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 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.
Accepté le :
DOI : 10.1051/cocv/2016007
Mots-clés : Optical tomography, parameter identification, level set regularization, numerical strategy
@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 = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1051/cocv/2016007/
, Optical tomography in medical imaging. Inverse Probl. 15 (1999) R41–R93. | DOI | MR | Zbl
and . 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
and , 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).
and , Level-set of type for recovering shape and contrast in inverse problems. Inverse Probl. Sci. Eng. 20 (2012) 517–587. | DOI | MR | Zbl
and , Corrigendum: Level-set of 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.
, and , On multiple level-set regularization methods for inverse problems. Inverse Probl. 25 (2009) 035004. | DOI | MR | Zbl
, and , 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
and , 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
, and , 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
and , On the regularity of solutions to elliptic equations. Rend. Mat. Appl. 19 (1999) 471–488. | MR | Zbl
, and , Recent advances in diffuse optical imaging. Phys. Med. Biol. 50 (2005) R1–R43. | DOI
, On uniqueness in diffuse optical tomography. Inverse Probl. 25 (2009) 055010. | DOI | MR | Zbl
, and , 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
, , , and . Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data. Inverse Probl. 15 (1999) 1375–1391. | DOI | MR | Zbl
, An -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189–206. | Numdam | MR | Zbl
, , and , Multichannel photon counting instrument for spatially resolved near infrared spectroscopy. Rev. Sci. Instrum. 70 (1999) 193–201. | DOI
, and , Stochastic algorithms for inverse problems involving pdes and many measurements. SIAM J. Sci. Comput. 36 (2014) s3–s22. | DOI | MR | Zbl
, A level-set approach for inverse problems involving obstacles. ESAIM: COCV 1 (1995/96) 17–33. | Numdam | MR | Zbl
and , Application of temporal filters to time resolved data in optical tomography. Phys. Med. Biol. 44 (1999) 1699–717. | DOI
, , and , The finite element model for the propagation of light in scattering media: boundary and source conditions. Med. Phys. 22 (1995) 1779–1792. | DOI
, , and , Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography. Inverse Probl. 28 (2012) 084009. | DOI | MR | Zbl
and , On level set regularization for highly ill-posed distributed parameter estimation problems. J. Comput. Phys. 216 (2006) 707–723. | DOI | MR | Zbl
, , and , Absorption and scattering images of heterogeneous scattering media can be simultaneously reconstructed by use of dc data. Appl. Optim. 41 (2002) 5427–5437. | DOI
, , , and , 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 :
