Complex Numbers [Lecture notes] by Peter M. Neumann

By Peter M. Neumann

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Conservative formulations of general relativistic kinetic theory. Phys. Rev. D, 68, 023006 1–26 (2003) A Finite Element Method for the Even-Parity Radiative Transfer Equation Using the PN Approximation Stephen Wright, Simon Arridge, and Martin Schweiger Department of Computer Science, University College London, Gower Street, London, WC1E 6BT 1 Introduction The concept of using optical radiation to penetrate highly scattering media, combined with image reconstruction methods to recover optical parameters inside the media, has been a recurrent idea for over a century.

K} are a set of basis functions for χh+ . Let us assume that the basis is developed separately for the spatial and angular terms fk (r, ˆ s) = ui (r)θj (ˆ s) i = 1, . . D ; j = 1, . . S ; K = D × S (16) 42 Stephen Wright et al. Since we require even parity for φh+ it is natural to choose θj to have even parity. The space χh+ is equipped with a norm ψ(r, ˆ s)φ(r, ˆ s) dr dˆ s ψ, φ := S2 (17) Ω Since φh+ is an approximation to φ+ it does not satisfy (14) exactly, but rather s · ∇ (Dˆ s · ∇) φh+ (r, ˆ s; ω) − q h+ (r, ˆ s; ω) + η h+ (r, ˆ s; ω) = e(r, ˆ s; ω) C+ − ˆ (18) s · ∇ (Dq − (r, ˆ s; ω)) where q h+ and η h+ are the projection into χh+ of q + and ˆ respectively.

A. H. L. A. DiMarzio, M. J. Gaudette, and Q. Zhang. Imaging the body with diffuse optical tomography. IEEE Sig. Proc. Magazine, 18(6):57–75, 2001. 6. M. R. Arridge, M. T. Delpy. An investigation of light transport through scattering bodies with non-scattering regions. Phys. Med. , 41:767–783, 1996. 7. O. Dorn. A transport-backtransport method for optical tomography. Inverse Problems, 14(5):1107–1130, 1998. 8. H. E. L. Barbour. Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissue.

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