Abstract

We compare the performance of linear and nonlinear methods for aligning the excitation and detection planes throughout volumes of large specimens in digitally scanned light sheet microscopy. An effective nonlinear method involves the registration of four corner extrema of the imaging volume via a projective transform. We show that this improves the light collection efficiency of the commonly used three-point affine registration by an average of 42% over a typical specimen volume. Accurate illumination/detection registration methods are now pertinent to biological research in view of current trends towards imaging large or expanded samples, at depth, with diffraction limited resolution.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (2)

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

2015 (3)

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

F. Chen, P. W. Tillberg, and E. S. Boyden, Science 347, 543 (2015).
[Crossref]

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

2014 (2)

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

G. Zheng, X. Ou, and C. Yang, Biomed. Opt. Express 5, 1 (2014).
[Crossref]

2013 (1)

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

2012 (2)

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

E. Baumgart and U. Kubitscheck, Opt. Express 20, 21805 (2012).
[Crossref]

2011 (1)

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

2010 (1)

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

2004 (1)

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

2003 (1)

B. Zitová and J. Flusser, Image Vis. Comput. 21, 977 (2003).
[Crossref]

Bao, Z.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

Baumgart, E.

Bene, F. D.

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

Betzig, E.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Boyden, E. S.

F. Chen, P. W. Tillberg, and E. S. Boyden, Science 347, 543 (2015).
[Crossref]

Brady, D. J.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Chen, B. C.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Chen, F.

F. Chen, P. W. Tillberg, and E. S. Boyden, Science 347, 543 (2015).
[Crossref]

Chhetri, R. K.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Chmielewski, A. K.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Choi, J. M.

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

Coleman, M.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Davidson, M. W.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Eliceiri, K. W.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

English, B. P.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Evers, J. F.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Feller, S. D.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Flickinger, D.

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

Flusser, J.

B. Zitová and J. Flusser, Image Vis. Comput. 21, 977 (2003).
[Crossref]

Fraser, S. E.

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

Gehm, M. E.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Golish, D. R.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Hammer, J. A.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Harvey, B. K.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Huisken, J.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

Janetopoulos, C.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Ji, N.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Kaminski, C. F.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Keller, P. J.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

Khairy, K.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

King, J.

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

Kittle, D. S.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Koos, D. S.

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

Kubitscheck, U.

Kyrsting, A.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Legant, W. R.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Lemon, W. C.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Liu, Z.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Mahou, P.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Marks, D. L.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Milkie, D. E.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Muresan, L.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Myers, E. W.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Ou, X.

Pitrone, P. G.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Preibisch, S.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Richie, C. T.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Royer, L. A.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Santella, A.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

Schindelin, J.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Schmidt, A. D.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

Shao, L.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Sofroniew, N. J.

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

Stack, R. A.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Stelzer, E. H. K.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

Stuyvenberg, L.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Sun, W.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Supatto, W.

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

Svoboda, K.

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

Swoger, J.

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

Tillberg, P. W.

F. Chen, P. W. Tillberg, and E. S. Boyden, Science 347, 543 (2015).
[Crossref]

Tomancak, P.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Truong, T. V.

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

Vera, E. M.

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Wan, Y.

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Wang, K.

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Wayland, M. T.

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Weber, M.

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

Wittbrodt, J.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

Wu, X. S.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

Yang, C.

Zheng, G.

Zitová, B.

B. Zitová and J. Flusser, Image Vis. Comput. 21, 977 (2003).
[Crossref]

Biomed. Opt. Express (1)

eLife (1)

N. J. Sofroniew, D. Flickinger, J. King, and K. Svoboda, eLife 5, e14472 (2016).
[Crossref]

Image Vis. Comput. (1)

B. Zitová and J. Flusser, Image Vis. Comput. 21, 977 (2003).
[Crossref]

Nat. Biotechnol. (1)

L. A. Royer, W. C. Lemon, R. K. Chhetri, Y. Wan, M. Coleman, E. W. Myers, and P. J. Keller, Nat. Biotechnol. 34, 1267 (2016).
[Crossref]

Nat. Commun. (1)

K. Wang, W. Sun, C. T. Richie, B. K. Harvey, E. Betzig, and N. Ji, Nat. Commun. 6, 7276 (2015).
[Crossref]

Nat. Methods (3)

T. V. Truong, W. Supatto, D. S. Koos, J. M. Choi, and S. E. Fraser, Nat. Methods 8, 757 (2011).
[Crossref]

P. G. Pitrone, J. Schindelin, L. Stuyvenberg, S. Preibisch, M. Weber, K. W. Eliceiri, J. Huisken, and P. Tomancak, Nat. Methods 10, 598 (2013).
[Crossref]

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. K. Stelzer, Nat. Methods 7, 637 (2010).
[Crossref]

Nature (1)

D. J. Brady, M. E. Gehm, R. A. Stack, D. L. Marks, D. S. Kittle, D. R. Golish, E. M. Vera, and S. D. Feller, Nature 486, 386 (2012).
[Crossref]

Opt. Express (1)

Sci. Rep. (1)

A. K. Chmielewski, A. Kyrsting, P. Mahou, M. T. Wayland, L. Muresan, J. F. Evers, and C. F. Kaminski, Sci. Rep. 5, 9385 (2015).
[Crossref]

Science (3)

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, and B. P. English, Science 346, 1257998 (2014).
[Crossref]

F. Chen, P. W. Tillberg, and E. S. Boyden, Science 347, 543 (2015).
[Crossref]

J. Huisken, J. Swoger, F. D. Bene, J. Wittbrodt, and E. H. K. Stelzer, Science 305, 1007 (2004).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. (a) Schematic of the light sheet optics using a large NA detection objective. The beam is scanned in x to create a virtual sheet. (b)–(e) For the best image quality, the illumination planes (shown as red when not perfectly registered to the illumination planes) must be registered to the detection planes (green). (c) The linear registration using coordinates (1,2) is sufficient to register one plane. The affine registration (d) is commonly used in DSLM systems and provides reasonable performance by registering three points, [typically (1,2) and the midpoint of (3,4)]. However, the projective registration (e) using four control points (14) provides superior performance due to decreased out-of-focus fluorescence.
Fig. 2.
Fig. 2. Scan lens characterization: (a) shows the illumination profile in the x z plane for 400 scan positions, with a three-point registration. Each of the beam positions in (a) was localized by fitting a 2D Gaussian. The identified positions, using (b) a three point and (c) a four point, show that the positional discrepancy of the three-point method is largely fixed by the four-point registration; mean average discrepancy values being reduced for the affine versus homographic correction of 0.3673 μm and 0.2494 μm, respectively, with standard deviation values also dropping from 0.1467 μm to 0.1133 μm.
Fig. 3.
Fig. 3. In situ characterization: (a), (b) ratios of intensity maxima of localized fluorescent bead images were compared in 3D observation volumes using three- and four-point corrections. (a) On average, a 42% increase in contrast was obtained for the four-point case, with benefits increasing as the depth increases (b). (c) Corresponding graph for a beam scanned through a dye solution. The four-point correction results in greater light capture efficiency over the three-point correction, which again becomes more significant with depth. Here the brightness of the beam’s image indicated how well it was registered to the imaging plane. (d) Transgenic zebrafish expressing mCherry:beta-actinCAAX. The contrast of features is substantially improved by the four-point registration, and the images afford greater detail and clarity, as shown by inset line profiles.

Equations (8)

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r = ( r ˜ 1 r ˜ 3 r ˜ 2 r ˜ 3 ) .
( x 1 x 2 x 3 z 1 z 2 z 3 1 1 1 ) ( λ μ ν ) = ( x 4 z 4 1 ) ,
A = ( λ x 1 μ x 2 ν x 3 λ z 1 μ z 2 ν z 3 λ μ ν ) .
A ( 100 ) T λ ( x 1 , z 1 , 1 ) T A ( 010 ) T μ ( x 2 , z 2 , 1 ) T A ( 001 ) T ν ( x 3 , z 3 , 1 ) T A ( 111 ) T    ( x 4 , z 4 , 1 ) T .
( a b c ) = ( V x 1 V x 2 V x 3 V z 1 V z 2 V z 3 1 1 1 ) 1 ( V x 4 V x 4 1 ) .
B = ( a x 1 b x 2 c x 3 a z 1 b z 2 c z 3 a b c ) .
( V ˜ x V ˜ z k ) = ( a V x 1 b V x 2 c V x 3 a V z 1 b V z 2 c V z 3 a b c ) ( 0 1 0 z 1 z 2 z 3 1 1 1 ) 1 ( x z 1 ) ,
( V x V z ) = 1 k ( V ˜ x V ˜ z )

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