Abstract

A novel dual-frequency pattern is developed which combines a high-frequency sinusoid component with a unit-frequency sinusoid component, where the high-frequency component is used to generate robust phase information, and the unit-frequency component is used to reduce phase unwrapping ambiguities. With our proposed pattern scheme, phase unwrapping can overcome the major shortcomings of conventional spatial phase unwrapping: phase jumping and discontinuities. Compared with conventional temporal phase unwrapped approaches, the proposed pattern scheme can achieve higher quality phase data using a less number of patterns. To process data in real time, we also propose and develop look-up table based fast and accurate algorithms for phase generation and 3-D reconstruction. Those fast algorithms can be applied to our pattern scheme as well as traditional phase measuring profilometry. For a 640×480 video stream, we can generate phase data at 1063.8 frames per second and full 3-D coordinate point clouds at 228.3 frames per second. These achievements are 25 and 10 times faster than previously reported studies.

© 2010 Optical Society of America

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References

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2009 (1)

2008 (2)

S. Li, W. Chen, and X. Su, "Reliability-guided phase unwrapping in wavelet-transform profilometry," Appl. Opt. 47, 3369-3377 (2008).
[CrossRef] [PubMed]

S. Y. Chen, Y. F. Li, and J. Zhang, "Vision processing for realtime 3-d data acquisition based on coded structured light," IEEE Trans. Image Processing 17, 167-176 (2008).
[CrossRef]

2007 (4)

2006 (5)

2005 (1)

2004 (1)

J. Salvi, J. Pages, and J. Batlle, "Pattern codification strategies in structured light systems," Pattern Recognition 37, 827-849 (2004).
[CrossRef]

2003 (3)

1998 (1)

M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geoscience and Remote Sensing 36, 813-821 (1998).
[CrossRef]

1997 (2)

1993 (1)

X. Su, G. von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

1990 (1)

1989 (1)

M. Halioua and H. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers in Eng. 11, 185-215 (1989).
[CrossRef]

1983 (1)

Alexeenko, I.

Batlle, J.

J. Salvi, J. Pages, and J. Batlle, "Pattern codification strategies in structured light systems," Pattern Recognition 37, 827-849 (2004).
[CrossRef]

Chen, S. Y.

S. Y. Chen, Y. F. Li, and J. Zhang, "Vision processing for realtime 3-d data acquisition based on coded structured light," IEEE Trans. Image Processing 17, 167-176 (2008).
[CrossRef]

Chen, W.

Costantini, M.

M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geoscience and Remote Sensing 36, 813-821 (1998).
[CrossRef]

Dubey, S. K.

Gool, L. V.

T. P. Koninckx and L. V. Gool, "Real-time range acquisition by adaptive structured light," IEEE Trans. Pattern Anal. Mach Intell. 28, 432-445 (2006).
[CrossRef] [PubMed]

Guan, C.

Guo, H.

Hahn, J.

Halioua, M.

M. Halioua and H. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers in Eng. 11, 185-215 (1989).
[CrossRef]

Hassebrook, L. G.

Hossain, M. M.

Huang, P. S.

P. S. Huang and S. Zhang, "Fast three-step phase-shifting algorithm," Appl. Opt. 45, 5086-5091 (2006).
[CrossRef] [PubMed]

S. Zhang and P. S. Huang, "High-resolution, real-time three-dimensional shape measurement," Opt. Eng. 45, 123601 (2006).
[CrossRef]

Huntley, J. M.

Kim, E.-H.

Kim, H.

Koninckx, T. P.

T. P. Koninckx and L. V. Gool, "Real-time range acquisition by adaptive structured light," IEEE Trans. Pattern Anal. Mach Intell. 28, 432-445 (2006).
[CrossRef] [PubMed]

Lau, D. L.

Lee, B.

Li, J.

Li, J.-L.

Li, S.

Li, X.

Li, Y. F.

S. Y. Chen, Y. F. Li, and J. Zhang, "Vision processing for realtime 3-d data acquisition based on coded structured light," IEEE Trans. Image Processing 17, 167-176 (2008).
[CrossRef]

Liu, G.

Liu, H.

Mehta, D. S.

Mutoh, K.

Osten, W.

Pages, J.

J. Salvi, J. Pages, and J. Batlle, "Pattern codification strategies in structured light systems," Pattern Recognition 37, 827-849 (2004).
[CrossRef]

Pedrini, G.

Royer, D.

Saldner, H. O.

Salvi, J.

J. Salvi, J. Pages, and J. Batlle, "Pattern codification strategies in structured light systems," Pattern Recognition 37, 827-849 (2004).
[CrossRef]

Shakher, C.

Shi, Y.

Su, H.-J.

Su, W.-H.

Su, X.

S. Li, W. Chen, and X. Su, "Reliability-guided phase unwrapping in wavelet-transform profilometry," Appl. Opt. 47, 3369-3377 (2008).
[CrossRef] [PubMed]

X. Su, G. von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

Su, X.-Y.

Takeda, M.

Tiziani, H. J.

von Bally, G.

X. Su, G. von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

Vukicevic, D.

X. Su, G. von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

Wizinowich, P. L.

Yau, S.-T.

Zhang, J.

S. Y. Chen, Y. F. Li, and J. Zhang, "Vision processing for realtime 3-d data acquisition based on coded structured light," IEEE Trans. Image Processing 17, 167-176 (2008).
[CrossRef]

Zhang, S.

Appl. Opt. (8)

IEEE Trans. Geoscience and Remote Sensing (1)

M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geoscience and Remote Sensing 36, 813-821 (1998).
[CrossRef]

IEEE Trans. Image Processing (1)

S. Y. Chen, Y. F. Li, and J. Zhang, "Vision processing for realtime 3-d data acquisition based on coded structured light," IEEE Trans. Image Processing 17, 167-176 (2008).
[CrossRef]

IEEE Trans. Pattern Anal. Mach Intell. (1)

T. P. Koninckx and L. V. Gool, "Real-time range acquisition by adaptive structured light," IEEE Trans. Pattern Anal. Mach Intell. 28, 432-445 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

X. Su, G. von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993).
[CrossRef]

Opt. Eng. (2)

S. Zhang and S.-T. Yau, "High-speed three-dimensional shape measurement system using a modified two-plusone phase-shifting algorithm," Opt. Eng. 46, 113603 (2007).
[CrossRef]

S. Zhang and P. S. Huang, "High-resolution, real-time three-dimensional shape measurement," Opt. Eng. 45, 123601 (2006).
[CrossRef]

Opt. Express (6)

Opt. Lasers in Eng. (1)

M. Halioua and H. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers in Eng. 11, 185-215 (1989).
[CrossRef]

Pattern Recognition (1)

J. Salvi, J. Pages, and J. Batlle, "Pattern codification strategies in structured light systems," Pattern Recognition 37, 827-849 (2004).
[CrossRef]

Other (3)

S. Zhang, D. Royer, and S. T. Yau, "High-resolution, real-time-geometry video acquisition system," ACM SIGGRAPH (2006).

Y. Wang, Q. Hao, A. Fatehpuria, D. L. Lau, and L. G. Hassebrook, "Data acquisition and quality analysis of 3-dimensional fingerprints," in IEEE conference on Biometrics, Identity and Security, Tampa, Florida, (2009).

S. F. Frisken, R. N. Perry, A. P. Rockwood, and T. R. Jones, "Adaptively sampled distance fields: A general representation of shape for computer graphics," in Proceedings of the 27th annual conference on Computer graphics and interactive techniques, 249-254 (2000).

Supplementary Material (3)

» Media 1: MOV (1376 KB)     
» Media 2: MOV (4037 KB)     
» Media 3: MOV (3925 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

PMP system diagram.

Fig. 2.
Fig. 2.

Cross-section of the proposed pattern strategy for N = 5, n = 0, fh = 16, Ap = 127.5, Bp 1 = 102, and Bp 2 = 25.5.

Fig. 3.
Fig. 3.

Experimental setup.

Fig. 4.
Fig. 4.

Visualization of the scanned object (top left), one of captured images with projected pattern, Ic 0 (top center), Ac (top right), Bc 1 (bottom left), binarized Bc 1 with threshold of 10 (bottom center), and Bc 2 (bottom right).

Fig. 5.
Fig. 5.

The (top) visualization and (bottom) cross-section at the 250th column of the (left) wrapped phase ϕh filtered by binarized Bc 1, the (center) base phase ϕu filtered by Bc 1, and the (right) final unwrapped phase filtered by Bc 1 (right).

Fig. 6.
Fig. 6.

the reconstructed 3-D point clouds with texture (top) and depth rendering (bottom) in front view (left), side view (center) and top view (right)

Fig. 7.
Fig. 7.

Sample video clips.(Media 1), (Media 2), (Media 3)

Fig. 8.
Fig. 8.

Sample point clouds, using dual-frequency pattern scheme for N = 6, live show of various hand poses. (top) Front view and (middle) Top view.

Fig. 9.
Fig. 9.

Radius of best-fit sphere (top) and distance between the camera and the center of the ball (bottom) for 295 measurements.

Tables (1)

Tables Icon

Table 1. Processing time and rate, in milliseconds and frames per second (in parentheses), respectively, for various stages of PMP processing by means of the equations and LUT described in this paper.

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

I n p ( x p , y p ) = A p ( x p , y p ) + B p ( x p , y p ) cos ( 2 πf y p 2 πn N ) .
I n c ( x c , y c ) = A c ( x c , y c ) + B c ( x c , y c ) cos ( ϕ ( x c , y c ) 2 πn N ) .
A c = 1 N n = 0 N 1 I n c
B c = 2 N { [ n = 0 N 1 I n c sin ( 2 πn N ) ] 2 + [ n = 0 N 1 I n c cos ( 2 πn N ) ] 2 } 0.5 ,
ϕ = tan 1 [ n = 0 N 1 I n c sin ( 2 πn N ) n = 0 N 1 I n c cos ( 2 πn N ) ] .
M c = m 11 c m 12 c m 13 c m 14 c m 21 c m 22 c m 23 c m 24 c m 31 c m 32 c m 33 c m 34 c and M p = m 11 p m 12 p m 13 p m 14 p m 21 p m 22 p m 23 p m 24 p m 31 p m 32 p m 33 p m 34 p .
X w Y w Z w = m 11 c m 31 c x c , m 12 c m 32 c x c , m 13 c m 33 c x c m 21 c m 31 c y c , m 22 c m 32 c y c , m 23 c m 33 c y c m 21 p m 31 p y p , m 22 p m 32 p y p , m 23 p m 33 p y p 1 m 14 c m 34 c x c m 24 c m 34 c y c m 24 p m 34 p y p .
I n p = A p + B 1 p cos ( 2 π f h y p 2 πn N ) + B 2 p cos ( 2 π f u y p 4 πn N ) .
I n c = A c + B 1 c cos ( ϕ h 2 πn N ) + B 2 c cos ( ϕ u 4 πn N ) ,
B m c = 2 N { [ n = 0 N 1 I n c sin ( m 2 πn N ) ] 2 + [ n = 0 N 1 I n c cos ( m 2 πn N ) ] 2 } 0.5 ,
( ϕ h , ϕ u ) = ( tan 1 [ n = 0 N 1 I n c sin ( 2 πn N ) n = 0 N 1 I n c cos ( 2 πn N ) ] , tan 1 [ n = 0 N 1 I n c sin ( 4 πn N ) n = 0 N 1 I n c cos ( 4 πn N ) ] , )
B c = 1 3 [ 3 ( I 1 c I 2 c ) 2 + ( 2 I 0 c I 1 c I 2 c ) 2 ] 0.5
B c = 1 2 [ ( I 1 c I 3 c ) 2 + ( I 0 c I 2 c ) 2 ] 0.5
B c = 1 6 [ 3 ( I 1 c + I 2 c I 4 c I 5 c ) 2 + ( 2 I 0 c 2 I 3 c + I 1 c I 2 c I 4 c + I 5 c ) 2 ] 0.5
MLUT [ U , V ] = 1 3 [ 3 V 2 + U 2 ] 0.5 ,
V = I 1 c I 2 c and U = 2 I 0 c I 1 c I 2 c ;
MLUT [ U , V ] = 1 2 [ V 2 + U 2 ] 0.5 ,
V = I 1 c I 3 c and U = I 0 c I 2 c ;
MLUT [ U , V ] = 1 6 [ 3 V 2 + U 2 ] 0.5 ,
V = I 1 c + I 2 c I 4 c I 5 c and U = 2 I 0 c 2 I 3 c + I 1 c I 2 c I 4 c + I 5 c .
B 1 c = 1 6 [ 3 ( I 1 c + I 2 c I 4 c I 5 c ) 2 + ( 2 I 0 c 2 I 3 c + I 1 c I 2 c I 4 c + I 5 c ) 2 ] 0.5
B 2 c = 1 6 [ 3 ( I 1 c I 2 c + I 4 c I 5 c ) 2 + ( 2 I 0 c + 2 I 3 c + I 1 c I 2 c I 4 c I 5 c ) 2 ] 0.5 .
ϕ = tan 1 [ 3 0.5 ( I 1 c I 2 c ) 2 I 0 c I 1 c I 2 c ]
ϕ = tan 1 [ I 1 c I 3 c I 0 c I 2 c ]
ϕ = tan 1 [ 3 0.5 ( I 1 c + I 2 c I 4 c I 5 c ) 2 I 0 c 2 I 3 c + I 1 c I 2 c I 4 c + I 5 c ]
PLUT [ U , V ] = tan 1 [ 3 0.5 V U ] ,
PLUT [ U , V ] = tan 1 [ V U ] ,
PLUT [ U , V ] = tan 1 [ 3 0.5 V U ] ,
( ϕ h , ϕ u ) = ( tan 1 [ 3 0.5 ( I 1 c + I 2 c I 4 c I 5 c ) 2 I 0 c 2 I 3 c + I 1 c I 2 c I 4 c + I 5 c ] , tan 1 [ 3 0.5 ( I 1 c I 2 c + I 4 c I 5 c ) 2 I 0 c + 2 I 3 c I 1 c I 2 c I 4 c I 5 c ] )
PLUT [ U , V ] = tan 1 [ 3 0.5 V U ] ,
V = I 1 c I 2 c + I 4 c I 5 c and U = 2 I 0 c + 2 I 3 c I 1 c I 2 c I 4 c I 5 c .
X w = M x ( x c , y c ) + N x ( x c , y c ) T
Y w = M y ( x c , y c ) + N y ( x c , y c ) T
Z w = M z ( x c , y c ) + N z ( x c , y c ) T
Z w = M z ( x c , y c ) + N z ( x c , y c ) T ,
X w = E x ( x c , y c ) Z w + F x ( x c , y c ) and Y w = E y ( x c , y c ) Z w + F y ( x c , y c ) ,
E x ( x c , y c ) = ( m 22 c m 33 c m 23 c m 32 c ) x c + ( m 13 c m 32 c m 12 c m 33 c ) y c + ( m 12 c m 23 c m 13 c m 22 c ) ( m 21 c m 32 c m 22 c m 31 c ) x c + ( m 12 c m 31 c m 11 c m 32 c ) y c + ( m 11 c m 22 c m 12 c m 21 c ) ,
F x ( x c , y c ) = ( m 22 c m 34 c m 24 c m 32 c ) x c + ( m 14 c m 32 c m 12 c m 34 c ) y c + ( m 12 c m 24 c m 14 c m 22 c ) ( m 21 c m 32 c m 22 c m 31 c ) x c + ( m 12 c m 31 c m 11 c m 32 c ) y c + ( m 11 c m 22 c m 12 c m 21 c ) ,
E y ( x c , y c ) = ( m 23 c m 31 c m 21 c m 33 c ) x c + ( m 11 c m 33 c m 13 c m 31 c ) y c + ( m 13 c m 21 c m 11 c m 23 c ) ( m 21 c m 32 c m 22 c m 31 c ) x c + ( m 12 c m 31 c m 11 c m 32 c ) y c + ( m 11 c m 22 c m 12 c m 21 c ) , and
F y ( x c , y c ) = ( m 21 c m 32 c m 22 c m 31 c ) x c + ( m 12 c m 31 c m 11 c m 32 c ) y c + ( m 11 c m 22 c m 12 c m 21 c ) ( m 21 c m 32 c m 22 c m 31 c ) x c + ( m 12 c m 31 c m 11 c m 32 c ) y c + ( m 11 c m 22 c m 12 c m 21 c ) .
M x ( x c , y c ) = ( a x 1 x c + a x 2 y c + a x 3 d 1 x c + d 2 y c + d 3 ) ( c 1 x c + c 2 y c + c 3 d 1 x c + d 2 y c + d 3 ) , N x ( x c , y c ) = b x 1 x c + b x 2 y c + b x 3 d 1 x c + d 2 y c + d 3 M x ( x c , y c ) ,
M y ( x c , y c ) = ( a y 1 x c + a y 2 y c + a y 3 d 1 x c + d 2 y 2 + d 3 ) ( c 1 x c + c 2 y c + c 3 d 1 x c + d 2 y c + d 3 ) , N y ( x c , y c ) = b y 1 x c + b y 2 y c + b y 3 d 1 x c + d 2 y c + d 3 M y ( x c , y c ) ,
M z ( x c , y c ) = ( a z 1 x c + a z 2 y c + a z 3 d 1 x c + d 2 y 2 + d 3 ) ( c 1 x c + c 2 y c + c 3 d 1 x c + d 2 y c + d 3 ) , N z ( x c , y c ) = b z 1 x c + b z 2 y c + b z 3 d 1 x c + d 2 y c + d 3 M z ( x c , y c ) ,
a x 1 = m 22 c m 34 c m 33 p + m 23 c m 32 c m 34 p + m 24 c m 33 c m 32 p m 22 c m 33 c m 34 p m 23 c m 34 c m 32 p m 24 c m 32 c m 33 p ,
a x 2 = m 12 c m 33 c m 34 p + m 13 c m 34 c m 32 p + m 14 c m 32 c m 33 p m 12 c m 34 c m 33 p m 13 c m 32 c m 34 p m 14 c m 33 c m 32 p ,
a x 3 = m 12 c m 24 c m 33 p + m 13 c m 22 c m 34 p + m 14 c m 23 c m 32 p m 12 c m 23 c m 34 p m 13 c m 24 c m 32 p m 14 c m 22 c m 33 p ,
a y 1 = m 21 c m 33 c m 34 p + m 23 c m 34 c m 31 p + m 24 c m 31 c m 33 p m 21 c m 34 c m 33 p m 23 c m 31 c m 34 p m 24 c m 33 c m 31 p ,
a y 2 = m 11 c m 34 c m 33 p + m 13 c m 31 c m 34 p + m 14 c m 33 c m 31 p m 11 c m 33 c m 34 p m 13 c m 34 c m 31 p m 14 c m 31 c m 33 p ,
a y 3 = m 11 c m 23 c m 34 p + m 13 c m 24 c m 31 p + m 14 c m 21 c m 33 p m 11 c m 24 c m 33 p m 13 c m 21 c m 34 p m 14 c m 23 c m 31 p ,
a z 1 = m 21 c m 34 c m 32 p + m 22 c m 31 c m 34 p + m 24 c m 32 c m 31 p m 21 c m 32 c m 34 p m 22 c m 34 c m 31 p m 24 c m 31 c m 32 p ,
a z 2 = m 11 c m 32 c m 34 p + m 12 c m 34 c m 31 p + m 14 c m 31 c m 32 p m 11 c m 34 c m 32 p m 12 c m 31 c m 34 p m 14 c m 32 c m 31 p ,
a z 3 = m 11 c m 24 c m 32 p + m 12 c m 21 c m 34 p + m 14 c m 22 c m 31 p m 11 c m 22 c m 34 p m 12 c m 24 c m 31 p m 14 c m 21 c m 32 p ,
b x 1 = m 22 c m 33 c m 24 p + m 23 c m 34 c m 22 p + m 24 c m 32 c m 23 p m 22 c m 34 c m 23 p m 23 c m 32 c m 24 p m 24 c m 33 c m 22 p ,
b x 2 = m 12 c m 34 c m 23 p + m 13 c m 32 c m 24 p + m 14 c m 33 c m 22 p m 12 c m 33 c m 24 p m 13 c m 34 c m 22 p m 14 c m 32 c m 23 p ,
b x 3 = m 12 c m 23 c m 24 p + m 13 c m 24 c m 22 p + m 14 c m 22 c m 23 p m 12 c m 24 c m 23 p m 13 c m 22 c m 24 p m 14 c m 23 c m 22 p ,
b y 1 = m 21 c m 34 c m 23 p + m 23 c m 31 c m 24 p + m 24 c m 33 c m 21 p m 21 c m 33 c m 24 p m 23 c m 34 c m 21 p m 24 c m 31 c m 23 p ,
b y 2 = m 11 c m 33 c m 24 p + m 13 c m 34 c m 21 p + m 14 c m 31 c m 23 p m 11 c m 34 c m 23 p m 13 c m 31 c m 24 p m 14 c m 33 c m 21 p ,
b y 3 = m 11 c m 24 c m 23 p + m 13 c m 21 c m 24 p + m 14 c m 23 c m 21 p m 11 c m 23 c m 24 p m 13 c m 24 c m 21 p m 14 c m 21 c m 23 p ,
b z 1 = m 21 c m 32 c m 24 p + m 22 c m 34 c m 21 p + m 24 c m 31 c m 22 p m 21 c m 34 c m 22 p m 22 c m 31 c m 24 p m 24 c m 32 c m 21 p ,
b z 2 = m 11 c m 34 c m 22 p + m 12 c m 31 c m 24 p + m 14 c m 32 c m 21 p m 11 c m 32 c m 24 p m 12 c m 34 c m 21 p m 14 c m 31 c m 22 p ,
b z 3 = m 11 c m 22 c m 24 p + m 12 c m 24 c m 21 p + m 14 c m 21 c m 22 p m 11 c m 24 c m 22 p m 12 c m 21 c m 21 p m 14 c m 22 c m 21 p ,
c 1 = m 21 c m 32 c m 33 p + m 22 c m 33 c m 31 p + m 23 c m 31 c m 32 p m 21 c m 33 c m 32 p m 22 c m 31 c m 33 p m 23 c m 32 c m 31 p ,
c 2 = m 11 c m 33 c m 32 p + m 12 c m 31 c m 33 p + m 13 c m 32 c m 31 p m 11 c m 32 c m 33 p m 12 c m 33 c m 31 p m 13 c m 31 c m 32 p ,
c 3 = m 11 c m 22 c m 33 p + m 12 c m 23 c m 31 p + m 13 c m 21 c m 32 p m 11 c m 23 c m 32 p m 12 c m 21 c m 33 p m 13 c m 22 c m 31 p ,
d 1 = m 21 c m 33 c m 22 p + m 22 c m 31 c m 23 p + m 23 c m 32 c m 21 p m 21 c m 32 c m 23 p m 22 c m 33 c m 21 p m 23 c m 31 c m 22 p ,
d 2 = m 11 c m 32 c m 23 p + m 12 c m 33 c m 21 p + m 13 c m 31 c m 22 p m 11 c m 33 c m 22 p m 12 c m 31 c m 23 p m 13 c m 32 c m 21 p ,
d 3 = m 11 c m 23 c m 22 p + m 12 c m 21 c m 23 p + m 13 c m 22 c m 21 p m 11 c m 22 c m 23 p m 12 c m 23 c m 21 p m 13 c m 21 c m 22 p .

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