Abstract

The retrieval of the backscatter cross section in lidar data is of great interest in remote sensing. For the numerical calculation of the backscatter cross section, a deconvolution has to be performed; its determination is therefore an ill-posed problem. Most of the common techniques, such as the well-known method of Gaussian decomposition, make implicit assumptions on both the emitted laser pulse and the scatterers. It is well understood that a land surface is quite complicated, and in many cases it cannot be composed of pure Gaussian function combinations. Therefore the assumption of Gaussian decomposition of waveforms may be invalid sometimes. In such cases an inversion method might be a natural choice. We propose a regularizing method with a posteriori choice of the regularizing parameter for solving the problem. The proposed method can alleviate difficulties in numerical computation and can suppress the propagation of noise. Numerical evidence is given of the success of the approach presented for recovering the backscatter cross section in lidar data.

© 2009 Optical Society of America

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References

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  1. H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
    [CrossRef]
  2. M. Flood, “Laser altimetry: from science to commercial lidar mapping,” Photogramm. Eng. Remote Sens. 67, 1209-1217 (2001).
  3. W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
    [CrossRef]
  4. W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).
  5. Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
    [CrossRef]
  6. Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
    [CrossRef]
  7. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  8. M. Z. Nashed, “Perturbations and approximations for generalized inverses and linear operator equations,” in Generalized Inverses and ApplicationsM.Z.Nashed, ed. (Academic, 1976), pp. 325-396.
  9. Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, “Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space,” Appl. Opt. 45, 7456-7467 (2006).
    [CrossRef] [PubMed]
  10. Y. F. Wang and T. Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Probl. 17, 281-291 (2001).
    [CrossRef]
  11. Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).
  12. C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).
  13. Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Educational Press, Beijing, 2007).

2008 (1)

Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
[CrossRef]

2007 (1)

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

2006 (2)

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, “Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space,” Appl. Opt. 45, 7456-7467 (2006).
[CrossRef] [PubMed]

2005 (1)

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

2004 (2)

C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

2001 (2)

M. Flood, “Laser altimetry: from science to commercial lidar mapping,” Photogramm. Eng. Remote Sens. 67, 1209-1217 (2001).

Y. F. Wang and T. Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Probl. 17, 281-291 (2001).
[CrossRef]

Ahlberg, S.

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Briese, C.

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

Chang, H. C.

H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
[CrossRef]

Ducic, V.

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

Fan, S. F.

Feng, X.

Flood, M.

M. Flood, “Laser altimetry: from science to commercial lidar mapping,” Photogramm. Eng. Remote Sens. 67, 1209-1217 (2001).

Ge, L. L.

H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
[CrossRef]

Grimm, A.

C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).

Guan, Y. N.

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, “Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space,” Appl. Opt. 45, 7456-7467 (2006).
[CrossRef] [PubMed]

Hug, C.

C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).

Kraus, K.

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

Li, X. W.

Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
[CrossRef]

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Melzer, T.

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

Milne, T.

H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
[CrossRef]

Nashed, M. Z.

M. Z. Nashed, “Perturbations and approximations for generalized inverses and linear operator equations,” in Generalized Inverses and ApplicationsM.Z.Nashed, ed. (Academic, 1976), pp. 325-396.

Nashed, Z.

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Persson, Å.

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

Rizos, C.

H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
[CrossRef]

Söderman, U.

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

Studnicka, N.

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Töpel, J.

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

Ullrich, A.

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).

Wagner, W.

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

Wang, Y. F.

Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
[CrossRef]

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Y. F. Wang, S. F. Fan, X. Feng, G. J. Yan, and Y. N. Guan, “Regularized inversion method for retrieval of aerosol particle size distribution function in W1,2 space,” Appl. Opt. 45, 7456-7467 (2006).
[CrossRef] [PubMed]

Y. F. Wang and T. Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Probl. 17, 281-291 (2001).
[CrossRef]

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Educational Press, Beijing, 2007).

Xiao, T. Y.

Y. F. Wang and T. Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Probl. 17, 281-291 (2001).
[CrossRef]

Yan, G. J.

Yang, C. C.

Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
[CrossRef]

Yang, H.

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Zhang, H.

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Zhao, F.

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Appl. Opt. (1)

Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. (3)

Å. Persson, U. Söderman, J. Töpel, and S. Ahlberg, “Visualization and analysis of full-waveform airborne laser scanner data,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 103-108 (2005).

C. Hug, A. Ullrich, and A. Grimm, “LITEMAPPER-5600-a waveform digitising lidar terrain and vegetation mapping system,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 36, 24-29 (2004).

W. Wagner, A. Ullrich, T. Melzer, C. Briese, and K. Kraus, “From single-pulse to full-waveform airborne laser scanners: potential and practical challenges,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 35, 201-206 (2004).

Inverse Probl. (1)

Y. F. Wang and T. Y. Xiao, “Fast realization algorithms for determining regularization parameters in linear inverse problems,” Inverse Probl. 17, 281-291 (2001).
[CrossRef]

ISPRS J. Photogramm. Remote Sens. (1)

W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitising airborne laser scanner,” ISPRS J. Photogramm. Remote Sens. 60, 100-112 (2006).
[CrossRef]

J. Geophys. Res. (1)

Y. F. Wang, C. C. Yang, and X. W. Li, “A regularizing kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval using smoothness constraint,” J. Geophys. Res. 113, D13101 (2008).
[CrossRef]

Photogramm. Eng. Remote Sens. (1)

M. Flood, “Laser altimetry: from science to commercial lidar mapping,” Photogramm. Eng. Remote Sens. 67, 1209-1217 (2001).

Remote Sens. Environ. (1)

Y. F. Wang, X. W. Li, Z. Nashed, F. Zhao, H. Yang, Y. N. Guan, and H. Zhang, “Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval,” Remote Sens. Environ. 111, 36-50 (2007).
[CrossRef]

Other (4)

H. C. Chang, L. L. Ge, C. Rizos, and T. Milne, “Validation of DEMs derived from radar interferometry, airborne laser scanning, and photogrammetry by using GPS-RTK,” in Proceedings of 2004 IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 5, pp. 2815-2818.
[CrossRef]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

M. Z. Nashed, “Perturbations and approximations for generalized inverses and linear operator equations,” in Generalized Inverses and ApplicationsM.Z.Nashed, ed. (Academic, 1976), pp. 325-396.

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Educational Press, Beijing, 2007).

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Figures (17)

Fig. 1
Fig. 1

Synthetic emitted laser pulse.

Fig. 2
Fig. 2

Synthetic cross sections.

Fig. 3
Fig. 3

Synthetic recorded waveform with and without addition of noise of level 1.

Fig. 4
Fig. 4

Zoomed display of the synthetic recorded waveform with and without addition of noise of level 1.

Fig. 5
Fig. 5

Synthetic recorded waveform with and without addition of noise of level 2.

Fig. 6
Fig. 6

Zoomed display of the synthetic recorded waveform with and without addition of noise of level 2.

Fig. 7
Fig. 7

Comparison of the true and recovered cross sections in the case of noise of level 1.

Fig. 8
Fig. 8

Comparison of the true and recovered cross sections in the case of noise of level 2.

Fig. 9
Fig. 9

Comparison of the true and recovered cross sections using least-squares fitting method in the case of noise of level 2.

Fig. 10
Fig. 10

First emitted laser pulse.

Fig. 11
Fig. 11

Recorded echo waveform of the laser pulse shown in Fig. 10 (dotted curve) and its reconstruction using the cross section shown in Fig. 12 (solid curve).

Fig. 12
Fig. 12

Backscatter cross section in time domain of the waveforms in Figs. 10, 11 calculated by regularized inversion.

Fig. 13
Fig. 13

Backscatter cross section in time domain of the waveforms in Figs. 10, 11 calculated by least-squares fitting.

Fig. 14
Fig. 14

Second emitted laser pulse.

Fig. 15
Fig. 15

Recorded echo waveform of the laser pulse shown in Fig. 14 (dotted curve) and its reconstruction using the cross section shown in Fig. 16 (solid curve).

Fig. 16
Fig. 16

Backscatter cross section in time domain of the waveforms in Figs. 14, 15 calculated by regularized inversion.

Fig. 17
Fig. 17

Backscatter cross section in time domain of the waveforms in Figs. 14, 15 calculated by least-squares fitting.

Tables (1)

Tables Icon

Table 1 Comparison of RMSEs of Our Regularizing Algorithm with the Least-Squares Fitting Method for Different Noise Levels

Equations (32)

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

P r ( t ) = D r 2 4 π R 4 β t 2 P t ( t 2 R v g ) σ ,
P r ( t ) = i = 1 N D r 2 4 π R i 4 β t 2 P t ( t ) σ i ( t ) Γ ( t ) ,
h ( t ) = ( f g ) ( t ) ,
h ( t ) = h eff ( t ) + n ( t ) .
f g eff = h eff ,
F g = h ,
F g h 2 min .
g ̃ = k ( h , v k ) σ k u k = k [ ( h eff , v k ) σ k u k + ( n , v k ) σ k u k ]
n min .
min J [ g ] F g h 2 ,
s.t. c ( g ) Δ ,
min F g h L 2 2 + ν Γ ( g ) ,
[ g 1 ( τ ) , g 2 ( τ ) ] W 1 , 2 Ω [ g 1 ( τ ) g 2 ( τ ) + i , j = 1 n g 1 τ i g 2 τ j ] d τ 1 d τ 2 d τ n ,
L = [ 1 + 1 s t 2 1 s t 2 0 0 1 s t 2 1 + 2 s t 2 1 s t 2 0 0 1 s t 2 1 + 2 s t 2 1 s t 2 0 0 1 s t 2 1 + 1 s t 2 ] .
Ψ ( ν ) = F g ν h 2 δ 2 ,
ν k + 1 = ν k 2 Ψ ( ν k ) Ψ ( ν k ) + [ Ψ ( ν k ) 2 2 Ψ ( 2 ν k ) Ψ ( ν k ) ] 1 2 .
Ψ ( ν ) = ν β ( ν ) ,
Ψ ( ν ) = β ( ν ) 2 ν [ L 1 2 d g ν d ν 2 + ( L g ν , d 2 g ν d ν 2 ) ] ,
( F * F + ν L ) g ν = F * h ,
( F * F + ν L ) d g ν d ν = L g ν ,
( F * F + ν L ) d 2 g ν d ν 2 = 2 L d g ν d ν .
J ν [ g t ] 1 2 F g t h t 2 + ν 2 L 1 2 g t 2
f lp ( x ) = 31.25 x 3 + 206.25 x 2 356.25 x + 218.75 .
g cs ( x ) = 8 x 3 10 x 2 + 3 x + 1 6 .
h w f ( x ) = f lp ( x ) g cs ( x ) .
h w f = h w f true + δ { rand [ size ( h w f true ) ] } ,
rmse = 1 l i [ h comp ( x i ) h meas ( x i ) ] 2 i [ h comp ( x i ) ] 2 ,
J ν [ g t ] 1 2 F g t h ν 2 + ν 2 L 1 2 g t 2 .
J ν [ g t + ρ ϕ ] = 1 2 ( F g t h t 2 + ν L 1 2 g t 2 ) + ρ 2 [ ( F g t h t , F ϕ ) + ( F ϕ , F g t h t ) + ν ( L 1 2 g t , L 1 2 ϕ ) + ν ( L 1 2 ϕ , L 1 2 g t ) ] + ρ 2 2 ( F ϕ + ν L 1 2 ϕ 2 ) .
d d ρ J ν [ f t + ρ ϕ ] ρ = 0 = ( F g t h t , F ϕ ) + ν ( L g t , ϕ ) = [ ( F T F + ν L ) x F T h t , ϕ ] ,
grad g t { J ν [ g t ] } = ( F T F + ν L ) g t F T h t .
g t = ( F T F + ν L ) 1 F T h t .

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