Abstract

A new model for turbulence-corrupted imagery is proposed based on the theory of optimal mass transport. By describing the relationship between photon density and the phase of the traveling wave, and combining it with a least action principle, the model suggests a new class of methods for approximately recovering the solution of the photon density flow created by a turbulent atmosphere. Both coherent and incoherent imagery are used to validate and compare the model to other methods typically used to describe this type of data. Given its superior performance in describing experimental data, the new model suggests new algorithms for a variety of atmospheric imaging and wave propagation applications.

© 2018 Optical Society of America

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2017 (3)

2016 (4)

2015 (3)

2014 (2)

2013 (3)

X. Zhu and P. Milanfar, “Removing atmospheric turbulence via space-invariant deconvolution,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 157–170 (2013).
[Crossref]

Y. Lou, S.-H. Kang, S. Soatto, and A. L. Bertozzi, “Video stabilization of atmospheric turbulence distortion,” Inverse Probl. Imaging 7, 839–861 (2013).
[Crossref]

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[Crossref]

2012 (2)

Y. Mao and J. Gilles, “Non rigid geometric distortions correction-application to atmospheric turbulence stabilization,” Inverse Probl. Imaging 6, 531–546 (2012).
[Crossref]

M. T. Velluet, M. Vorontsov, P. B. W. Schwering, G. Marchi, S. Nicolas, and J. Riker, “Turbulence characterization and image processing data sets from a NATO RTO SET 165 trial in Dayton, Ohio, USA,” Proc. SPIE 8380, 83800J (2012).
[Crossref]

2011 (2)

G. Huang, Z. Zhong, W. Zou, and S. A. Burns, “Lucky averaging: quality improvement of adaptive optics scanning laser ophthalmoscope images,” Opt. Lett. 36, 3786–3788 (2011).
[Crossref]

S. Jin, P. Markowich, and C. Sparber, “Mathematical and computational methods for semiclassical Schrödinger equations,” Acta Numer. 20, 121–209 (2011).
[Crossref]

2009 (1)

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).
[Crossref]

2007 (1)

J. Bec and K. Khanin, “Burgers turbulence,” Phys. Rep. 447, 1–66 (2007).
[Crossref]

2006 (1)

H. Liu, S. Osher, and R. Tsai, “Multi-valued solution and level set methods in computational high frequency wave propagation,” Commun. Comput. Phys. 1, 765–804 (2006).

2005 (1)

A. C. Fannjiang and K. Solna, “Propagation and time reversal of wave beams in atmospheric turbulence,” SIAM 3, 522–558 (2005).
[Crossref]

2004 (2)

J. Rubinstein and G. Wolansky, “A variational principle in optics,” J. Opt. Soc. Am. A 21, 2164–2172 (2004).
[Crossref]

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60, 225–240 (2004).
[Crossref]

2003 (1)

J.-D. Benamou, O. Lafitte, R. Sentis, and I. Solliec, “A geometric optics method for high-frequency electromagnetic fields computations near fold caustics–part I,” J. Comput. Appl. Math. 156, 93–125 (2003).
[Crossref]

2002 (1)

T. Corpetti, E. Mémin, and P. Pérez, “Dense estimation of fluid flows,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 365–380 (2002).
[Crossref]

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

1995 (1)

1994 (2)

R. P. Leland, “White noise in atmospheric optics,” Acta Applicandae Mathematicae 35, 103–130 (1994).
[Crossref]

F. Dalaudier, A. S. Gurvich, V. Kan, and C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[Crossref]

1991 (1)

J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci. 26, 381–393 (1991).
[Crossref]

1989 (1)

Y. Brenier, “The least action principle and the related concept of generalized flows for incompressible perfect fluids,” J. Am. Math. Soc. 2, 225–255 (1989).
[Crossref]

1986 (1)

1985 (1)

F. P. Wheeler, “Aspects of the simulation of wave propagation through a clear turbulent medium,” J. Comput. Appl. Math. 12–13, 625–633 (1985).
[Crossref]

1983 (1)

R. Salmon, “Practical use of Hamilton’s principle,” J. Fluid Mech. 132, 431–444 (1983).
[Crossref]

1982 (1)

N. Woolf, “High resolution imaging from the ground,” Annu. Rev. Astron. Astrophys. 20, 367–398 (1982).
[Crossref]

1981 (1)

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[Crossref]

1978 (1)

1975 (1)

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics. I. The Markov approximation, its validity and application to angular broadening,” Astrophys. J. 196, 695–707 (1975).
[Crossref]

1967 (1)

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

1966 (1)

1963 (1)

T. Anderson, “Asymptotic theory for principal component analysis,” Ann. Math. Stat. 34, 122–148 (1963).
[Crossref]

Ahmed, N.

Anderson, D. R.

K. P. Burnham and D. R. Anderson, Model Selection and Inference: A Practical Information-Theoretic Approach (Springer-Verlag, 1998).

Anderson, J. G.

Anderson, T.

T. Anderson, “Asymptotic theory for principal component analysis,” Ann. Math. Stat. 34, 122–148 (1963).
[Crossref]

Angenent, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60, 225–240 (2004).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Asundi, A.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

Bao, C.

Barbastathis, G.

Bastiaans, M. J.

Bec, J.

J. Bec and K. Khanin, “Burgers turbulence,” Phys. Rep. 447, 1–66 (2007).
[Crossref]

Benamou, J.-D.

J.-D. Benamou, O. Lafitte, R. Sentis, and I. Solliec, “A geometric optics method for high-frequency electromagnetic fields computations near fold caustics–part I,” J. Comput. Appl. Math. 156, 93–125 (2003).
[Crossref]

Bertozzi, A. L.

Y. Lou, S.-H. Kang, S. Soatto, and A. L. Bertozzi, “Video stabilization of atmospheric turbulence distortion,” Inverse Probl. Imaging 7, 839–861 (2013).
[Crossref]

Boyd, R. W.

Brady, D. J.

Brenier, Y.

Y. Brenier, “The least action principle and the related concept of generalized flows for incompressible perfect fluids,” J. Am. Math. Soc. 2, 225–255 (1989).
[Crossref]

Brown, W. P.

Burnham, K. P.

K. P. Burnham and D. R. Anderson, Model Selection and Inference: A Practical Information-Theoretic Approach (Springer-Verlag, 1998).

Burns, S. A.

Cao, Y.

Cattell, L.

J. M. Nichols, A. T. Watnik, T. Doster, S. Park, A. Kanaev, L. Cattell, and G. K. Rohde, “An optimal transport model for imaging in atmospheric turbulence,” arXiv:1705.01050 (2017).

Chen, Q.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

Corpetti, T.

T. Corpetti, E. Mémin, and P. Pérez, “Dense estimation of fluid flows,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 365–380 (2002).
[Crossref]

Cremers, D.

D. Mitzel, T. Pock, T. Schoenemann, and D. Cremers, “Video super resolution using duality based TV-L1 optical flow,” Lect. Notes Comput. Sci. 5748, 432–441 (2009).
[Crossref]

Dagobert, T.

J. Gilles, T. Dagobert, and C. De Franchis, “Atmospheric turbulence restoration by diffeomorphic image registration and blind deconvolution,” in Advanced Concepts for Intelligent Vision Systems (Springer, 2008), pp. 400–409.

Dalaudier, F.

F. Dalaudier, A. S. Gurvich, V. Kan, and C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[Crossref]

De Franchis, C.

J. Gilles, T. Dagobert, and C. De Franchis, “Atmospheric turbulence restoration by diffeomorphic image registration and blind deconvolution,” in Advanced Concepts for Intelligent Vision Systems (Springer, 2008), pp. 400–409.

Dockery, G. D.

J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci. 26, 381–393 (1991).
[Crossref]

Dolinar, S.

Doster, T.

T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56, 3386–3396 (2017).
[Crossref]

J. M. Nichols, A. T. Watnik, T. Doster, S. Park, A. Kanaev, L. Cattell, and G. K. Rohde, “An optimal transport model for imaging in atmospheric turbulence,” arXiv:1705.01050 (2017).

T. Doster and A. T. Watnik, “Measuring multiplexed OAM modes with convolutional neural networks,” in Lasers Congress 2016 (ASSL, LSC, LAC) (Optical Society of America, 2016), pp. LTh3B.2.

Duncan, M. D.

Erkmen, B. I.

Fannjiang, A. C.

A. C. Fannjiang and K. Solna, “Propagation and time reversal of wave beams in atmospheric turbulence,” SIAM 3, 522–558 (2005).
[Crossref]

Feizulin, Z. I.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Feller, S. D.

Fried, D. L.

Fu, S.

Furhad, M. H.

Gao, C.

Gehm, M. E.

Gilles, J.

Y. Mao and J. Gilles, “Non rigid geometric distortions correction-application to atmospheric turbulence stabilization,” Inverse Probl. Imaging 6, 531–546 (2012).
[Crossref]

J. Gilles, T. Dagobert, and C. De Franchis, “Atmospheric turbulence restoration by diffeomorphic image registration and blind deconvolution,” in Advanced Concepts for Intelligent Vision Systems (Springer, 2008), pp. 400–409.

Gladysz, S.

Gureyev, T. E.

Gurvich, A. S.

F. Dalaudier, A. S. Gurvich, V. Kan, and C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[Crossref]

Haker, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60, 225–240 (2004).
[Crossref]

Hardie, R. C.

R. C. Hardie and D. A. LeMaster, “On the simulation and mitigation of anisoplanatic optical turbulence for long range imaging,” Proc. SPIE 102040, 102040B (2017).
[Crossref]

Horn, B. K. P.

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[Crossref]

Hou, W.

Huang, G.

Huang, H.

Jackson, J.

J. Jackson, Classical Electrodynamics (Wiley, 1975).

Jin, S.

S. Jin, P. Markowich, and C. Sparber, “Mathematical and computational methods for semiclassical Schrödinger equations,” Acta Numer. 20, 121–209 (2011).
[Crossref]

Johnson, A.

Jokipii, J. R.

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics. I. The Markov approximation, its validity and application to angular broadening,” Astrophys. J. 196, 695–707 (1975).
[Crossref]

Judd, K. P.

Kan, V.

F. Dalaudier, A. S. Gurvich, V. Kan, and C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[Crossref]

Kanaev, A.

J. M. Nichols, A. T. Watnik, T. Doster, S. Park, A. Kanaev, L. Cattell, and G. K. Rohde, “An optimal transport model for imaging in atmospheric turbulence,” arXiv:1705.01050 (2017).

Kanaev, A. V.

Kang, S.-H.

Y. Lou, S.-H. Kang, S. Soatto, and A. L. Bertozzi, “Video stabilization of atmospheric turbulence distortion,” Inverse Probl. Imaging 7, 839–861 (2013).
[Crossref]

Khanin, K.

J. Bec and K. Khanin, “Burgers turbulence,” Phys. Rep. 447, 1–66 (2007).
[Crossref]

Kim, J.

Kolouri, S.

S. Kolouri, A. B. Tosun, J. A. Ozolek, and G. K. Rohde, “A continuous linear optimal transport approach for pattern analysis in image datasets,” Pattern Recogn. 51, 453–462 (2016).
[Crossref]

S. Kolouri, S. Park, M. Thorpe, D. Slepcev, and G. K. Rohde, “Transport-based analysis, modeling, and learning from signal and data distributions,” arXiv:1609.04767 (2016).

Kravtsov, Y. A.

Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[Crossref]

Kuttler, J. R.

J. R. Kuttler and G. D. Dockery, “Theoretical description of the parabolic approximation/Fourier split-step method of representing electromagnetic propagation in the troposphere,” Radio Sci. 26, 381–393 (1991).
[Crossref]

Lafitte, O.

J.-D. Benamou, O. Lafitte, R. Sentis, and I. Solliec, “A geometric optics method for high-frequency electromagnetic fields computations near fold caustics–part I,” J. Comput. Appl. Math. 156, 93–125 (2003).
[Crossref]

Lambert, A.

Lavery, M. P. J.

Lee, L. C.

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics. I. The Markov approximation, its validity and application to angular broadening,” Astrophys. J. 196, 695–707 (1975).
[Crossref]

Leland, R. P.

R. P. Leland, “White noise in atmospheric optics,” Acta Applicandae Mathematicae 35, 103–130 (1994).
[Crossref]

LeMaster, D. A.

R. C. Hardie and D. A. LeMaster, “On the simulation and mitigation of anisoplanatic optical turbulence for long range imaging,” Proc. SPIE 102040, 102040B (2017).
[Crossref]

Li, L.

Li, S.

Liu, H.

H. Liu, S. Osher, and R. Tsai, “Multi-valued solution and level set methods in computational high frequency wave propagation,” Commun. Comput. Phys. 1, 765–804 (2006).

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Y. Lou, S.-H. Kang, S. Soatto, and A. L. Bertozzi, “Video stabilization of atmospheric turbulence distortion,” Inverse Probl. Imaging 7, 839–861 (2013).
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Y. Mao and J. Gilles, “Non rigid geometric distortions correction-application to atmospheric turbulence stabilization,” Inverse Probl. Imaging 6, 531–546 (2012).
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M. T. Velluet, M. Vorontsov, P. B. W. Schwering, G. Marchi, S. Nicolas, and J. Riker, “Turbulence characterization and image processing data sets from a NATO RTO SET 165 trial in Dayton, Ohio, USA,” Proc. SPIE 8380, 83800J (2012).
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Figures (11)

Fig. 1.
Fig. 1. Illustration of the photon/intensity transport problem. Intensity is transported in the transverse plane as the associated EM field moves through space from z=0 to z=Z. Changes in propagation direction are due to variations of refraction index in the medium. Moreover, the velocity is given by the transverse gradient of the phase perturbations. The implementation of the transport model described in Section 3 assumes the intensity is being transported along constant velocity paths, i.e., straight lines with negligible diffraction.
Fig. 2.
Fig. 2. Path taken by the light, y(z), as it moves in the direction of increasing index gradient given by n(x,z)=1αy. The exact path and the geometric optics path coincide almost exactly over this range and for these values of α. Values for α consistent with the Kolmogorov turbulence spectrum are often at most O(107), resulting in millimeters of transverse displacement over such distances. For comparison, we also show the constant velocity approximation, obtained by minimizing the kinetic energy via Eq. (10).
Fig. 3.
Fig. 3. Estimated map, f(x,z), between a clean and turbulence-corrupted image of a coherent Laguerre–Gauss beam subjected to Kolmogorov turbulence (implemented via phase screens [2]).
Fig. 4.
Fig. 4. Estimated map, f(x,n), between two frames taken from a turbulent video sequence. The scene was naturally illuminated; hence, the imagery is appropriately modeled as incoherent.
Fig. 5.
Fig. 5. Comparison between the optical flow and transport models described in this section. Shown is the mean square error associated with frame-to-frame reconstruction showing that, as expected, the transport approach is able to obtain better matches between frames. Frame exemplars can be seen in Fig. 4.
Fig. 6.
Fig. 6. Mean square error of frame reconstruction of individual frames using both optical flow and transport models, as a function of the number of principal components used in each model, respectively. Results shown for video illustrated in Fig. 4.
Fig. 7.
Fig. 7. Frame taken from a turbulent video sequence. The scene was naturally illuminated; hence, the imagery is appropriately modeled as incoherent.
Fig. 8.
Fig. 8. Comparison between the optical flow and transport models. Shown are the mean square error associated with frame-to-frame reconstruction showing that, as expected, the transport approach is able to obtain better matches between frames. An example frame is shown in Fig. 7.
Fig. 9.
Fig. 9. Top left, region of the first frame of the video shown in Fig. 7 displaying a target bar pattern. Top center, the mean frame obtained by registering subsequent frames to the first frame using optical flow, Top right, the mean frame after warping subsequent frames with optimal transport. The region bounded by the red box is shown in more detail in the middle row. The bottom row shows the intensity profile along the dashed blue line. Using the optimal transport model, the contrast in the mean target bar pattern almost perfectly matches that of the original first frame. The estimated optimal transport model is able to better match the frames than is optical flow.
Fig. 10.
Fig. 10. Mean square error of frame reconstruction of individual frames using both optical flow and transport models, as a function of the number of principal components used in each model, respectively. In terms of accuracy and parsimony, the transport model performs extremely well relative to optical flow. Results shown for video illustrated in Fig. 7.
Fig. 11.
Fig. 11. Graphical depiction of the model for the electric field. The light is linearly polarized and therefore oscillates, with frequency ω and amplitude ρ(x,z)1/2, at a point in the image plane. The polarization angle (defined with respect to the x^1 direction) is given by γ; the electric field is therefore pointing in the direction given by γ={cos(γ),sin(γ)}. The phase of the plane wave accumulates at a rate k0 and is given by k0z. Perturbations to the EM field caused by the variations in refractive index are captured in the additional phase term ϕ(x,z).

Equations (43)

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2E(x,t)+(E(x,t)·n(x)n(x))n2(x)c022E(x,t)t2=0,
E(x,z,t)=ρ(x,z)1/2ei(ωtk0zϕ(x,z)){cosγ,sinγ,0}
ρ(x,z)z+X·(ρ(x,z)v(x,z))=0,
v(x,z)z+(v(x,z)·X)v(x,z)=Xη(x,z)+3k02X|Xη(x,z)|2.
k0ϕ(x,z)z+12|Xϕ(x,z)|2=k02η(x,z)+3|Xη(x,z)|2,
kW(x,k)dkW(x,k)dk=Xϕ˜(x,z)k0v˜(x,z),
W(x,k)=Γ(x+x2,xx2,ω)exp(i2πkx)dxdω,
det(Jf(x0,z))ρ(xz)=ρ(x0),
AZε00ZΩ0ρ(x0){|v(xz)|22[η(xz)+3k02|Xη(xz)|2]}dx0dz,
Eη[V(xz)]=Zε0Ω0ρ(x0)Eη[η(xz)+3k02|Xη(xz)|2]dx0=3Zε0k02Ω0ρ(x0)Eη[|Xη(xz)|2]dx0,
dp(0,Z)2=inffΩ0f(x0,Z)x02ρ(x0)dx=infv0ZT(xz)dzinfvA
v(xz)(f(x0,Z)x0)/Z=Xϕ(xz)/k0,
yGO(z)=1α(1cosh[αz]).
d2y(z)dz2α2(1+6k02α2)y+α(1+6k02α2)=0.
y(z)=1α(1cosh[αz1+6k02α2]).
E(x,z,t)=ρ(x,z)1/2ei(ωtk0zϕ){cos(γ),sin(γ),0},
2ρ[γ·Xn(x,z)]{ϕx,ϕy,k0+zϕ2ρ}+[k0zρ+·(ρϕ)]{γ,0}={0,0,0},
[γ·Xn(x,z)][k0+zϕ]=0.
[γ·Xn(x,z)]=0.
[k0zρ+·(ρϕ)]γ={0,0}.
k0zρ+·(ρϕ)=0,
[k0zϕ+12|ϕ|2k02η122ρ1/2ρ1/2]cos(γ)={xxnn,xynn}·γ,[k0zϕ+12|ϕ|2k02η122ρ1/2ρ1/2]sin(γ)={xynn,yynn}·γ.
[k0zϕ+12|Xϕ|2k02η12X2ρ1/2ρ1/22TTnn]γ=0.
k0zϕ+12|Xϕ|2=k02η+12X2ρ1/2ρ1/2+2TTnn.
k0zϕ+12|Xϕ|2=k02η+34|Xlog(n2)|2k02η+3|Xη|2.
v(x,z)z+(v(x,z)·X)v(x,z)=Xη(x,z)+3k02X|Xη(x,z)|2,
X·E(x,z)=E(x,z)·Xn2(x)n2(x),
[Xρ2ρ+2Xnn]·γ={0,0}.
Xρ2ρ=2Xnn.
X·Xρ2ρ=2X·Xnn,
X2ρ2ρ(Xρ)22ρ2=2X·(Xnn)X2ρ2ρ(Xρ)22ρ2=2[X2nn|Xn|2n2].
X2ρ2ρ(Xρ)24ρ2=X2(ρ1/2)ρ1/2,
X2ρ2ρ(Xρ)22ρ2=X2ρ2ρ(Xρ)24ρ2(Xρ)24ρ2=X2ρ1/2ρ1/2(Xρ)24ρ2,
X2ρ1/2ρ1/2=(Xρ)24ρ22[X2nn|Xn|2n2].
X2ρ1/2ρ1/2=2X2nn+6|Xn|2n2=2X2nn+6|Xlog(n)|2=2X2nn+32|Xlog(n2)|2,
12X2ρ1/2ρ1/2=X2nn+34|Xlog(n2)|2=2TTnn+3|Xη|2,
OPL=z+ϕ2k0,
L(x,z)=0zdl(ζ)=0z(dζ2+dx(ζ)2)1/2dζ=0z(1+(dx(ζ)dζ)2)1/2dζ0zdζ+120z(dx(ζ)dζ)2dζ,
L(x,z)=z+ϕ2k0=0zdζ+120z(dx(ζ)dζ)2dζ,
ϕ2k0=120z(dx(ζ)dζ)2dζ.
1k0dϕdz=(dx(z)dz)2.
1k0dϕdx(z)=dx(z)dzvx(x,z),
1k0{dϕdx(z)x^,dϕdy(z)y^}{vx(x,z),vy(x,z)}1k0Xϕ=v(x,z).