Abstract

Depth from defocus (DFD) based on optical methods is an effective method for depth reconstruction from 2D optical images. However, due to optical diffraction, optical path deviation occurs, which results in blurring imaging. Blurring, in turn, results in inaccurate depth reconstructions using DFD. In this paper, a nanoscale depth reconstruction method using defocus with optical diffraction is proposed. A blurring model is proposed by considering optical diffraction, leading to a much higher accuracy in depth reconstruction. Firstly, Fresnel diffraction in an optical system is analyzed, and a relationship between intensity distribution and depth information is developed. Secondly, a blurring imaging model with relative blurring and heat diffusion is developed through curving fitting of a numerical model. In this way, a new DFD method with optical diffraction is proposed. Finally, experimental results show that this new algorithm is more effective for depth reconstruction on the nanoscale.

© 2014 Optical Society of America

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References

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  1. A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9(4), 523–531 (1987).
    [Crossref] [PubMed]
  2. P. N. Vinay and C. Subhasis, “On defocus, diffusion and depth estimation,” Pattern Recognit. Lett. 28(3), 311–319 (2007).
    [Crossref]
  3. S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
    [Crossref]
  4. P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
    [Crossref] [PubMed]
  5. P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
    [Crossref]
  6. A. R. FitzGerrell, E. R. Dowski, and W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36(23), 5796–5804 (1997).
    [Crossref] [PubMed]
  7. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59(10), 1314–1321 (1969).
    [Crossref]
  8. C. Mair and C. J. Goodman, “Diffraction-limited depth-from-defocus,” Electron. Lett. 36(24), 2012–2013 (2000).
    [Crossref]
  9. Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
    [Crossref]
  10. Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).
  11. R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
    [Crossref]
  12. I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
    [Crossref] [PubMed]
  13. H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
    [Crossref] [PubMed]
  14. P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
    [Crossref]
  15. R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance 1(1), 13–26 (1997).

2013 (1)

R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
[Crossref]

2012 (2)

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
[Crossref]

2011 (1)

Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).

2008 (1)

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

2007 (1)

P. N. Vinay and C. Subhasis, “On defocus, diffusion and depth estimation,” Pattern Recognit. Lett. 28(3), 311–319 (2007).
[Crossref]

2005 (1)

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

2003 (1)

P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
[Crossref]

2000 (1)

C. Mair and C. J. Goodman, “Diffraction-limited depth-from-defocus,” Electron. Lett. 36(24), 2012–2013 (2000).
[Crossref]

1998 (1)

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

1997 (2)

R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance 1(1), 13–26 (1997).

A. R. FitzGerrell, E. R. Dowski, and W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36(23), 5796–5804 (1997).
[Crossref] [PubMed]

1996 (1)

S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
[Crossref]

1987 (1)

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9(4), 523–531 (1987).
[Crossref] [PubMed]

1969 (1)

Burger, M.

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

Cathey, W. T.

Dera, P.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Dong, Z. L.

Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
[Crossref]

Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).

Dowski, E. R.

Dubrovinskaia, N.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Dubrovinsky, L.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Favaro, P.

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
[Crossref]

Fitzgerald, J. P. S.

R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
[Crossref]

FitzGerrell, A. R.

Fujita, J.

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Goodman, C. J.

C. Mair and C. J. Goodman, “Diffraction-limited depth-from-defocus,” Electron. Lett. 36(24), 2012–2013 (2000).
[Crossref]

Kantor, A.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Kantor, I.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Konenkamp, R.

R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
[Crossref]

Kouznetsov, D.

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Kurnosov, A.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Lagnado, R.

R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance 1(1), 13–26 (1997).

Mair, C.

C. Mair and C. J. Goodman, “Diffraction-limited depth-from-defocus,” Electron. Lett. 36(24), 2012–2013 (2000).
[Crossref]

Mennucci, A.

P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
[Crossref]

Navar, S. K.

S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
[Crossref]

Noguchi, M.

S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
[Crossref]

Oberst, H.

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Osher, S.

R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance 1(1), 13–26 (1997).

Osher, S. J.

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

Pentland, A. P.

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9(4), 523–531 (1987).
[Crossref] [PubMed]

Prakapenka, V.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Shimizu, F.

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Shimizu, K.

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Sinogeikin, S.

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Soatto, S.

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
[Crossref]

Stokseth, P. A.

Subhasis, C.

P. N. Vinay and C. Subhasis, “On defocus, diffusion and depth estimation,” Pattern Recognit. Lett. 28(3), 311–319 (2007).
[Crossref]

Vinay, P. N.

P. N. Vinay and C. Subhasis, “On defocus, diffusion and depth estimation,” Pattern Recognit. Lett. 28(3), 311–319 (2007).
[Crossref]

Wang, P.

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

Wang, W.

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

Wang, Z. J.

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

Watanabe, M.

S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
[Crossref]

Wei, Y. J.

Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
[Crossref]

Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).

Word, R. C.

R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
[Crossref]

Wu, C. D.

Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
[Crossref]

Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).

Xu, Y. G.

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. C. Word, J. P. S. Fitzgerald, and R. Konenkamp, “Direct imaging of optical diffraction in photoemission electron microscopy,” Appl. Phys. Lett. 103(2), 021118 (2013).
[Crossref]

Electron. Lett. (1)

C. Mair and C. J. Goodman, “Diffraction-limited depth-from-defocus,” Electron. Lett. 36(24), 2012–2013 (2000).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9(4), 523–531 (1987).
[Crossref] [PubMed]

S. K. Navar, M. Watanabe, and M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18(12), 1186–1198 (1996).
[Crossref]

P. Favaro, S. Soatto, M. Burger, and S. J. Osher, “Shape from defocus via diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 518–531 (2008).
[Crossref] [PubMed]

IET Computer Vision (1)

Y. J. Wei, Z. L. Dong, and C. D. Wu, “Depth measurement using single camera with fixed camera parameters,” IET Computer Vision 6(1), 29–39 (2012).
[Crossref]

Int. J. Comput. Vis. (1)

P. Favaro, A. Mennucci, and S. Soatto, “Observing shape from blurred images,” Int. J. Comput. Vis. 52(1), 25–43 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

Journal of Computational Finance (1)

R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance 1(1), 13–26 (1997).

Optical Society of America (1)

P. Wang, Y. G. Xu, W. Wang, and Z. J. Wang, “Analytic expression for Fresnel diffraction,” Optical Society of America 15(3), 684–688 (1998).
[Crossref]

Pattern Recognit. Lett. (1)

P. N. Vinay and C. Subhasis, “On defocus, diffusion and depth estimation,” Pattern Recognit. Lett. 28(3), 311–319 (2007).
[Crossref]

Phys. Rev. Lett. (1)

H. Oberst, D. Kouznetsov, K. Shimizu, J. Fujita, and F. Shimizu, “Fresnel diffraction mirror for an atomic wave,” Phys. Rev. Lett. 94(1), 013203 (2005).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

I. Kantor, V. Prakapenka, A. Kantor, P. Dera, A. Kurnosov, S. Sinogeikin, N. Dubrovinskaia, and L. Dubrovinsky, “A new diamond anvil cell design for X-ray diffraction and optical measurements,” Rev. Sci. Instrum. 83(12), 125102 (2012).
[Crossref] [PubMed]

Science China. Technol. Soc. (1)

Y. J. Wei, C. D. Wu, and Z. L. Dong, “Global depth reconstruction of nano grid with singly fixed camera,” Science China. Technol. Soc. 54(4), 1044–1052 (2011).

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Figures (13)

Fig. 1
Fig. 1 Schematic of circular aperture diffraction.
Fig. 2
Fig. 2 Diagrammatic sketch of optical path.
Fig. 3
Fig. 3 Distribution curve along optical axis when λ = 600nm, sinu = 0.
Fig. 4
Fig. 4 Fitted Gaussian curve with a fixed l.
Fig. 5
Fig. 5 Depth variation and blurring degree comparison.
Fig. 6
Fig. 6 Nano grid scanned by AFM.
Fig. 7
Fig. 7 The blurred images of a static nano grid.
Fig. 8
Fig. 8 Reconstructed 3D depth with/without diffraction.
Fig. 9
Fig. 9 Reconstructed 3D shape with/without diffraction.
Fig. 10
Fig. 10 Arbitrary section shape of nano grid.
Fig. 11
Fig. 11 The blurred images of dynamitic AFM cantilever.
Fig. 12
Fig. 12 Reconstructed 3D depth with/without diffraction.
Fig. 13
Fig. 13 Arbitrary section shape of AFM cantilever.

Equations (36)

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

E ˜ P = A R + b exp [ i k ( R + b + ρ 2 2 b ) ] n = 1 ( i R + b R a ρ ) n J n ( 2 π λ a b ρ )
E ˜ P = exp [ i k ( x + ρ 2 2 b ) ] { i J 1 ( 2 π λ 1 y sin u ) π λ 1 y sin u + λ π sin 2 u n = 2 J n ( 2 π λ 1 y sin u ) ( i y 1 sin u ) n x n 1 }
E ˜ P = exp [ i k ( x + ρ 2 2 b ) ] B exp ( i β )
I P = E ˜ P E ˜ P * = B 2
l = x / m
1 u + 1 v = 1 f
r g = a v 2 | 1 f 1 v 1 s |
σ 2 = γ 2 r g 2
s = 1 ( 1 f 1 v ± 2 σ a v γ )
σ = f ( r d ) = F d ( D i f f r a c t i o n , s )
σ = a l 2 + b l + c
a l 2 + b l + c σ = 0
l = b ± b 2 4 a ( c σ ) 2 a
s = s 0 + l = s 0 + b ± b 2 4 a ( c σ ) 2 a
E ( y , z ) = h ( y , z , r d ) I ( u , v ) d u d v
E ( y , z ) = h ( y , z ) I ( y , z )
{ u ˙ ( y , z , t ) = ε Δ u ( y , z , t ) t ( 0 , ) u ( y , z , 0 ) = I ( y , z )
Δ u = 2 u y 2 + 2 u z 2
σ 2 = 2 t ε
{ u ˙ ( y , z , t ) = ( ε ( y , z ) u ( y , z , t ) ) t ( 0 , ) u ( y , z , 0 ) = I ( y , z )
= [ y z ] T , = y + z
σ 2 ( y , z ) = 2 t ε ( y , z )
E 2 ( y , z ) = h ( y , z , σ 2 2 ) I ( u , v ) d u d v = 1 2 π σ 2 2 e x p ( ( y u ) 2 + ( z v ) 2 2 σ 2 2 ) I ( u , v ) d u d v = 1 2 π ( σ 2 2 σ 1 2 ) exp ( ( y u ) 2 + ( z v ) 2 2 ( σ 2 2 σ 1 2 ) ) d u d v 1 2 π σ 1 2 exp ( ( u y ˜ ) 2 + ( v z ˜ ) 2 2 σ 1 2 ) I ( y ˜ , z ˜ ) d y ˜ d z ˜ = 1 2 π Δ σ 2 exp ( ( y u ) 2 + ( z v ) 2 2 Δ σ 2 ) E 1 ( u , v ) d u d v = h ( y , z , Δ σ 2 ) E 1 ( u , v ) d u d v = h ( y , z , Δ σ 2 ) E 1 ( u , v )
{ u ˙ ( y , z , t ) = ( ε ( y , z ) u ( y , z , t ) ) t ( 0 , ) u ( y , z , 0 ) = E 1 ( y , z ) u ( y , z , Δ t ) = E 2 ( y , z )
Δ σ 2 = σ 2 2 σ 1 2 = ( a l 2 2 + b l 2 + c ) 2 ( a l 1 2 + b l 1 + c ) 2
ε = Δ σ 2 2 Δ t = ( a l 2 2 + b l 2 + c ) 2 ( a l 1 2 + b l 1 + c ) 2 2 Δ t
a l 2 2 + b l 2 + c = ± Δ σ 2 + ( a l 1 2 + b l 1 + c ) 2
c = c ± ( a l 1 2 + b l 1 + c ) 2 + Δ σ 2
s = s 0 + b ± b 2 4 a c 2 a
s ˜ = arg min s 2 ( y , z ) ( u ( y , z , Δ t ) E 2 ( y , z ) ) 2 d y d z
s ˜ = arg min s 2 ( y , z ) ( u ( y , z , Δ t ) E 2 ( y , z ) ) 2 d y d z + α s 2 ( y , z ) 2 + α k s 2 ( y , z ) 2
F ( s ) = ( u ( y , z , Δ t ) E 2 ( y , z ) ) 2 d y d z + α s 2 + α k s 2
s ˜ = arg min s F ( s ) s . t . E q . ( 1 4 ) , E q . ( 29 )
s t = F ' ( s )
ϕ = s ˜ s 1
E a v e = 1 n k = 1 n | H k H ˜ k |

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