwrapping using a hybrid genetic algorithm," Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, "Robust, simple, and fast algorithm for phase unwrapping," Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Chen, W.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Clegg, D. B.

Costanitini, M.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

Cumming, I.

W. Xu and I. Cumming, "A region-growing algorithm for InSAR phase unwrapping," IEEE Trans. Geosci. Remote Sens. 34, 2044-2046 (1996).

Cusack, R.

R. Cusack and N. Papadakis, "New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2001).
[CrossRef]

R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
[CrossRef]

Gdeisat, M. A.

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Two-dimensional phase unwrapping using a hybrid genetic algorithm," Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

D. C. Ghiglia and L. A. Romero, "Minimum Lp-norm two-dimensional phase unwrapping," J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

Goldrein, H. T.

R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
[CrossRef]

Herráez, M. A.

Huntley, J. M.

J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

R. Cusack, J. M. Huntley, and H. T. Goldrein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
[CrossRef]

Jenkinson, M.

M. Jenkinson, "Fast, automated, N-dimensional phase unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
[CrossRef] [PubMed]

Karout, S. A.

Lalor, M. J.

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Two-dimensional phase unwrapping using a hybrid genetic algorithm," Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, "Robust, simple, and fast algorithm for phase unwrapping," Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

Lilley, F.

F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

Malvarosa, F.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

Milillo, G.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

Minati, L.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

Papadakis, N.

R. Cusack and N. Papadakis, "New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2001).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

M. D. Pritt and J. S. Shipman, "Least-square two-dimensional phase unwrapping using FFTs," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Romero, L. A.

Shipman, J. S.

M. D. Pritt and J. S. Shipman, "Least-square two-dimensional phase unwrapping using FFTs," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Su, X.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Xu, W.

W. Xu and I. Cumming, "A region-growing algorithm for InSAR phase unwrapping," IEEE Trans. Geosci. Remote Sens. 34, 2044-2046 (1996).

Zhang, Q.

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (3)

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, "A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements," IEEE Trans. Geosci. Remote Sens. 40, 1741-1743 (2002).
[CrossRef]

M. D. Pritt and J. S. Shipman, "Least-square two-dimensional phase unwrapping using FFTs," IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

W. Xu and I. Cumming, "A region-growing algorithm for InSAR phase unwrapping," IEEE Trans. Geosci. Remote Sens. 34, 2044-2046 (1996).

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

Magn. Reson. Med. (1)

M. Jenkinson, "Fast, automated, N-dimensional phase unwrapping algorithm," Magn. Reson. Med. 49, 193-197 (2003).
[CrossRef] [PubMed]

Neuroimage (1)

R. Cusack and N. Papadakis, "New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images," Neuroimage 16, 754-764 (2001).
[CrossRef]

Opt. Eng. (1)

F. Lilley, M. J. Lalor, and D. R. Burton, "Robust fringe analysis system for human body shape measurement," Opt. Eng. 39, 187-195 (2000).
[CrossRef]

Opt. Lasers Eng. (1)

X. Su, W. Chen, Q. Zhang, and Y. Chao, "Dynamic 3D-shape measurement method based on FTP," Opt. Lasers Eng. 36, 49-64 (2001).
[CrossRef]

Proc. SPIE (1)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path," Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

Other (2)

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, "Three-dimensional phase unwrapping algorithms: a comparison," presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).

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Figures (13)

Fig. 1
Fig. 1

(Color online) Definition of the edges and their qualities.

Fig. 2
Fig. 2

(Color online) Demonstration of the unwrapping path of the proposed algorithm.

Fig. 3
Fig. 3

Results for the last frame ( t = 99 ) of the simulated spherical object. (a) Wrapped phase, the unwrapped phase using the (b) flood-fill algorithm, (c) Cusack algorithm, (d) Huntley algorithm, and (e) proposed algorithm.

Fig. 4
Fig. 4

Three-dimensional representation for the first frame ( t = 0 ) of the simulated steep surface object.

Fig. 5
Fig. 5

Results for the first frame ( t = 0 ) of the simulated steep surface object. (a) Wrapped phase, the unwrapped phase using the (b) flood-fill algorithm, (c) Cusack algorithm, (d) Huntley algorithm, and (e) proposed algorithm.

Fig. 6
Fig. 6

Results for the last frame ( t = 99 ) of the simulated steep surface object. (a) Wrapped phase, the unwrapped phase using the (b) flood-fill algorithm, (c) Cusack algorithm, (d) Huntley algorithm, and (e) proposed algorithm.

Fig. 7
Fig. 7

Effect of quality map on the unwrapping path for a simulated spherical object, (a) wrapped-phase map at frame number 75 ( t = 75 ) , unwrapped-phase map for the same frame resulting from using: (b) pseudocorrelation, (c) maximum gradient, (d) phase derivative variance, (e) second difference using 26 neighbors, and (f) second difference using only six orthogonal neighbors.

Fig. 8
Fig. 8

Results of frame 25 of the mannequin's chest: (a) wrapped phase, (b) unwrapped phase using the (b) Cusack algorithm, (c) Huntley algorithm, and (d) proposed algorithm.

Fig. 9
Fig. 9

Results for frames 0, 25, 35, and 49 of the RANDO dummy's face shown in rows 1, 2, 3, and 4, respectively, (column a) represents the wrapped-phase maps for these particular frames, the unwrapped-phase maps of these frames resulting from using the (column b) Cusack algorithm, (column c) Huntley algorithm, and (column d) proposed algorithm.

Fig. 10
Fig. 10

Three-dimensional view for the unwrapped-phase maps for frame 49 of the RANDO dummy's face using the (a) Cusack algorithm, (b) Huntley algorithm, and (c) proposed algorithm.

Fig. 11
Fig. 11

Three different frames of fringe patterns for a patient undergoing radiotherapy treatment for breast cancer. (a) Frame 0, (b) frame (15), and (c) frame (24).

Fig. 12
Fig. 12

Results showing a region from a female human thorax, taken from a real clinical patient undergoing treatment for breast cancer. Wrapped-phase images of frames 0, 15, and 24 are shown in (a), (b), and (c), respectively. The unwrapped-phase for those frames are shown underneath, using the (d)–(f) Cusack algorithm, (g)–(i) Huntley algorithm, and (j)–(l) proposed algorithm.

Fig. 13
Fig. 13

Three-dimensional view for the unwrapped-phase maps for frame 24, showing a clinical patient's breast using the (a) Cusack algorithm, (b) Huntley algorithm, and (c) proposed algorithm.

Tables (1)

Tables Icon

Table 1 Execution Times in Seconds for the Proposed Algorithm, the Cusack Algorithm, and the 3D Noise Immune Algorithm (Huntley Algorithm)

Equations (23)

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Q m , n , l = 1 27 ( ( cos ( ψ i , j , k ) ) 2 + ( sin ( ψ i , j , k ) ) 2 ) 1 / 2 ,
V m , n , l = 1 27 ( ( Δ i , j , k x Δ i , j , k x ¯ ) 2 + ( Δ i , j , k y Δ i , j , k y ¯ ) 2 + ( Δ i , j , k z Δ i , j , k z ¯ ) 2 ) ,
Δ i , j , k x = γ [ ψ i + 1 , j , k ψ i , j , k ] ,
Δ i , j , k y = γ [ ψ i , j + 1 , k ψ i , j , k ] ,
Δ i , j , k z = γ [ ψ i , j , k + 1 ψ i , j , k ] ,
Q m , n , l = 1 V m , n , l .
M m , n , l = max { max { | Δ i , j , k x | } , max { | Δ i , j , k y | } , max { | Δ i , j , k z | } } .
S D i , j , k = H 2 ( i , j , k ) + V 2 ( i , j , k ) + N 2 ( i , j , k ) + n = 1 10 D n 2 ( i , j , k ) ,
H ( i , j , k ) = γ [ φ ( i 1 , j , k ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i + 1 , j , k ) ] ,
V ( i , j , k ) = γ [ φ ( i , j 1 , k ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i , j + 1 , k ) ] ,
N ( i , j , k ) = γ [ φ ( i , j , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i , j , k + 1 ) ] ,
D 1 ( i , j , k ) = γ [ φ ( i 1 , j 1 , k ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i + 1 , j + 1 , k ) ] ,
D 2 ( i , j , k ) = γ [ φ ( i + 1 , j 1 , k ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i 1 , j + 1 , k ) ] ,
D 3 ( i , j , k ) = γ [ φ ( i 1 , j 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i + 1 , j + 1 , k + 1 ) ] ,
D 4 ( i , j , k ) = γ [ φ ( i , j 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i , j + 1 , k + 1 ) ] ,
D 5 ( i , j , k ) = γ [ φ ( i + 1 , j 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i 1 , j + 1 , k + 1 ) ] ,
    D 6 ( i , j , k ) = γ [ φ ( i 1 , j , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i + 1 , j , k + 1 ) ] ,
D 7 ( i , j , k ) = γ [ φ ( i 1 , j + 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i + 1 , j 1 , k + 1 ) ] ,
D 8 ( i , j , k ) = γ [ φ ( i + 1 , j , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i 1 , j , k + 1 ) ] ,
D 9 ( i , j , k ) = γ [ φ ( i , j + 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i , j 1 , k + 1 ) ] ,
D 10 ( i , j , k ) = γ [ φ ( i + 1 , j + 1 , k 1 ) φ ( i , j , k ) ] γ [ φ ( i , j , k ) φ ( i 1 , j 1 , k + 1 ) ] .
z i , j , t = 6.12 [ ( 1 x 1 2 ) exp ( x 1 2 ( y 1 + 1 ) 2 ) ] 20.6 [ ( x 1 5 x 1 3 y 1 5 ) exp ( x 1 2 y 1 2 ) ] 0.68 [ exp ( ( x 1 + 1 ) 2 y 1 2 ) ] + 0.1 ( x 2 + y 2 ) + ( 0.01 t ) ,
N o i s e ( a t f r a m e t ) = N o i s e ( a t f r a m e t 1 ) + 256 × 256 G a u s s i a n n o i s e ( m e a n = 0 , σ = 0.165 ) .

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