Abstract

Optical diffraction tomography relies on solving an inverse scattering problem governed by the wave equation. Classical reconstruction algorithms are based on linear approximations of the forward model (Born or Rytov), which limits their applicability to thin samples with low refractive-index contrasts. More recent works have shown the benefit of adopting nonlinear models. They account for multiple scattering and reflections, improving the quality of reconstruction. To reduce the complexity and memory requirements of these methods, we derive an explicit formula for the Jacobian matrix of the nonlinear Lippmann-Schwinger model which lends itself to an efficient evaluation of the gradient of the data-fidelity term. This allows us to deploy efficient methods to solve the corresponding inverse problem subject to sparsity constraints.

© 2017 Optical Society of America

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References

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2017 (2)

U. S. Kamilov, “A parallel proximal algorithm for anisotropic total variation minimization,” IEEE Trans. Image Process. 26, 539–548 (2017).
[Crossref]

D. Jin, R. Zhou, Z. Yaqoob, and P. T. So, “Tomographic phase microscopy: Principles and applications in bioimaging,” J. Opt. Soc. Am. B 34, B64–B77 (2017).
[Crossref]

2016 (1)

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

2015 (2)

2013 (3)

Y. Nesterov, “Gradient methods for minimizing composite functions,” Math. Prog. 140, 125–161 (2013).
[Crossref]

H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

S. Lefkimmiatis, J. Ward, and M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Image Process. 22, 1873–1888 (2013).
[Crossref] [PubMed]

2012 (1)

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

2011 (3)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vision 40, 120–145 (2011).
[Crossref]

Y. Sung and R. R. Dasari, “Deterministic regularization of three-dimensional optical diffraction tomography,” J. Opt. Soc. Am. A 28, 1554–1561 (2011).
[Crossref]

2010 (1)

J. A. Schmalz, G. Schmalz, T. E. Gureyev, and K. M. Pavlov, “On the derivation of the Green’s function for the Helmholtz equation using generalized functions,” Am. J. Phys. 78, 181–186 (2010).
[Crossref]

2009 (4)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref] [PubMed]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717 (2007).
[Crossref] [PubMed]

2006 (1)

P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Model Simul. 4, 1168–1200 (2006).
[Crossref]

2005 (2)

Jean-Michel Geffrin, Pierre Sabouroux, and Christelle Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl. 21, 6(2005).
[Crossref]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

2002 (1)

A. Abubakar and P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495 (2002).
[Crossref]

1998 (1)

1983 (1)

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Math. Dokl. 27, 372–376 (1983).

1981 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1962 (1)

J.-J. Moreau, “Fonctions convexes duales et points proximaux dans un espace hilbertien,” C. R. Acad. Sci. Ser. A Math. 255, 2897–2899 (1962).

Abubakar, A.

A. Abubakar and P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495 (2002).
[Crossref]

Attouch, H.

H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

Badizadegan, K.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref] [PubMed]

Belkebir, K.

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

Bolte, J.

H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

Bottou, L.

L. Bottou, “Large-scale machine learning with stochastic gradient descent,” in “Proceedings of COMPSTAT’2010: 19th International Conference on Computational StatisticsParis France, August 22–27, 2010 Keynote, Invited and Contributed Papers,” Y. Lechevallier and G. Saporta, eds. (Physica-Verlag HD, Heidelberg, 2010), pp. 177–186.

Boufounos, P. T.

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

H.-Y. Liu, U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “Compressive imaging with iterative forward models,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2017), pp. 6025–6029.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” arXiv preprint arXiv:1705.04281 (2017).

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Chambolle, A.

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vision 40, 120–145 (2011).
[Crossref]

Chaumet, P. C.

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

Chen, B.

Choi, W.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 (Springer Science & Business Media, 2012).

Combettes, P. L.

P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Model Simul. 4, 1168–1200 (2006).
[Crossref]

Dasari, R. R.

Devaney, A.

Devaney, A. J.

A. J. Devaney, Mathematical Foundations of Imaging, Tomography and Wavefield Inversion (Cambridge University Press, 2012).
[Crossref]

Donati, L.

M. Unser, E. Soubies, F. Soulez, M. McCann, and L. Donati, “GlobalBioIm: A unifying computational framework for solving inverse problems,” in “Proceedings of the OSA Imaging and Applied Optics Congress on Computational Optical Sensing and Imaging (COSI’17),” (San Francisco CA, USA, 2017). Paper no. CTu1B.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Eyraud, Christelle

Jean-Michel Geffrin, Pierre Sabouroux, and Christelle Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl. 21, 6(2005).
[Crossref]

Fang-Yen, C.

Feld, M. S.

Geffrin, Jean-Michel

Jean-Michel Geffrin, Pierre Sabouroux, and Christelle Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl. 21, 6(2005).
[Crossref]

Goy, A.

Gureyev, T. E.

J. A. Schmalz, G. Schmalz, T. E. Gureyev, and K. M. Pavlov, “On the derivation of the Green’s function for the Helmholtz equation using generalized functions,” Am. J. Phys. 78, 181–186 (2010).
[Crossref]

Jin, D.

Jin, K. H.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[Crossref]

Kamilov, U.

Kamilov, U. S.

U. S. Kamilov, “A parallel proximal algorithm for anisotropic total variation minimization,” IEEE Trans. Image Process. 26, 539–548 (2017).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

H.-Y. Liu, U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “Compressive imaging with iterative forward models,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2017), pp. 6025–6029.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” arXiv preprint arXiv:1705.04281 (2017).

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 (Springer Science & Business Media, 2012).

Lee, K.

Lee, S.

Lefkimmiatis, S.

S. Lefkimmiatis, J. Ward, and M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Image Process. 22, 1873–1888 (2013).
[Crossref] [PubMed]

Lim, J.

Liu, D.

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

H.-Y. Liu, U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “Compressive imaging with iterative forward models,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2017), pp. 6025–6029.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” arXiv preprint arXiv:1705.04281 (2017).

Liu, H.-Y.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” arXiv preprint arXiv:1705.04281 (2017).

H.-Y. Liu, U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “Compressive imaging with iterative forward models,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2017), pp. 6025–6029.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717 (2007).
[Crossref] [PubMed]

Mansour, H.

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

H.-Y. Liu, U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “Compressive imaging with iterative forward models,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2017), pp. 6025–6029.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” arXiv preprint arXiv:1705.04281 (2017).

McCann, M.

M. Unser, E. Soubies, F. Soulez, M. McCann, and L. Donati, “GlobalBioIm: A unifying computational framework for solving inverse problems,” in “Proceedings of the OSA Imaging and Applied Optics Congress on Computational Optical Sensing and Imaging (COSI’17),” (San Francisco CA, USA, 2017). Paper no. CTu1B.

Moreau, J.-J.

J.-J. Moreau, “Fonctions convexes duales et points proximaux dans un espace hilbertien,” C. R. Acad. Sci. Ser. A Math. 255, 2897–2899 (1962).

Mudry, E.

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

Nesterov, Y.

Y. Nesterov, “Gradient methods for minimizing composite functions,” Math. Prog. 140, 125–161 (2013).
[Crossref]

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Math. Dokl. 27, 372–376 (1983).

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717 (2007).
[Crossref] [PubMed]

Papadopoulos, I.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Park, Y.

Pavlov, K. M.

J. A. Schmalz, G. Schmalz, T. E. Gureyev, and K. M. Pavlov, “On the derivation of the Green’s function for the Helmholtz equation using generalized functions,” Am. J. Phys. 78, 181–186 (2010).
[Crossref]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Pock, T.

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vision 40, 120–145 (2011).
[Crossref]

Psaltis, D.

Sabouroux, Pierre

Jean-Michel Geffrin, Pierre Sabouroux, and Christelle Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl. 21, 6(2005).
[Crossref]

Schmalz, G.

J. A. Schmalz, G. Schmalz, T. E. Gureyev, and K. M. Pavlov, “On the derivation of the Green’s function for the Helmholtz equation using generalized functions,” Am. J. Phys. 78, 181–186 (2010).
[Crossref]

Schmalz, J. A.

J. A. Schmalz, G. Schmalz, T. E. Gureyev, and K. M. Pavlov, “On the derivation of the Green’s function for the Helmholtz equation using generalized functions,” Am. J. Phys. 78, 181–186 (2010).
[Crossref]

Sentenac, A.

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

Shin, S.

Shoreh, M.

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[Crossref]

So, P. T.

Soubies, E.

M. Unser, E. Soubies, F. Soulez, M. McCann, and L. Donati, “GlobalBioIm: A unifying computational framework for solving inverse problems,” in “Proceedings of the OSA Imaging and Applied Optics Congress on Computational Optical Sensing and Imaging (COSI’17),” (San Francisco CA, USA, 2017). Paper no. CTu1B.

Soulez, F.

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Figures (7)

Fig. 1
Fig. 1

Optical diffraction tomography. A sample of refractive index n(x) is immersed in a medium of index nb and illuminated by an incident plane wave (wave vector k). The interaction of the wave with the object produces forward and backward scattered waves. The forward scattered wave is recorded in the detector plane. Optionally, a second detector plane may record the backward scattered wave (see Section 5).

Fig. 2
Fig. 2

Predicted evolution of ΔMem as function of the number N of points for two values of KNAGD and NThreads. The vertical dashed lines give examples of 2D and 3D volumes for a range of values of N. Finally, the three crosses correspond to values of ΔMem measured experimentally.

Fig. 3
Fig. 3

Forward-model solution for a bead with radius 3λ and a contrast of 1 using CG (bottom-left) and NAGD (bottom-right), as well as the Mie solution (top-right). The setting used for this experiment is presented in the top-left panel. The colormap is the same for each figure.

Fig. 4
Fig. 4

Evolution of the number of iterations k ε 0 needed to let the relative error Eq. (28) fall below ε0 = 10−2 as function of bead radius (left) and bead contrast (right).

Fig. 5
Fig. 5

Sheep-Logan phantom and refractive indices of the gray levels. The contrast is 20%.

Fig. 6
Fig. 6

Reconstructions obtained by the proposed method and by SEAGLE for the (256 × 256) ODT problem with μ = 3.3 · 10−2. The colormap is identical to that of Fig. 5. For comparison, we provide the TV-regularized Rytov reconstruction with μ = 3 · 10−3.

Fig. 7
Fig. 7

Reconstructions (permittivity) obtained by the proposed method and by SEAGLE for the FoamDielExt target of the Institut Fresnel’s database [34] with μ = 1.6 · 10−2. The SNR values (computed from the experimentally measured permittivity of the ground truth) are 25.13 dB (Ours) and 25.15 dB (SEAGLE) while the computing times are respectively of 6 min and 93 min.

Tables (3)

Tables Icon

Algorithm 1 Accelerated forward-backward splitting.

Tables Icon

Table 1 Proposed method vs. SEAGLE [6, 7] in terms of running time and memory consumption. The reconstructed refractive-index maps are presented in Fig. 6.

Tables Icon

Algorithm 2 ADMM for solving Eq. (29).

Equations (34)

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u in ( x , t ) = Re ( u 0 e i k · x i ω t ) ,
2 u ( x , t ) n 2 ( x ) c 2 2 u t 2 ( x , t ) = 0 ,
2 u ( x ) + k 0 2 n 2 ( x ) u ( x ) = 0 ,
2 u sc ( x ) + k 0 2 n b 2 u sc ( x ) = f ( x ) u ( x ) ,
u sc ( x ) = Ω g ( x x ) f ( x ) u ( x ) d x ,
g ( x ) = { 1 4 H 0 ( 1 ) ( k 0 n b x ) , D = 2 , 1 4 π e i k 0 n b x x , D = 3 .
u ( x ) = u in ( x ) + Ω g ( x x ) f ( x ) u ( x ) d x .
u p = u p in + G diag ( f ) u p ,
u p ( f ) = arg min u N 1 2 ( I G diag ( f ) ) u u p in 2 2 .
y p = G ˜ diag ( f ) u p ( f ) + u p in | Γ ,
f ^ { arg min f N ( D ( f ) + μ ( f ) ) } ,
D ( f ) = p = 1 P D p ( f ) ,
D p ( f ) = 1 2 G ˜ diag ( f ) u p ( f ) y p sc 2 2 ,
( f ) = i 0 ( f ) + f 2 , 1 = i 0 ( f ) + n = 1 N d = 1 D ( d f ) n 2
D ( f ) = p = 1 P D p ( f ) ,
D p ( f ) = Re ( J h p H ( f ) G ˜ H ( G ˜ diag ( f ) u p ( f ) y p sc ) ) ,
h p : f diag ( f ) u p ( f ) .
D ( f ) p ω D p ( f ) ,
J h p ( f ) = ( I + diag ( f ) ( I G diag ( f ) ) 1 G ) diag ( u p ( f ) ) .
d h p ( f ; v ) = lim ε 0 diag ( f + ε v ) u p ( f + ε v ) diag ( f ) u p ( f ) ε = diag ( u p ( f ) ) v + lim ε 0 diag ( f ) u p ( f + ε v ) u p ( f ) ε .
u p in = ( I G diag ( f + ε v ) ) u p ( f + ε v ) = ( I G diag ( f ) ) u p ( f + ε v ) ε G diag ( v ) u p ( f + ε v )
( I G diag ( f ) ) u p ( f ) = u p in .
( I G diag ( f ) ) ( u p ( f + ε v ) u p ( f ) ) = ε G diag ( v ) u p ( f + ε v ) .
d h p ( f ; v ) = ( I + diag ( f ) ( I G diag ( f ) ) 1 G ) diag ( u p ( f ) ) v
J h p ( f ) = ( I + diag ( f ) ( I G diag ( f ) ) 1 G ) diag ( u p ( f ) ) ,
Δ Mem = N × K NAGD × 16 [ bytes ] ,
Δ Mem = N × K NAGD × N Threads × 16 [ bytes ] .
ε k = u k u Mie u Mie 2 .
prox μ ( v ) = arg min f N ( 1 2 f v 2 2 + μ f TV + i 0 ( f ) )
prox μ ( v ) = arg min f N ( 1 2 f v 2 2 + μ q 1 2 , 1 + i 0 ( q 2 ) ) , s . t . q 1 = f , q 2 = f ,
( f , q 1 , q 2 , w 1 , w 2 ) = 1 2 f v 2 2 + ρ 1 2 f q 1 + w 1 ρ 1 2 2 + ρ 2 2 f q 2 + w 2 ρ 2 2 2 + μ q 1 2 , 1 + i 0 ( q 2 ) ,
q N , [ prox i 0 ( q ) ] n = ( q n ) + ,
q N × D , [ prox γ 2 , 1 ( q ) ] n , d = q n , d ( 1 γ q n , . 2 ) + ,
( x ) + : = max ( x , 0 ) , x .

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