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A wavelet multi-scale method for the inverse problem of diffuse optical tomography. (English) Zbl 1317.65194
Summary: This paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium. This is the so-called diffuse optical tomography problem. Such a nonlinear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is developed, implemented, and validated. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Numerical results show that the proposed method brings more stability with respect to the ordinary L-BFGS method and enhances the reconstructed images for most of initial guesses of optical properties.

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R25 Ill-posed problems for PDEs
35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65T60 Numerical methods for wavelets
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
FreeFem++; L-BFGS
Full Text: DOI
[1] Arridge, S. R., Optical tomography in medical imaging, Inverse Problems, 15, 2, R41, (1999) · Zbl 0926.35155
[2] Gibson, A.; Hebden, J.; Arridge, S. R., Recent advances in diffuse optical imaging, Phys. Med. Biol., 50, 4, R1, (2005)
[3] Arridge, S.; Schweiger, M., A gradient-based optimisation scheme for optical tomography, Opt. Express, 2, 6, 213-226, (1998)
[4] Schweiger, M.; Arridge, S. R.; Nissilä, I., Gauss-Newton method for image reconstruction in diffuse optical tomography, Phys. Med. Biol., 50, 10, 2365, (2005)
[5] Niu, H.; Guo, P.; Ji, L.; Zhao, Q.; Jiang, T., Improving image quality of diffuse optical tomography with a projection-error-based adaptive regularization method, Opt. Express, 16, 17, 12423-12434, (2008)
[6] Dehghani, H.; Eames, M. E.; Yalavarthy, P. K.; Davis, S. C.; Srinivasan, S.; Carpenter, C. M.; Pogue, B. W.; Paulsen, K. D., Near infrared optical tomography using NIRFAST: algorithm for numerical model and image reconstruction, Commun. Numer. Methods Eng., 25, 6, 711-732, (2009) · Zbl 1163.92022
[7] Tarvainen, T.; Kolehmainen, V.; Arridge, S.; Kaipio, J., Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model, J. Quant. Spectrosc. Radiat. Transfer, 112, 2600-2608, (2011)
[8] Klose, A. D.; Hielscher, A. H., Quasi-Newton methods in optical tomographic image reconstruction, Inverse Problems, 19, 2, 387, (2003) · Zbl 1022.65142
[9] Ren, K.; Bal, G.; Hielscher, A. H., Frequency domain optical tomography based on the equation of radiative transfer, SIAM J. Sci. Comput., 28, 4, 1463-1489, (2006) · Zbl 1200.65089
[10] Balima, O.; Boulanger, J.; Charette, A.; Marceau, D., New developments in frequency domain optical tomography. part II: application with a L-BFGS associated to an inexact line search, J. Quant. Spectrosc. Radiat. Transfer, 112, 7, 1235-1240, (2011)
[11] Balima, O.; Favennec, Y.; Rousse, D., Optical tomography reconstruction algorithm with the finite element method: an optimal approach with regularization tools, J. Comput. Phys., 251, 461-479, (2013) · Zbl 1349.65592
[12] Y. Favennec, F. Dubot, B. Rousseau, D.R. Rousse, Mixing regularization tools for enhancing regularity in optical tomography applications, in: O. Fudym, J.L. Battaglia, G.S. Dulikravich (Eds.), IPDO 2013: 4th Inverse Problems, Design and Optimization Symposium, Albi, 2013. · Zbl 1317.65194
[13] Zacharopoulos, A. D.; Arridge, S. R.; Dorn, O.; Kolehmainen, V.; Sikora, J., Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method, Inverse Problems, 22, 5, 1509, (2006) · Zbl 1105.78012
[14] Arridge, S. R.; Dorn, O.; Kolehmainen, V.; Schweiger, M.; Zacharopoulos, A., Parameter and structure reconstruction in optical tomography, (Journal of Physics: Conference Series, vol. 135, (2008), IOP Publishing), 012001
[15] Arridge, S.; Kaipio, J.; Kolehmainen, V.; Schweiger, M.; Somersalo, E.; Tarvainen, T.; Vauhkonen, M., Approximation errors and model reduction with an application in optical diffusion tomography, Inverse Problems, 22, 1, 175, (2006) · Zbl 1138.65042
[16] Kaipio, J.; Somersalo, E., Statistical and computational inverse problems, vol. 160, (2005), Springer
[17] Cao, N.; Nehorai, A.; Jacobs, M., Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm, Opt. Express, 15, 21, 13695-13708, (2007)
[18] Liu, J., A multiresolution method for distributed parameter estimation, SIAM J. Sci. Comput., 14, 2, 389-405, (1993) · Zbl 0773.65059
[19] Ding, L.; Han, B.; Liu, J.-q., A wavelet multiscale method for inversion of Maxwell equations, Appl. Math. Mech., 30, 1035-1044, (2009) · Zbl 1180.35505
[20] Zhang, X.; Liu, K.; Liu, J., The wavelet multiscale method for inversion of porosity in the fluid-saturated porous media, Appl. Math. Comput., 180, 2, 419-427, (2006) · Zbl 1101.76043
[21] He, Y.; Han, B., A wavelet adaptive-homotopy method for inverse problem in the fluid-saturated porous media, Appl. Math. Comput., 208, 1, 189-196, (2009) · Zbl 1156.74020
[22] Fu, H.; Han, B.; Gai, G., A wavelet multiscale-homotopy method for the inverse problem of two-dimensional acoustic wave equation, Appl. Math. Comput., 190, 1, 576-582, (2007) · Zbl 1122.65386
[23] Lei, J.; Liu, S.; Li, Z.; Sun, M.; Wang, X., A multi-scale image reconstruction algorithm for electrical capacitance tomography, Appl. Math. Model., 35, 6, 2585-2606, (2011) · Zbl 1219.78124
[24] Fu, H.; Han, B.; Liu, H., A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations, Neurocomputing, 104, 138-145, (2013)
[25] Zhao, J.; Liu, T.; Liu, S., Identification of space-dependent permeability in nonlinear diffusion equation from interior measurements using wavelet multiscale method, Inverse Probl. Sci. Eng., 22, 4, 507-529, (2014) · Zbl 1308.65162
[26] Nath, S. K.; Vasu, R.; Pandit, M., Wavelet based compression and denoising of optical tomography data, Opt. Commun., 167, 1, 37-46, (1999)
[27] Zhu, W.; Wang, Y.; Deng, Y.; Yao, Y.; Barbour, R. L., A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography, IEEE Trans. Med. Imaging, 16, 2, 210-217, (1997)
[28] Zhu, W.; Wang, Y.; Yao, Y.; Chang, J.; Graber, H. L.; Barbour, R. L., Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method, J. Opt. Soc. Amer. A, 14, 4, 799-807, (1997)
[29] Zhu, W.; Wang, Y.; Zhang, J., Total least-squares reconstruction with wavelets for optical tomography, J. Opt. Soc. Amer. A, 15, 10, 2639-2650, (1998)
[30] Modest, M., Radiative heat transfer, (2003), McGraw-Hill, Inc.
[31] Howell, J.; Siegel, R.; Mengüç, M., Thermal radiation heat transfer, (2011), CRC Press
[32] Tarvainen, T., Computational methods for light transport in optical tomography, (2006), University of Kuopio, (Ph.D. thesis)
[33] Klose, A.; Netz, U.; Beuthan, J.; Hielscher, A., Optical tomography using the time-independent equation of radiative transfer—part 1: forward model, J. Quant. Spectrosc. Radiat. Transfer, 72, 691-713, (2002)
[34] Brattka, V.; Yoshikawa, A., Towards computability of elliptic boundary value problems in variational formulation, J. Complexity, 22, 6, 858-880, (2006) · Zbl 1126.03052
[35] Arridge, S. R.; Schweiger, M., Photon-measurement density functions. part 2: finite-element-method calculations, Appl. Opt., 34, 34, 8026-8037, (1995)
[36] Nocedal, J., Updating quasi-Newton matrices with limited storage, Math. Comp., 35, 151, 773-782, (1980) · Zbl 0464.65037
[37] F. Dubot, Y. Favennec, B. Rousseau, D.R. Rousse, Paramétrisation des variables de contrôle et méthodes de recherche linéaire dans un code d’inversion de l’approximation de diffusion basé sur le L-BFGS, in: Comptes-Rendus Congrès de la Société Française de Thermique, Lyon, France, 2014.
[38] Antoniou, A.; Lu, W.-S., Practical optimization: algorithms and engineering applications, (2007), Springer · Zbl 1128.90001
[39] Fletcher, R., Practical methods of optimization, vol. 1, unconstrained optimization, 126, (1980), British Library Cataloguing in Publication Data · Zbl 0439.93001
[40] Alifanov, O., Solution of an inverse problem of heat conduction by iteration methods, J. Eng. Phys. Thermophys., 26, 4, 471-476, (1974)
[41] McBride, T. O.; Pogue, B. W.; Jiang, S.; Oesterberg, U. L.; Paulsen, K. D., A parallel-detection frequency-domain near-infrared tomography system for hemoglobin imaging of the breast in vivo, Rev. Sci. Instrum., 72, 3, 1817-1824, (2001)
[42] Mallat, S., A wavelet tour of signal processing, (1999), Academic press · Zbl 0998.94510
[43] Addison, P. S., The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance, (2010), CRC Press · Zbl 1081.42025
[44] Hecht, F., New development in freefem++, J. Numer. Math., 20, 3-4, 251-266, (2012) · Zbl 1266.68090
[45] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1998), Springer · Zbl 0893.35138
[46] Flannery, B. P.; Press, W. H.; Teukolsky, S. A.; Vetterling, W., Numerical recipes in C, (1992), Press Syndicate of the University of Cambridge New York · Zbl 0778.65003
[47] Williams, J. R.; Amaratunga, K., A discrete wavelet transform without edge effects using wavelet extrapolation, J. Fourier Anal. Appl., 3, 4, 435-449, (1997) · Zbl 0880.42021
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