August  2010, 4(3): 411-427. doi: 10.3934/ipi.2010.4.411

On forward and inverse models in fluorescence diffuse optical tomography

1. 

Institute for Mathematics and Scientific Computing, Karl-Franzens University Graz, Heinrichstraße 36, 8010 Graz, Austria

2. 

Institute of Medical Engineering, Graz University of Technology, Kronesgasse 5/II, 8010 Graz, Austria

3. 

Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

Received  December 2009 Revised  June 2010 Published  July 2010

This paper investigates forward and inverse problems in fluorescence optical tomography, with the aim to devise stable methods for the tomographic image reconstruction.
   We analyze solvability of a standard nonlinear forward model and two approximations by reduced models, which provide certain advantages for a theoretical as well as numerical treatment of the inverse problem. Important properties of the forward operators, that map the unknown fluorophore concentration on virtual measurements, are derived; in particular, the ill-posedness of the reconstruction problem is proved, and uniqueness issues are discussed.
   For the stable solution of the inverse problem, we consider Tikhonov-type regularization methods, and we prove that the forward operators have all the properties, that allow to apply standard regularization theory. We also investigate the applicability of nonlinear regularization methods, i.e., TV-regularization and a method of levelset-type, which are better suited for the reconstruction of localized or piecewise constant solutions.
   The theoretical results are supported by numerical tests, which demonstrate the viability of the reduced models for the treatment of the inverse problem, and the advantages of nonlinear regularization methods for reconstructing localized fluorophore distributions.
Citation: Herbert Egger, Manuel Freiberger, Matthias Schlottbom. On forward and inverse models in fluorescence diffuse optical tomography. Inverse Problems and Imaging, 2010, 4 (3) : 411-427. doi: 10.3934/ipi.2010.4.411
[1]

Meghdoot Mozumder, Tanja Tarvainen, Simon Arridge, Jari P. Kaipio, Cosimo D'Andrea, Ville Kolehmainen. Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography. Inverse Problems and Imaging, 2016, 10 (1) : 227-246. doi: 10.3934/ipi.2016.10.227

[2]

Shui-Nee Chow, Ke Yin, Hao-Min Zhou, Ali Behrooz. Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography. Inverse Problems and Imaging, 2014, 8 (1) : 79-102. doi: 10.3934/ipi.2014.8.79

[3]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[4]

Philipp Hungerländer, Barbara Kaltenbacher, Franz Rendl. Regularization of inverse problems via box constrained minimization. Inverse Problems and Imaging, 2020, 14 (3) : 437-461. doi: 10.3934/ipi.2020021

[5]

Yan Liu, Wuwei Ren, Habib Ammari. Robust reconstruction of fluorescence molecular tomography with an optimized illumination pattern. Inverse Problems and Imaging, 2020, 14 (3) : 535-568. doi: 10.3934/ipi.2020025

[6]

Abhishake Rastogi. Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4111-4126. doi: 10.3934/cpaa.2020183

[7]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems and Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074

[8]

Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems and Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051

[9]

Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems and Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421

[10]

Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107

[11]

Bruno Sixou, Cyril Mory. Kullback-Leibler residual and regularization for inverse problems with noisy data and noisy operator. Inverse Problems and Imaging, 2019, 13 (5) : 1113-1137. doi: 10.3934/ipi.2019050

[12]

Ye Zhang, Bernd Hofmann. Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 229-256. doi: 10.3934/ipi.2020062

[13]

Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems and Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395

[14]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems and Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271

[15]

I-Kun Chen, Daisuke Kawagoe. Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography. Inverse Problems and Imaging, 2019, 13 (2) : 337-351. doi: 10.3934/ipi.2019017

[16]

Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems and Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397

[17]

Raymond H. Chan, Haixia Liang, Suhua Wei, Mila Nikolova, Xue-Cheng Tai. High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Problems and Imaging, 2015, 9 (1) : 55-77. doi: 10.3934/ipi.2015.9.55

[18]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic and Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042

[19]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems and Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059

[20]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems and Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (154)
  • HTML views (0)
  • Cited by (7)

[Back to Top]