Abstract

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the emphasis is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of Partial Differential Equations (PDEs). The standard polynomial Finite Element Method (FEM) is a method of choice to solve the diffusion equation because it has no restriction in terms of neither the geometry nor the homogeneity of the system, but it is time consuming. In order to speed up computation time, this paper proposes an alternative numerical model, describing the diffusion operator in orthonormal basis of compactly supported wavelets. The discretization of the PDEs yields to matrices which are easily computed from derivative wavelet product integrals. Due to the shape of the wavelet basis, the studied domain is included in a regular fictitious domain. A validation study and a comparison with the standard FEM are conducted on synthetic data.

© 2009 Optical Society of America

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B. Kanmani, P. Bansal, and R. M. Vasu, "Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation," Proc. SPIE 6164, R1-R10 (2006).

2005

J. Baccou, and J. Liandrat, "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett. 18(12), 1325-1331 (2005).
[CrossRef]

2004

2003

R. Elaloufi, R. Carminati, and J. J. Greffet, "Definition of the diffusion coefficient in scattering and absorbing media," J. Opt. Soc. Am. A 20(4), 678-685, (2003).
[CrossRef]

F. Fedele, J. P. Laible, and M. J. Eppstein, "Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation," J. Comp. Phys. 187(2), 597-619 (2003).
[CrossRef]

R. Weissleder, and V. Ntziachristos, "Shedding light onto live molecular targets," Nature Med 9(1), 123-128 (2003).

2002

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

1999

S. K. Nath, R. M. Vasu, and M. Pandit, "Wavelet based compression and denoising of optical tomography data," Opt. Commun. 167(1-6), 37-46 (1999).
[CrossRef]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15(2), R41-R93 (1999).
[CrossRef]

1997

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

1996

S. Dumont, and F. Lebon, "Representation of plane elastostatics operators in Daubechies wavelets," Comp. Structures 60(4), 561-569 (1996).
[CrossRef]

M. Q. Chen, C. I. Hwang, and Y. P. Shih, "The computation of wavelet-Galerkin approximation on a bounded interval," Int. J. Num. Meth. Engin. 39(17), 2921-2944 (1996).
[CrossRef]

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

1995

P. Charton, and V. Perrier, "Produits rapides matrice-vecteur en bases d'ondelettes: application à la résolution numérique d'équations aux dérivées partielles," Math. Modelling Numerical Anal. 29(6), 701-747 (1995).

O. V. Vasilyev, S. Paolucci, and M. Sen, "A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain," J.Comp. Phys. 120(1), 33-47 (1995).
[CrossRef]

1994

1993

B. Sahiner, and A. E. Yagle, "Image-Reconstruction From Projections Under Wavelet Constraints," IEEE Trans. Signal Proc. 41(12), 3579-3584 (1993).
[CrossRef]

1992

S. Jaffard, "Wavelet Methods For Fast Resolution Of Elliptic Problems," SIAM J. Num. Anal. 29(4), 965-986 (1992).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, "The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue-Temporal And Frequency-Analysis," Phys. Med. Bio. 37(7), 1531-1560 (1992).
[CrossRef]

G. Beylkin, "On The Representation Of Operators In Bases Of Compactly Supported Wavelets," SIAM J. Numerical Anal. 29(6), 1716-1740 (1992).
[CrossRef]

1989

S. Mallat, "A Theory of Multiresolution Signal Decomposition: the Wavelet Representation," IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674-693 (1989).
[CrossRef]

1984

A. Grossmann, and J. Morlet, "Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape," SIAM J. Math. Anal. 15(4), 723-736 (1984).
[CrossRef]

Arridge, S. R.

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15(2), R41-R93 (1999).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, "The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue-Temporal And Frequency-Analysis," Phys. Med. Bio. 37(7), 1531-1560 (1992).
[CrossRef]

Baccou, J.

J. Baccou, and J. Liandrat, "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett. 18(12), 1325-1331 (2005).
[CrossRef]

Bangerth, W.

Bansal, P.

B. Kanmani, P. Bansal, and R. M. Vasu, "Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation," Proc. SPIE 6164, R1-R10 (2006).

Barbour, R. L.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

Beylkin, G.

G. Beylkin, "On The Representation Of Operators In Bases Of Compactly Supported Wavelets," SIAM J. Numerical Anal. 29(6), 1716-1740 (1992).
[CrossRef]

Bonnet, S.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

Carminati, R.

Charton, P.

P. Charton, and V. Perrier, "Produits rapides matrice-vecteur en bases d'ondelettes: application à la résolution numérique d'équations aux dérivées partielles," Math. Modelling Numerical Anal. 29(6), 701-747 (1995).

Chen, M. Q.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, "The computation of wavelet-Galerkin approximation on a bounded interval," Int. J. Num. Meth. Engin. 39(17), 2921-2944 (1996).
[CrossRef]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, "The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue-Temporal And Frequency-Analysis," Phys. Med. Bio. 37(7), 1531-1560 (1992).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, "The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue-Temporal And Frequency-Analysis," Phys. Med. Bio. 37(7), 1531-1560 (1992).
[CrossRef]

Deng, Y. N.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

Dumont, S.

S. Dumont, and F. Lebon, "Representation of plane elastostatics operators in Daubechies wavelets," Comp. Structures 60(4), 561-569 (1996).
[CrossRef]

Elaloufi, R.

Eppstein, M. J.

F. Fedele, J. P. Laible, and M. J. Eppstein, "Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation," J. Comp. Phys. 187(2), 597-619 (2003).
[CrossRef]

Fedele, F.

F. Fedele, J. P. Laible, and M. J. Eppstein, "Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation," J. Comp. Phys. 187(2), 597-619 (2003).
[CrossRef]

Feng, T. C.

Glowinski, R.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

Greffet, J. J.

Grossmann, A.

A. Grossmann, and J. Morlet, "Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape," SIAM J. Math. Anal. 15(4), 723-736 (1984).
[CrossRef]

Haskell, R. C.

Hwang, C. I.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, "The computation of wavelet-Galerkin approximation on a bounded interval," Int. J. Num. Meth. Engin. 39(17), 2921-2944 (1996).
[CrossRef]

Jaffard, S.

S. Jaffard, "Wavelet Methods For Fast Resolution Of Elliptic Problems," SIAM J. Num. Anal. 29(4), 965-986 (1992).
[CrossRef]

Joshi, A.

Kanmani, B.

B. Kanmani, P. Bansal, and R. M. Vasu, "Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation," Proc. SPIE 6164, R1-R10 (2006).

Laible, J. P.

F. Fedele, J. P. Laible, and M. J. Eppstein, "Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation," J. Comp. Phys. 187(2), 597-619 (2003).
[CrossRef]

Lebon, F.

S. Dumont, and F. Lebon, "Representation of plane elastostatics operators in Daubechies wavelets," Comp. Structures 60(4), 561-569 (1996).
[CrossRef]

Liandrat, J.

J. Baccou, and J. Liandrat, "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett. 18(12), 1325-1331 (2005).
[CrossRef]

Mallat, S.

S. Mallat, "A Theory of Multiresolution Signal Decomposition: the Wavelet Representation," IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674-693 (1989).
[CrossRef]

McAdams, M. S.

Morlet, J.

A. Grossmann, and J. Morlet, "Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape," SIAM J. Math. Anal. 15(4), 723-736 (1984).
[CrossRef]

Nath, S. K.

S. K. Nath, R. M. Vasu, and M. Pandit, "Wavelet based compression and denoising of optical tomography data," Opt. Commun. 167(1-6), 37-46 (1999).
[CrossRef]

Ntziachristos, V.

R. Weissleder, and V. Ntziachristos, "Shedding light onto live molecular targets," Nature Med 9(1), 123-128 (2003).

Pan, T. W.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

Pandit, M.

S. K. Nath, R. M. Vasu, and M. Pandit, "Wavelet based compression and denoising of optical tomography data," Opt. Commun. 167(1-6), 37-46 (1999).
[CrossRef]

Paolucci, S.

O. V. Vasilyev, S. Paolucci, and M. Sen, "A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain," J.Comp. Phys. 120(1), 33-47 (1995).
[CrossRef]

Perrier, V.

P. Charton, and V. Perrier, "Produits rapides matrice-vecteur en bases d'ondelettes: application à la résolution numérique d'équations aux dérivées partielles," Math. Modelling Numerical Anal. 29(6), 701-747 (1995).

Peyrin, F.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

Prost, R.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

Sahiner, B.

B. Sahiner, and A. E. Yagle, "Image-Reconstruction From Projections Under Wavelet Constraints," IEEE Trans. Signal Proc. 41(12), 3579-3584 (1993).
[CrossRef]

Sen, M.

O. V. Vasilyev, S. Paolucci, and M. Sen, "A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain," J.Comp. Phys. 120(1), 33-47 (1995).
[CrossRef]

Sevick-Muraca, E. M.

Shih, Y. P.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, "The computation of wavelet-Galerkin approximation on a bounded interval," Int. J. Num. Meth. Engin. 39(17), 2921-2944 (1996).
[CrossRef]

Svaasand, L. O.

Tsay, T. T.

Turjman, F.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

Vasilyev, O. V.

O. V. Vasilyev, S. Paolucci, and M. Sen, "A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain," J.Comp. Phys. 120(1), 33-47 (1995).
[CrossRef]

Vasu, R. M.

B. Kanmani, P. Bansal, and R. M. Vasu, "Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation," Proc. SPIE 6164, R1-R10 (2006).

S. K. Nath, R. M. Vasu, and M. Pandit, "Wavelet based compression and denoising of optical tomography data," Opt. Commun. 167(1-6), 37-46 (1999).
[CrossRef]

Wang, Y.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

Weissleder, R.

R. Weissleder, and V. Ntziachristos, "Shedding light onto live molecular targets," Nature Med 9(1), 123-128 (2003).

Wells, R. O.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

Yagle, A. E.

B. Sahiner, and A. E. Yagle, "Image-Reconstruction From Projections Under Wavelet Constraints," IEEE Trans. Signal Proc. 41(12), 3579-3584 (1993).
[CrossRef]

Yao, Y. Q.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

Zhou, X. D.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

Zhu, W.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

Appl. Math. Lett.

J. Baccou, and J. Liandrat, "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett. 18(12), 1325-1331 (2005).
[CrossRef]

Comp. Structures

S. Dumont, and F. Lebon, "Representation of plane elastostatics operators in Daubechies wavelets," Comp. Structures 60(4), 561-569 (1996).
[CrossRef]

IEEE Trans. Image Proc.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, "Multiresolution Reconstruction in Fan-Beam Tomography," IEEE Trans. Image Proc. 11(3), 169-176 (2002).
[CrossRef]

IEEE Trans. Med. Imaging

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, "A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography," IEEE Trans. Med. Imaging 16(2), 210-217 (1997).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

S. Mallat, "A Theory of Multiresolution Signal Decomposition: the Wavelet Representation," IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674-693 (1989).
[CrossRef]

IEEE Trans. Signal Proc.

B. Sahiner, and A. E. Yagle, "Image-Reconstruction From Projections Under Wavelet Constraints," IEEE Trans. Signal Proc. 41(12), 3579-3584 (1993).
[CrossRef]

Int. J. Num. Meth. Engin.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, "The computation of wavelet-Galerkin approximation on a bounded interval," Int. J. Num. Meth. Engin. 39(17), 2921-2944 (1996).
[CrossRef]

Inverse Problems

S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15(2), R41-R93 (1999).
[CrossRef]

J. Comp. Phys.

F. Fedele, J. P. Laible, and M. J. Eppstein, "Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation," J. Comp. Phys. 187(2), 597-619 (2003).
[CrossRef]

J. Comput. Phys.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, "Wavelet and finite element solutions for the Neumann problem using fictitious domains," J. Comput. Phys. 126(1), 40-51 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J.Comp. Phys.

O. V. Vasilyev, S. Paolucci, and M. Sen, "A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain," J.Comp. Phys. 120(1), 33-47 (1995).
[CrossRef]

Math. Modelling Numerical Anal.

P. Charton, and V. Perrier, "Produits rapides matrice-vecteur en bases d'ondelettes: application à la résolution numérique d'équations aux dérivées partielles," Math. Modelling Numerical Anal. 29(6), 701-747 (1995).

Nature Med

R. Weissleder, and V. Ntziachristos, "Shedding light onto live molecular targets," Nature Med 9(1), 123-128 (2003).

Opt. Commun.

S. K. Nath, R. M. Vasu, and M. Pandit, "Wavelet based compression and denoising of optical tomography data," Opt. Commun. 167(1-6), 37-46 (1999).
[CrossRef]

Opt. Express

Phys. Med. Bio.

S. R. Arridge, M. Cope, and D. T. Delpy, "The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue-Temporal And Frequency-Analysis," Phys. Med. Bio. 37(7), 1531-1560 (1992).
[CrossRef]

Proc. SPIE

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Figures (4)

Fig. 1.
Fig. 1.

Daubechies scaling function of order M=3.

Fig. 2.
Fig. 2.

(a) Schema of a fictitious domain (rectangle Ω’) larger than the studied domain Ω in 2D. (b) Schema of the discretized approximation of edges ∂Ω. (c) Schema of a fictitious domain in 1D and corresponding scaling functions Φ j,k =2-j/2Φ)./2 j -k) with j=0 and k=-4 to 7.

Fig. 3.
Fig. 3.

(a)(b) Schemas of the 2D geometries: black points: positions of the source; gray points: detectors; ellipses: 2 simulated fluorophores; gray grid: regular grid for a scale 2–3; large rectangle: fictitious larger domain; (c) schemas of the 3D geometry: ellipsoid: simulated fluorophore. (d) Example of result for the forward problem (ux for the source #8 (surrounded in (a)), in the first simulation), by wavelet-Galerkin method, analytical modelling and FEM. (e) (h) Results of the rectangular simulation. (f)(i) Results of the irregular geometry simulation. g) (j) Results of the cube simulation. (e) (f) (g) Results after reconstruction, with a forward modelling by wavelet-Galerkin method and a reconstruction by ART; (h) (i) (j) Results with a forward modelling by standard FEM method and a reconstruction by ART.

Fig. 4.
Fig. 4.

Reconstruction of the second simulation (a) with connection coefficients accurately calculated on Ω; (b) without accurate connection coefficients on Ω.

Tables (5)

Tables Icon

Table 1. Numerical methods by the Finite Elements Methods.

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Table 2. Description of the 3 simulations

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Table A-1. Daubechies scaling filter coefficients, order M=3.

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Table A-2. Values of Γ1,1 k,l [-∞;+∞]=∫+∞-∞Φ(1)(x-k) Φ(1)(x-l)dx, with Φ the Daubechies scaling function of order M=3. This filter is symmetrical, only the value |k-l| is considered.

Tables Icon

Table A-3. Values of Γ0,0 k,l [0;1]=∫1 0Φ(x-k)Φ(x-l)dx, with Φ the Daubechies scaling function of order M=3.

Equations (31)

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{.(D(r)ux(r,rs))+μa(r)ux(r,rs)=q(rs).(D(r)um(r,r,rs))+μa(r)um(r,r,rs)=β(r)ux(r,rs) .
{n.(Dux(r,rs))+ξxux(r,rs)rΩn.(Dum(r,r,rs))+ξmum(r,r,rs)=0rΩ
umobs (rd,rs)=Ωum(rd,r,rs)dr=Ωgm(r,rd)β(r)ux(r,rs)dr
. (Dgm(r,rd))+μagm(r,rd)=q(rd)
. (Du)+μau=q
u˜ = k=1Nukfkinterp.
q˜ = k=1Nqkfkinterp.
Ω . (Du˜)+μau˜q˜).flproj=0l=1,,N.
A u = Pq q
umobs=W β ,
Φj,k = 2j2 Φ (.2jk)
Φ = 2 nZh(n)Φ(2.n).
D k=1NukΩ(2fkinterp)flproj+μak=1NukΩfkinterpflproj=k=1NqkΩfkinterpflprojl=1,,N
ζ k=1NukΩfkfl+Dk=1NukΩfkfl+μak=1NukΩfkfl=k=1NqkΩfkfll=1,,N
u˜j = k=0Nxpuj,kΦj,k;q˜=k=0Nxpqj,kΦj,k
qj,k = Ωq Φj,k .
ζ k=0Nxpuj,kΩΦj,kΦj,l+Dk=0Nxpuj,kΩΦj,kΦj,l+μak=0Nxpuj,kΩΦj,kΦj,l
= k=pNxpqj,kΩΦj,kΦj,l,l=1,,N.
Γj,k,lm1,m2 [;+]=2j(1+m1+m2)+Φj,k(m1)(x)Φj,l(m2)(x)dx
Γk,l0,0 [;+]=+Φ(xk)Φ(xl)dx=δk,l.
Γk,lm1,m2 [0;n]=0nΦ(m1)(xk)Φ(m2)(xl)dx.
Γk,l0,0 [0;n]=p=0n1Γkp,lp0,0[0;1].
Φj,kx,ky (x,y)=Φj,kx(x)Φj,ky(y)
= 2j Φ (2jxkx)Φ(2jyky),
kx = 0 , , Nx p ,
ky = 0 , , Ny p .
ΩΦkx,ky(x,y)Φlx,ly(x,y)dxdy=Φ(kx)Φ(lx)y=0LyΦky(y)Φly(y)dy
+ Φ (Lxkx)Φ(Lxlx)y=0LyΦky(y)Φly(y)dy
+ Φ (ky)Φ(ly)x=0LxΦkx(x)Φlx(x)dx
+ Φ (Lyky)Φ(Lyly)x=0LxΦkx(x)Φlx(x)dx.
ε = umobsumrec2umobs2

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