Abstract

For an electrohydrodynamic (EHD) jet, variables such as the direction of the meniscus and the ejection stability need to be analyzed. Thus, the EHD jet should be observed three-dimensionally (3D) because the variables can only be obtained in the 3D field, especially in unstable modes. However, if the 3D field is reconstructed from multi-directional binary images, eliminating reconstruction errors caused by invisible areas is almost impossible, even when using a tomographic technique. To solve this problem, a new 3D reconstruction method including an ellipse estimation was developed in this study. The method was verified by numerical simulation and applied to estimate the jetting flow rate and the direction of an ethanol droplet ejected from a nozzle according to a voltage.

© 2017 Optical Society of America

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References

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  1. J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
    [Crossref] [PubMed]
  2. W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
    [Crossref] [PubMed]
  3. A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008).
    [Crossref]
  4. I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
    [Crossref]
  5. I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
    [Crossref]
  6. A. Jaworek and A. Krupa, “Main modes of electrohydrodynamic spraying of liquids,” in Third International Conference on Multiphase Flow (1998).
  7. A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999).
    [Crossref]
  8. A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996).
    [Crossref]
  9. C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
    [Crossref]
  10. X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Comparative study on basis functions for projection matrix of three-dimensional tomographic reconstruction for analysis of dropletbehavior from electrohydrodynamic jet,” Appl. Opt. 51(24), 5834–5844 (2012).
    [Crossref] [PubMed]
  11. X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Analysis of electrohydrodynamic jetting behaviors using three-dimensional shadowgraphic tomography,” Appl. Opt. 52(19), 4494–4504 (2013).
    [Crossref] [PubMed]
  12. X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015).
    [Crossref]
  13. C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009).
    [Crossref]

2015 (1)

X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015).
[Crossref]

2013 (1)

2012 (1)

2011 (1)

C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
[Crossref]

2009 (1)

C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009).
[Crossref]

2008 (2)

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008).
[Crossref]

2007 (1)

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

1999 (1)

A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999).
[Crossref]

1996 (1)

A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996).
[Crossref]

1987 (2)

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
[Crossref]

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
[Crossref]

Adair, K.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Alleyne, A. G.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Atkinson, C.

C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009).
[Crossref]

Bailey, A. I.

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
[Crossref]

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
[Crossref]

Barton, K.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Chen, C.

C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
[Crossref]

Chiu, Y. H.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Dressler, R. A.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Ferreira, P. M.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Georgiadis, J. G.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Gim, Y.

X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015).
[Crossref]

Gordon, M. S.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Hardy, M.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Hayati, I.

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
[Crossref]

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
[Crossref]

Jaworek, A.

A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008).
[Crossref]

A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999).
[Crossref]

A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996).
[Crossref]

A. Jaworek and A. Krupa, “Main modes of electrohydrodynamic spraying of liquids,” in Third International Conference on Multiphase Flow (1998).

Kang, S. J.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Kim, Y. J.

C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
[Crossref]

Ko, H. S.

X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015).
[Crossref]

X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Analysis of electrohydrodynamic jetting behaviors using three-dimensional shadowgraphic tomography,” Appl. Opt. 52(19), 4494–4504 (2013).
[Crossref] [PubMed]

X. H. Nguyen, S.-H. Lee, and H. S. Ko, “Comparative study on basis functions for projection matrix of three-dimensional tomographic reconstruction for analysis of dropletbehavior from electrohydrodynamic jet,” Appl. Opt. 51(24), 5834–5844 (2012).
[Crossref] [PubMed]

C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
[Crossref]

Krupa, A.

A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999).
[Crossref]

A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996).
[Crossref]

A. Jaworek and A. Krupa, “Main modes of electrohydrodynamic spraying of liquids,” in Third International Conference on Multiphase Flow (1998).

Landman, U.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Lee, C. Y.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Lee, S.-H.

Levandier, D. J.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Luedtke, W. D.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Mukhopadhyay, D. K.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Nguyen, X. H.

Park, J. U.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Rogers, J. A.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Sobczyk, A. T.

A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008).
[Crossref]

Sok, S.

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Soria, J.

C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009).
[Crossref]

Strano, M. S.

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Tadros, T. H. F.

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
[Crossref]

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
[Crossref]

Appl. Opt. (2)

Exp. Fluids (2)

C. Atkinson and J. Soria, “An efficient simultaneous reconstruction technique for tomographic particle image velocimetry,” Exp. Fluids 47(4-5), 553–568 (2009).
[Crossref]

A. Jaworek and A. Krupa, “Jet and drops formation in electrohydrodynamic spraying of liquids. A systematic approach,” Exp. Fluids 27(1), 43–52 (1999).
[Crossref]

Exp. Therm. Fluid Sci. (1)

C. Chen, Y. J. Kim, and H. S. Ko, “Three-dimensional tomographic reconstruction of unstable ejection phenomena of droplets for electrohydrodynamic jet,” Exp. Therm. Fluid Sci. 35(3), 433–441 (2011).
[Crossref]

J. Aerosol Sci. (1)

A. Jaworek and A. Krupa, “Generation and characteristics of the precession mode of EHD spraying,” J. Aerosol Sci. 27(1), 75–82 (1996).
[Crossref]

J. Col. Inter. Sci. (2)

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: I. Effect of electric field and the environment on pendant drops and factors affecting the formation of stable jets and atomization,” J. Col. Inter. Sci. 117(1), 205–221 (1987).
[Crossref]

I. Hayati, A. I. Bailey, and T. H. F. Tadros, “Investigations into the mechanisms of electrohydrodynamic spraying of liquids: II. Mechanism of stable jet formation and electrical forces acting on a liquid cone,” J. Col. Inter. Sci. 117(1), 222–230 (1987).
[Crossref]

J. Electrost. (1)

A. Jaworek and A. T. Sobczyk, “Electrospraying route to nanotechnology: an overview,” J. Electrost. 66(3-4), 197–219 (2008).
[Crossref]

J. Phys. Chem. A (1)

W. D. Luedtke, U. Landman, Y. H. Chiu, D. J. Levandier, R. A. Dressler, S. Sok, and M. S. Gordon, “Nanojets, electrospray, and ion field evaporation: molecular dynamics simulations and laboratory experiments,” J. Phys. Chem. A 112(40), 9628–9649 (2008).
[Crossref] [PubMed]

Nat. Mater. (1)

J. U. Park, M. Hardy, S. J. Kang, K. Barton, K. Adair, D. K. Mukhopadhyay, C. Y. Lee, M. S. Strano, A. G. Alleyne, J. G. Georgiadis, P. M. Ferreira, and J. A. Rogers, “High-resolution electrohydrodynamic jet printing,” Nat. Mater. 6(10), 782–789 (2007).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

X. H. Nguyen, Y. Gim, and H. S. Ko, “Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting,” Opt. Lasers Eng. 68, 235–243 (2015).
[Crossref]

Other (1)

A. Jaworek and A. Krupa, “Main modes of electrohydrodynamic spraying of liquids,” in Third International Conference on Multiphase Flow (1998).

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Figures (5)

Fig. 1
Fig. 1

Twisted phantom in 3D field; (a) isosurface of twisted phantom, (b) binary projection image of phantom at 0°, (c) 45°, and (d) −45°.

Fig. 2
Fig. 2

Reconstructed results in 3D field using MLOS estimation; (a) without ellipse estimation and (b) with ellipse estimation.

Fig. 3
Fig. 3

Schematic diagram of experimental setup.

Fig. 4
Fig. 4

Multi-spindle mode of ethanol ejection; (a) multi-directional projection images, (b) projection images by time, and (c) reconstructed result using ellipse estimation.

Fig. 5
Fig. 5

Precession mode of ethanol; (a) multi-directional projection images, (b) projection images by time, and (c) reconstructed result using ellipse estimation.

Equations (10)

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

( ( x x 0 )cost( y y 0 )sint+ x 0 ) 2 a 2 + ( ( x x 0 )sint+( y y 0 )cost+ y 0 ) 2 b 2 =1
L n = 2 a 2 m ( α n ,w ) 2 + b 2 1+m ( α n ,w ) 2 m( α n ,w )= 1 tan( α n +w )
 4 b 2 =( L i 2 4 a 2 )m ( α i ,w ) 2 + L i 2 =( L j 2 4 a 2 )m ( α j ,w ) 2 + L j 2
[ m ( α 1 ,w ) 2 +1 m ( α 1 ,w ) 2 m ( α 2 ,w ) 2 m ( α 2 ,w ) 2 +1 m ( α 1 ,w ) 2 m ( α 2 ,w ) 2 0 0 m ( α 2 ,w ) 2 +1 m ( α 2 ,w ) 2 m ( α 3 ,w ) 2 m ( α 3 ,w ) 2 +1 m ( α 2 ,w ) 2 m ( α 3 ,w ) 2 m ( α 1 ,w ) 2 +1 m ( α 3 ,w ) 2 m ( α 1 ,w ) 2 0 m ( α 3 ,w ) 2 +1 m ( α 3 ,w ) 2 m ( α 1 ,w ) 2 ][ L 1 2 L 2 2 L 3 2 ] =4[ a ( w ) 1 2 a ( w ) 2 2 a ( w ) 3 2 ]
Φ similarity = i=1 N | j=1 N a ( w ) j 2 N a ( w ) i 2 | N
A k =π a k b k
V= k=1 N A k
x 2 30+z + y 2 ( 40 30+z ) 2 10,{ x=20sin π 180 z60.5 y=20cos π 180 z60.5 0zπ
Φ avg = j=1 N v | f( x j , y j , z j ) f ^ ( x j , y i , z i ) | N v
Q V( t+Δt )V( t ) Δt

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