On the identification of piecewise constant coefficients in optical diffusion tomography by level set

Agnelli, J. P.; De Cezaro, A.; Leitão, A.; Marques Alves, M.

doi:10.1051/cocv/2016007

Agnelli, J. P.¹ ; De Cezaro, A.² ; Leitão, A.³ ; Marques Alves, M.³

¹ FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, X5000HUA, Córdoba, Argentina.
² Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil.
³ Department of Mathematics, Federal University of St. Catarina, P.O. Box 476, 88040-900 Florianópolis, Brazil.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 663-683.

Résumé

In this paper, we propose a level set regularization approach combined with a split strategy for the simultaneous identification of piecewise constant diffusion and absorption coefficients from a finite set of optical tomography data (Neumann-to-Dirichlet data). This problem is a high nonlinear inverse problem combining together the exponential and mildly ill-posedness of diffusion and absorption coefficients, respectively. We prove that the parameter-to-measurement map satisfies sufficient conditions (continuity in the $L^{1}$ topology) to guarantee regularization properties of the proposed level set approach. On the other hand, numerical tests considering different configurations bring new ideas on how to propose a convergent split strategy for the simultaneous identification of the coefficients. The behavior and performance of the proposed numerical strategy is illustrated with some numerical examples.

Reçu le : 2015-05-15
Accepté le : 2016-01-14

MR Zbl

DOI : 10.1051/cocv/2016007

Classification : 49N45, 65N21, 74J25
Mots-clés : Optical tomography, parameter identification, level set regularization, numerical strategy

Affiliations des auteurs :

Agnelli, J. P. ¹ ; De Cezaro, A. ² ; Leitão, A. ³ ; Marques Alves, M. ³

@article{COCV_2017__23_2_663_0,
     author = {Agnelli, J. P. and De Cezaro, A. and Leit\~ao, A. and Marques Alves, M.},
     title = {On the identification of piecewise constant coefficients in optical diffusion tomography by level set},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {663--683},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {2},
     year = {2017},
     doi = {10.1051/cocv/2016007},
     mrnumber = {3608098},
     zbl = {1358.49031},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2016007/}
}

TY  - JOUR
AU  - Agnelli, J. P.
AU  - De Cezaro, A.
AU  - Leitão, A.
AU  - Marques Alves, M.
TI  - On the identification of piecewise constant coefficients in optical diffusion tomography by level set
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 663
EP  - 683
VL  - 23
IS  - 2
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv/2016007/
DO  - 10.1051/cocv/2016007
LA  - en
ID  - COCV_2017__23_2_663_0
ER  -

%0 Journal Article
%A Agnelli, J. P.
%A De Cezaro, A.
%A Leitão, A.
%A Marques Alves, M.
%T On the identification of piecewise constant coefficients in optical diffusion tomography by level set
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 663-683
%V 23
%N 2
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2016007/
%R 10.1051/cocv/2016007
%G en
%F COCV_2017__23_2_663_0

Agnelli, J. P.; De Cezaro, A.; Leitão, A.; Marques Alves, M. On the identification of piecewise constant coefficients in optical diffusion tomography by level set. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 663-683. doi : 10.1051/cocv/2016007. https://www.numdam.org/articles/10.1051/cocv/2016007/

Bibliographie
Cité par

S.R. Arridge, Optical tomography in medical imaging. Inverse Probl. 15 (1999) R41–R93. | DOI | MR | Zbl

S.R. Arridge and W.R.B. Lionheart. Nonuniqueness in diffusion-based optical tomography. Opt. Lett. 23 (1998) 882–4. | DOI

S.R. Arridge and M. Schweiger, A general framework for iterative reconstruction algorithms in optical tomography, using a finite element method. In Computational radiology and imaging (Minneapolis, MN, 1997). Vol. 110 of IMA Volumes Math. Appl. Springer, New York (1999) 45–70. | MR | Zbl

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011). | MR | Zbl

M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design. European J. Appl. Math. 16 (2005) 263–301. | DOI | MR | Zbl

R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2. Springer-Verlag, Berlin (1988).

A. De Cezaro and A. Leitão, Level-set of $L^{2}$ type for recovering shape and contrast in inverse problems. Inverse Probl. Sci. Eng. 20 (2012) 517–587. | DOI | MR | Zbl

A. De Cezaro and A. Leitão, Corrigendum: Level-set of $L^{2}$ type for recovering shape and contrast in inverse problems. Inverse Probl. Sci. Eng. 21 (2013) 1–2. | Zbl

A. De Cezaro, A. Leitão and X.-C. Tai, On level-set type methods for recovering piecewise constant solutions of ill-posed problems. In Scale Space and Variational Methods in Computer Vision, edited by X.-C. Tai, K. Mørken, K. Lysaker and K.-A. Lie. Vol. 5667 of Lecture Notes Comput. Sci. Springer, Berlin (2009) 50–62.

A. De Cezaro, A. Leitão and X.-C. Tai, On multiple level-set regularization methods for inverse problems. Inverse Probl. 25 (2009) 035004. | DOI | MR | Zbl

A. De Cezaro, A. Leitão and X.-C. Tai, On piecewise constant level-set (pcls) methods for the identication of discontinuous parameters in ill-posed problems. Inverse Probl. 29 (2013) 015003. | DOI | MR | Zbl

O. Dorn and D. Lesselier, Level set methods for inverse scattering—some recent developments. Inverse Probl. 25 (2009) 125001. | DOI | MR | Zbl

H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems. Vol. 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1996). | MR | Zbl

L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

F. Frühauf, O. Scherzer and A. Leitão, Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators. SIAM J. Numer. Anal. 43 (2005) 767–786. | DOI | MR | Zbl

T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations. Rend. Mat. Appl. 19 (1999) 471–488. | MR | Zbl

A.P. Gibson, J.C. Hebden and S.R. Arridge, Recent advances in diffuse optical imaging. Phys. Med. Biol. 50 (2005) R1–R43. | DOI

B. Harrach, On uniqueness in diffuse optical tomography. Inverse Probl. 25 (2009) 055010. | DOI | MR | Zbl

J.C. Hebden, S.R Arridge and D.T. Delpy, Optical imaging in medicine: I. experimental techniques. Phys. Med. Biol. 42 (1997) 825. | DOI

V. Isakov, Inverse problems for partial differential equations. Vol. 127 of Applied Mathematical Sciences. Springer 2nd edition, New York (2006). | MR | Zbl

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems. Vol. 6 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2008). | MR | Zbl

V. Kolehmainen, S.R. Arridge, W.R.B. Lionheart, M. Vauhkonen and J.P. Kaipio. Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data. Inverse Probl. 15 (1999) 1375–1391. | DOI | MR | Zbl

N.G. Meyers, An $L^{p}$ -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189–206. | Numdam | MR | Zbl

V. Ntziachristos, X. Ma, A.G. Yodh and B. Chance, Multichannel photon counting instrument for spatially resolved near infrared spectroscopy. Rev. Sci. Instrum. 70 (1999) 193–201. | DOI

F. Roosta-Khorasani, Kees Van Den Doel and Uri M. Ascher, Stochastic algorithms for inverse problems involving pdes and many measurements. SIAM J. Sci. Comput. 36 (2014) s3–s22. | DOI | MR | Zbl

F. Santosa, A level-set approach for inverse problems involving obstacles. ESAIM: COCV 1 (1995/96) 17–33. | Numdam | MR | Zbl

M. Schweiger and S.R. Arridge, Application of temporal filters to time resolved data in optical tomography. Phys. Med. Biol. 44 (1999) 1699–717. | DOI

M. Schweiger, S.R. Arridge, M. Hiraoka and D.T. Delpy, The finite element model for the propagation of light in scattering media: boundary and source conditions. Med. Phys. 22 (1995) 1779–1792. | DOI

T. Tarvainen, B.T. Cox, J.P. Kaipo and S.R. Arridge, Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography. Inverse Probl. 28 (2012) 084009. | DOI | MR | Zbl

K. Van Den Doel and U.M. Ascher, On level set regularization for highly ill-posed distributed parameter estimation problems. J. Comput. Phys. 216 (2006) 707–723. | DOI | MR | Zbl

Yong Xu, Xuejun Gu, Taufiquar Khan and Huabei Jiang, Absorption and scattering images of heterogeneous scattering media can be simultaneously reconstructed by use of dc data. Appl. Optim. 41 (2002) 5427–5437. | DOI

A.D. Zacharopoulos, S.R. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method. Inverse Probl. 22 (2006) 1509–1532. | DOI | MR | Zbl

Cité par Sources :