zbMATH — the first resource for mathematics

Geometric reconstruction in bioluminescence tomography. (English) Zbl 1292.92009
Summary: In bioluminescence tomography the location as well as the radiation intensity of a photon source (marked cell clusters) inside an organism have to be determined given the outside photon count. This inverse source problem is ill-posed: it suffers not only from strong instability but also from non-uniqueness. To cope with these difficulties the source is modeled as a linear combination of indicator functions of measurable domains leading to a nonlinear operator equation. The solution process is stabilized by a Tikhonov like functional which penalizes the perimeter of the domains. For the resulting minimization problem existence of a minimizer, stability, and regularization property are shown. Moreover, an approximate variational principle is developed based on the calculated domain derivatives which states that there exist smooth almost stationary points of the Tikhonov like functional near to any of its minimizers. This is a crucial property from a numerical point of view as it allows to approximate the searched-for domain by smooth domains. Based on the theoretical findings numerical schemes are proposed and tested for star-shaped sources in 2D: computational experiments illustrate performance and limitations of the considered approach.
92C55 Biomedical imaging and signal processing
65K10 Numerical optimization and variational techniques
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35R30 Inverse problems for PDEs
Full Text: DOI
[1] K. Atkinson, <em>Theoretical Numerical Analysis</em>,, 3rd edition, (2009) · Zbl 1181.47078
[2] H. Attouch, <em>Variational Analysis in Sobolev and BV Space,</em>, MPS-SIAM Series on Optimization, (2006) · Zbl 1095.49001
[3] G. Bal, Inverse transport theory and applications,, Inverse Problems, 25, (2009) · Zbl 1178.35377
[4] M. Berger, <em>Differential Geometry: Manifolds, Curves and Surfaces</em>,, Graduate Texts in Mathematics, (1988) · Zbl 0629.53001
[5] M. Burger, A survey on level set methods for inverse problems and optimal design,, European J. Appl. Math., 16, 263, (2005) · Zbl 1091.49001
[6] F. Caubet, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid,, Inverse Probl. Imaging, 7, 123, (2013) · Zbl 1266.49076
[7] W. Cong, Practical reconstruction method for bioluminescence tomography,, Opt. Express, 13, 6756, (2005)
[8] C. H. Contag, It’s not just about anatomy: In vivo bioluminescence imaging as an eyepiece into biology,, Journal of Magnetic Resonance Imaging, 16, 378, (2002)
[9] A. De Cezaro, Level-set approaches of \(L_2\)-type for recovering shape and contrast in ill-posed problems,, Inverse Probl. Sci. Eng., 20, 571, (2012) · Zbl 1257.65029
[10] M. C. Delfour, <em>Shapes and Geometries: Analysis, Differential Calculus, and Optimization</em>,, Advances in Design and Control, (2001) · Zbl 1251.49001
[11] I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47, 324, (1974) · Zbl 0286.49015
[12] I. Ekeland, Nonconvex minization problems,, Bull. Am. Math. Soc., 1, 443, (1979) · Zbl 0441.49011
[13] W. Freeden, <em>Constructive Approximation on the Sphere</em>,, Numerical Mathematics and Scientific Computation, (1998) · Zbl 0896.65092
[14] E. Giusti, <em>Minimal Surfaces and Functions of Bounded Variation</em>,, Monographs in Mathematics, (1984) · Zbl 0545.49018
[15] W. Han, Mathematical theory and numerical analysis of bioluminescence tomography,, Inverse Problems, 22, 1659, (2006) · Zbl 1106.35124
[16] M. Hanke-Bourgeois, <em>Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens</em>,, 3rd edition, (2009) · Zbl 1155.65001
[17] H. Harbrecht, An efficient numerical method for a shape-identification problem arising from the heat equation,, Inverse Problems, 27, (2011) · Zbl 1219.65132
[18] F. Hettlich, <em>The Domain Derivative in Inverse Obstacle Problems</em>,, Habilitation thesis, (1999)
[19] M. Hinze, <em>Optimization with PDE Constraints,</em>, Mathematical Modelling: Theory and Applications, (2009) · Zbl 1167.49001
[20] C. T. Kelley, <em>Iterative Methods for Optimization</em>,, Frontiers in Applied Mathematics, (1999) · Zbl 0934.90082
[21] J. Nocedal, <em>Numerical Optimization</em>,, 2nd edition, (2006) · Zbl 1104.65059
[22] R. Ramlau, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data,, J. Comput. Phys., 221, 539, (2007) · Zbl 1114.68077
[23] R. Ramlau, Regularization of ill-posed Mumford-Shah models with perimeter penalization,, Inverse Problems, 26, (2010) · Zbl 1226.47105
[24] J. Simon, Differentiation with respect to the domain in boundary value problems,, Numer. Funct. Anal. Optim., 2, 649, (1980) · Zbl 0471.35077
[25] G. Wang, Uniqueness theorems in bioluminescence tomography,, Medical Physics 31 (2004), 31, 2289, (2004)
[26] R. Weissleder, Shedding light onto live molecular targets,, Nat. Med., 9, 123, (2003)
[27] H. Weyl, On the volume of tubes,, Amer. J. Math., 61, 461, (1939) · JFM 65.0796.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.