Abstract

Fluorescence diffuse optical tomography (fDOT) is an imaging modality that provides images of the fluorochrome distribution within the object of study. The image reconstruction problem is ill-posed and highly underdetermined and, therefore, regularisation techniques need to be used. In this paper we use a nonlinear anisotropic diffusion regularisation term that incorporates anatomical prior information. We introduce a split operator method that reduces the nonlinear inverse problem to two simpler problems, allowing fast and efficient solution of the fDOT problem. We tested our method using simulated, phantom and ex-vivo mouse data, and found that it provides reconstructions with better spatial localisation and size of fluorochrome inclusions than using the standard Tikhonov penalty term.

© 2011 OSA

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  39. B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2010 (5)

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol.12, 343–248 (2010).
[CrossRef]

A. Ale, R. B. Schulz, A. Sarantopoulos, and V. Ntziachristos, “Imaging performance of a hybrid x-ray computed tomography-fluorescence molecular tomography system using priors,” Med. Phys.37, 1976–1986 (2010).
[CrossRef] [PubMed]

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett.35, 763–765 (2010).
[CrossRef] [PubMed]

M. Freiberger, C. Clason, and H. Scharfetter, “Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach,” Appl. Opt.49, 3741–3747 (2010).
[CrossRef] [PubMed]

2009 (6)

A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express17, 3025–3035 (2009).
[CrossRef] [PubMed]

Y. Lin, H. Yan, O. Nalcioglu, and G. Gulsen, “Quantitative fluorescence tomography with functional and structural a priori information,” Appl. Opt.48, 1328–1336 (2009).
[CrossRef] [PubMed]

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt.14, 034044 (2009).
[CrossRef] [PubMed]

Q. Fang and D. Boas, “Tetrahedral mesh generation from volumetric binary and gray-scale images,” in “Proceedings of IEEE International Symposium on Biomedical Imaging,”(2009), pp. 1142–1145.
[CrossRef]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl.25, 123010 (2009).
[CrossRef]

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. OHara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt.14, 030501 (2009).
[CrossRef] [PubMed]

2008 (1)

A. Martin, J. A. J, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes.” Molec. Imaging7, 157–167 (2008).

2007 (4)

A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol.18, 87–95 (2007).
[CrossRef]

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577 (2007).
[CrossRef] [PubMed]

S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express15, 4066–4082 (2007).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express15, 6696–6716 (2007).
[CrossRef] [PubMed]

2006 (1)

V. Ntziachristos, “Fluorescence molecular imaging,” Ann. Rev. Biomed. Eng.8, 1–33 (2006).
[CrossRef]

2005 (5)

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A.12, 18252–18257 (2005).
[CrossRef]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225 (2005).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging24, 1377–1386 (2005).
[CrossRef] [PubMed]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Local diffusion regularization method for optical tomography reconstruction by using robust statistics,” Opt. Lett.30, 2439–2441 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol.13, 195–208 (2003).
[PubMed]

2001 (1)

2000 (1)

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A.97, 2767–2772 (2000).
[CrossRef] [PubMed]

1999 (1)

J. P. Kaipio, V. Kolehmainen, M. Vauhkonen, and E. Somersalo, “Inverse problems with structural prior information,” Inv. Probl.15, 713–729 (1999).
[CrossRef]

1998 (2)

J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process.7, 398–410 (1998).
[CrossRef]

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process.7, 421–432 (1998).
[CrossRef]

1997 (3)

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl.13, 729 (1997).
[CrossRef]

M. Hanke, “A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems,” Inv. Probl.13, 79 (1997).
[CrossRef]

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math.76, 167–188 (1997).
[CrossRef]

1996 (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters21, 158–160 (1996).
[CrossRef]

1993 (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput.14, 1487–1503 (1993).
[CrossRef]

1992 (1)

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal.29, 182–193 (1992).
[CrossRef]

1990 (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12, 629–639 (1990).
[CrossRef]

Aguirre, J.

J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” in “Nuclear Science Symposium Conference Record, 2008. NSS ’08. IEEE,”(2008), pp. 5412–5413.
[CrossRef]

Ale, A.

A. Ale, R. B. Schulz, A. Sarantopoulos, and V. Ntziachristos, “Imaging performance of a hybrid x-ray computed tomography-fluorescence molecular tomography system using priors,” Med. Phys.37, 1976–1986 (2010).
[CrossRef] [PubMed]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225 (2005).
[CrossRef] [PubMed]

Andersson-Engels, S.

Arridge, S.

Arridge, S. R.

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett.35, 763–765 (2010).
[CrossRef] [PubMed]

A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express17, 3025–3035 (2009).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl.25, 123010 (2009).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express15, 6696–6716 (2007).
[CrossRef] [PubMed]

A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol.18, 87–95 (2007).
[CrossRef]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

S. R. Arridge, V. Kolehmainen, and M. J. Schweiger, “Reconstruction and regularisation in optical tomography,” in “Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT),”(Pisa, Italy, 2007), pp. 1–18.

Axelsson, J.

Bakker, L.

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol.12, 343–248 (2010).
[CrossRef]

Bangerth, W.

Beek, M. V.

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol.12, 343–248 (2010).
[CrossRef]

Black, M. J.

M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust anisotropic diffusion,” IEEE Trans. Image Process.7, 421–432 (1998).
[CrossRef]

Boas, D.

Q. Fang and D. Boas, “Tetrahedral mesh generation from volumetric binary and gray-scale images,” in “Proceedings of IEEE International Symposium on Biomedical Imaging,”(2009), pp. 1142–1145.
[CrossRef]

D. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. thesis, University of Pennsylvania, Philadelphia (USA) (1996).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters21, 158–160 (1996).
[CrossRef]

Bremer, C.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol.13, 195–208 (2003).
[PubMed]

Brendel, B.

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol.12, 343–248 (2010).
[CrossRef]

Catteé, F.

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal.29, 182–193 (1992).
[CrossRef]

Chambolle, A.

A. Chambolle and P. Lions, “Image recovery via total variation minimization and related problems,” Num. Math.76, 167–188 (1997).
[CrossRef]

Chance, B.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A.97, 2767–2772 (2000).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Optics Letters21, 158–160 (1996).
[CrossRef]

Chatziioannou, A. F.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577 (2007).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225 (2005).
[CrossRef] [PubMed]

Choe, R.

Clason, C.

Coll, T.

F. Catteé, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Num. Anal.29, 182–193 (1992).
[CrossRef]

Corlu, A.

Correia, T.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt.14, 034044 (2009).
[CrossRef] [PubMed]

Davis, S. C.

Dehghani, H.

D. S. Kepshire, S. L. Gibbs-Strauss, J. A. OHara, M. Hutchins, N. Mincu, F. Leblond, M. Khayat, H. Dehghani, S. Srinivasan, and B. W. Pogue, “Imaging of glioma tumor with endogenous fluorescence tomography,” J. Biomed. Opt.14, 030501 (2009).
[CrossRef] [PubMed]

S. C. Davis, H. Dehghani, J. Wang, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Image-guided diffuse optical fluorescence tomography implemented with Laplacian-type regularization,” Opt. Express15, 4066–4082 (2007).
[CrossRef] [PubMed]

der Mark, M. V.

S. van de Ven, A. Wiethoff, T. Nielsen, B. Brendel, M. van der Voort, R. Nachabe, M. V. der Mark, M. V. Beek, L. Bakker, L. Fels, S. Elias, P. Luijten, and W. Mali, “A novel fluorescent imaging agent for diffuse optical tomography of the breast: First clinical experience in patients,” Molec. Imaging Biol.12, 343–248 (2010).
[CrossRef]

Desco, M.

J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” in “Nuclear Science Symposium Conference Record, 2008. NSS ’08. IEEE,”(2008), pp. 5412–5413.
[CrossRef]

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577 (2007).
[CrossRef] [PubMed]

Douiri, A.

A. Douiri, M. Schweiger, J. Riley, and S. R. Arridge, “Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information,” Meas. Sci. Technol.18, 87–95 (2007).
[CrossRef]

A. Douiri, M. Schweiger, J. Riley, and S. Arridge, “Local diffusion regularization method for optical tomography reconstruction by using robust statistics,” Opt. Lett.30, 2439–2441 (2005).
[CrossRef] [PubMed]

Durduran, T.

Egger, H.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion in fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

Elias, S.

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A. Ale, R. B. Schulz, A. Sarantopoulos, and V. Ntziachristos, “Imaging performance of a hybrid x-ray computed tomography-fluorescence molecular tomography system using priors,” Med. Phys.37, 1976–1986 (2010).
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G. Zacharakis, H. Kambara, H. Shih, J. Ripoll, J. Grimm, Y. Saeki, R. Weissleder, and V. Ntziachristos, “Volumetric tomography of fluorescent proteins through small animals in vivo,” Proc. Natl. Acad. Sci. U.S.A.12, 18252–18257 (2005).
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Figures (5)

Fig. 1
Fig. 1

Reconstructions of h (mm−1) from simulated data (image dimensions 32.5 mm × 88 mm) and PSNR values (dB): (a) the target, (b) reconstruction with Tikhonov prior, (c) reconstruction with anatomical prior and total variation, (d) reconstruction with anatomical prior and Perona-Malik function, (e) reconstruction with anatomical prior and Perona-Malik 2 function, (f) reconstruction with anatomical prior and Huber function, (g) reconstruction with anatomical prior and Tukey function, and (h) reconstruction with anatomical prior and exceedance function.

Fig. 2
Fig. 2

Weighted Perona-Malik prior (a) at the first iteration, where only the anatomical edges are visible, and (b) at the last iteration, where both anatomical and fluorescence target edges can be seen.

Fig. 3
Fig. 3

Normalised profile plots across the fluorescent target: (a) lateral direction and (b) longitudinal direction.

Fig. 4
Fig. 4

Fluorochrome concentrations (μM) reconstructed from phantom data using : (a) Tikhonov prior, (b) structural prior and total variation, (c) structural prior and Perona-Malik function, (d) structural prior and Perona-Malik 2 function, and (e) structural prior and exceedance function. Images have dimensions 150 mm × 150 mm.

Fig. 5
Fig. 5

Fluorochrome concentrations (μM) reconstructed from mouse data using : (a) Tikhonov prior, (b) structural prior and Perona-Malik 2. Images have dimensions 32 mm × 32 mm.

Tables (2)

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Table 1 Edge-Preserving Functions

Tables Icon

Algorithm 1 Two-Step Image Reconstruction Algorithm

Equations (39)

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[ κ ( r , λ e ) + μ a ( r , λ e ) ] U ( r , λ e ) = q ( r , λ e ) , r Ω
[ κ ( r , λ f ) + μ a ( r , λ f ) ] U ( r , λ f ) = U ( r , λ e ) h ( r , λ f ) , r Ω
U ( r , λ ) + 2 ζ κ ( r , λ ) U ( r , λ ) n = 0 , r Ω
U ( r s , r d , λ e ) = Ω q ( r , λ e ) G ( r s , r d , λ e ) d r , y e = Θ e ( r s , r d ) U ( r s , r d , λ e ) ,
U ( r s , r d , λ f ) = Ω h ( r , λ f ) U ( r s , r , λ e ) G ( r , r d , λ f ) d r , y f = Θ f ( r s , r d , λ f ) U ( r s , r d , λ f ) ,
y f y e = U ( r s , r d , λ f ) U ( r s , r d , λ e )
= 1 U ( r s , r d , λ e ) Ω h ( r , λ f ) U ( r s , r , λ e ) G ( r , r d , λ f ) d r ,
y f y e = P ( Ω Σ ) U ( r s , r d , λ e ) P ( Ω Σ ) U ( r s , r d , λ e ) ,
y f y e = 1 U ( r s , r d , λ e ) i = 1 N h ( r i , λ f ) U ( r s , r i , λ e ) G ( r i , r d , λ f ) v .
y ^ = y f y e = J h ,
y i = k M x × M y c i k ϕ i k
y ˜ = J ˜ h .
h t = [ g ( | h | ) h ] ,
[ ψ ( | h | ) | h | h ] = ψ ( | h | ) ( h | h | ) + Δ ( ψ ( | h | ) ) h | h | .
[ ψ ( | h | ) | h | h ] = ψ ( | h | ) | h | ( h ξ 1 ξ 1 + h ξ 2 ξ 2 ) + ψ ( | h | ) h η η ,
. [ ψ ( | h | ) | h | h ] = h ξ 1 ξ 1 + h ξ 2 ξ 2 + h η η .
h i k + 1 h i k Δ t = j ( i ) g j k + g i k 2 d 2 ( h j k h i k ) ,
h i k + 1 h i k Δ t = l = 1 m 1 ( h k ) h k
h k + 1 = ( I + Δ t l = 1 m L l ( h k ) ) h k ,
i j ( h k ) = { g j k + g i k 2 d x 2 j ( i ) Σ j ( i ) g j k + g i k 2 d x 2 j = i 0 e l s e
h k + 1 = 1 m l = 1 m ( I m Δ t 1 ( h k ) ) 1 h k ,
Minimise E ( h ) = 1 2 | | y ˜ J ˜ h | | 2 + α Ψ ( h ) ,
Ψ ( h ) = Ω ψ ( | h | ) d Ω .
E ( h ) = J ˜ T ( J ˜ h y ˜ ) + α ( h ) h ,
( h ) = [ g ( | h | ) ] .
Ψ W ( h ) = Ω W ( | x r e f | ) ψ ( | h | ) d Ω ,
W ( h ) = [ W ( | x r e f | ) g ( | h | ) ] .
h k + 1 = h k + τ ( J ˜ T J ˜ + α Ψ " ( h k ) ) 1 J ˜ T ( ( y ˜ J ˜ h k ) α Ψ ( h k ) ) ,
L T L = ( h ) , Γ = [ ( h ) ] 1 = L 1 ( L 1 ) T .
J = J ˜ L 1 , h = L h ,
Minimise E ( h ) = 1 2 | | y ˜ J h | | 2 + α | | h | | 2 .
h k + 1 = h k J T ( y ˜ + J h k )
h k + 1 = h k + Γ J ˜ T ( y ˜ J ˜ h k ) .
h k + 1 / 2 = h k + J ˜ T ( y ˜ J ˜ h k )
( h k + 1 ) h k + 1 = h k + 1 / 2
h k + 1 / 2 = h k + J ˜ T B  ( y ˜ J ˜ h k ) ,
B  = ( J ˜ J ˜ T + λ I ) 1 ,
P S N R = 10 log 10 max ( h t r u e ) M S E ,
M S E = Σ i = 1 X Σ j = 1 Y Σ k = 1 Z ( h t r u e h r e c o n ) 2 X × Y × Z .

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