Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation

Abstract The radiative transfer equation can be utilized in optical tomography in situations in which the more commonly applied diffusion approximation is not valid. In this paper, an image reconstruction method based on a frequency domain radiative transfer equation is developed. The approach is based on a total variation output regularized least squares method which is solved with a Gauss–Newton algorithm. The radiative transfer equation is numerically solved with a finite element method in which both the spatial and angular discretizations are implemented in piecewise linear bases. Furthermore, the streamline diffusion modification is utilized to improve the numerical stability. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the radiative transfer equation can be utilized in optical tomography and it can produce good quality images even in the presence of low-scattering regions.

[1]  S Arridge,et al.  Recovery of piecewise constant coefficients in optical diffusion tomography. , 2000, Optics express.

[2]  E. Aydin,et al.  A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method. , 2002, Medical physics.

[3]  M. Schweiger,et al.  Gauss–Newton method for image reconstruction in diffuse optical tomography , 2005, Physics in medicine and biology.

[4]  Guido Kanschat,et al.  Radiative transfer with finite elements. I. Basic method and tests , 2001 .

[5]  Jari P. Kaipio,et al.  Finite element model for the coupled radiative transfer equation and diffusion approximation , 2006 .

[6]  Stephen J. Wright,et al.  Reconstruction in optical tomography using the PN approximations , 2006 .

[7]  E D Aydin.  Three-dimensional photon migration through voidlike regions and channels. , 2007, Applied optics.

[8]  S R Arridge,et al.  An investigation of light transport through scattering bodies with non-scattering regions. , 1996, Physics in medicine and biology.

[9]  Guido Kanschat.  A Robust Finite Element Discretization for Radiative Transfer Problems with Scattering , 2007 .

[10]  S. Arridge.  Optical tomography in medical imaging , 1999 .

[11]  M. Schweiger,et al.  The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions. , 2000, Medical physics.

[12]  S. Arridge,et al.  Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions , 2005, Physics in medicine and biology.

[13]  Laurence S. Rothman,et al.  Journal of Quantitative Spectroscopy & Radiative Transfer 96 , 2005 .

[14]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[15]  G. C. Pomraning,et al.  Linear Transport Theory , 1967 .

[16]  Hyun Keol Kim,et al.  A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer , 2007 .

[17]  Simon R. Arridge,et al.  Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data , 2006 .

[18]  Oliver Dorn,et al.  A transport-backtransport method for optical tomography , 1998 .

[19]  S R Arridge,et al.  Application of temporal filters to time resolved data in optical tomography , 1999, Physics in medicine and biology.

[20]  L. C. Henyey,et al.  Diffuse radiation in the Galaxy , 1940 .

[21]  Jari P. Kaipio,et al.  A finite-element model of electron transport in radiation therapy and a related inverse problem , 1999 .

[22]  Akira Ishimaru,et al.  Wave propagation and scattering in random media , 1997 .

[23]  R. Fante,et al.  Reviews and abstracts - Wave propagation and scattering in random media - Vol.1 , 1978, IEEE Antennas and Propagation Society Newsletter.

[24]  Ville Kolehmainen,et al.  Hybrid radiative-transfer-diffusion model for optical tomography. , 2005, Applied optics.

[25]  Guillaume Bal,et al.  Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer , 2006, SIAM J. Sci. Comput..

[26]  Andreas H. Hielscher,et al.  Three-dimensional optical tomography with the equation of radiative transfer , 2000, J. Electronic Imaging.

[27]  A H Hielscher,et al.  Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer. , 1999, Medical physics.

[28]  K. D. Lathrop.  Remedies for Ray Effects , 1971 .

[29]  Alexander D. Klose,et al.  Optical tomography with the equation of radiative transfer , 2008 .

[30]  K D Paulsen,et al.  Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization. , 1996, Applied optics.

[31]  S. Arridge,et al.  Optical imaging in medicine: II. Modelling and reconstruction , 1997, Physics in medicine and biology.

[32]  Simon R. Arridge,et al.  Computational calibration method for optical tomography. , 2005 .

[33]  R. Alcouffe,et al.  Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues. , 1998, Physics in medicine and biology.

[34]  S. Jacques,et al.  Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media. , 1993, Journal of the Optical Society of America. A, Optics, image science, and vision.

[35]  Arridge,et al.  Optical tomography in the presence of void regions , 2000, Journal of the Optical Society of America. A, Optics, image science, and vision.

[36]  Alvin M. Weinberg.  Linear transport theory: by K. M. Case and P. W. Zweifel. 342 pages, diagrams, illustr. 6 × 9 in. Reading, Mass. Addison-Wesley Publ. Co., 1967. Price, $17.50 , 1968 .

[37]  M. Schweiger,et al.  Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head. , 1997, Applied optics.

[38]  K. D. Lathrop.  RAY EFFECTS IN DISCRETE ORDINATES EQUATIONS. , 1968 .

[39]  M Vauhkonen,et al.  Modelling the transport of ionizing radiation using the finite element method , 2005, Physics in medicine and biology.