| rdfs:comment
| - The forward problem, discussed above, involves calculating simulated data y (which may be CW, time-domain or frequency-domain data), given a forward operator F and knowledge of the sources q and internal optical properties x (which may include µa and µ's). The forward problem may therefore be formulated as: . (3) To reconstruct an image, it is necessary to solve the inverse problem, which is to calculate the internal optical properties x, given data y and sources q (Arridge 1999): . (4) . (5) . (6) . (7)
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| abstract
| - The forward problem, discussed above, involves calculating simulated data y (which may be CW, time-domain or frequency-domain data), given a forward operator F and knowledge of the sources q and internal optical properties x (which may include µa and µ's). The forward problem may therefore be formulated as: . (3) To reconstruct an image, it is necessary to solve the inverse problem, which is to calculate the internal optical properties x, given data y and sources q (Arridge 1999): . (4) This is a non-linear problem but it can be linearised if the actual optical properties x are close to an initial estimate x0 and the measured data y is close to the simulated measurements y0. This is typically the case in difference imaging where measurements are taken before and after a small change in the optical properties. Then we can expand (3) about x0 in a Taylor series: . (5) where and are the first and second-order Fréchet derivatives of F, respectively. The Fréchet derivative is a linear integral operator mapping functions in the image space to functions in the data space. In some cases, such as in section 3.3.1, the kernel of the integral operator is known in terms of the Green's functions. This approach was taken by Boas et al. (1994). More generally, the forward problem can be solved by a numeric technique by representing the Fréchet derivatives and by matrices J (the Jacobian) and H (the Hessian), respectively. Equation 5 can then be linearised by neglecting higher order terms and considering changes in the optical properties and data to give the linear problem (6): . (6) Linearising the change in intensity in this way gives rise to the Born approximation; linearising the change in log intensity instead is the Rytov approximation and may lead to improved images by reducing the dynamic range of y. Either way, image reconstruction consists of the problem of inverting the matrix J, or some form of normalised J. This may be large, underdetermined and ill-posed but standard matrix inversion methods can be used. These techniques differ in the way in which the matrix inversion is regularised to suppress the effect of measurement noise and modelling errors. Perhaps the most common techniques are truncated singular value decomposition, Tikhonov regularisation, and the algebraic reconstruction technique (ART) (Gaudette et al. 2000). The Moore-Penrose inverse J-1=JT(J JT)-1 offers a more efficient inversion if J is underdetermined and leads naturally to a Tikhonov-type formulation (7) where I is the identity matrix and λ is a regularisation parameter. J itself is calculated from the forward model, most efficiently using the adjoint method of Arridge and Schweiger (1995). The linear inverse problem can then be expressed as: . (7)
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