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.

Accepted:

DOI: 10.1051/cocv/2016007

Keywords: Optical tomography, parameter identification, level set regularization, numerical strategy

^{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 = {http://archive.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://archive.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://archive.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, Volume 23 (2017) no. 2, pp. 663-683. doi : 10.1051/cocv/2016007. http://archive.numdam.org/articles/10.1051/cocv/2016007/

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

,Nonuniqueness in diffusion-based optical tomography. Opt. Lett. 23 (1998) 882–4. | DOI

and .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

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

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

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

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

and ,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.

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

and ,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

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

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

,Optical imaging in medicine: I. experimental techniques. Phys. Med. Biol. 42 (1997) 825. | DOI

, and ,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

Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data. Inverse Probl. 15 (1999) 1375–1391. | DOI | MR | Zbl

, , , and .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

,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

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

,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

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