# Solution of the Perturbation Equation in Optical Tomography Using Weight
Functions as a Transform Basis

### Erh-Ya Lin *et al.* (1998)

#### Summary

This paper describes a new inverse solver for optical tomography. As with
prior studies, we employ an iterative perturbation approach, which at each
iteration requires the solution of a forward problem and an inverse problem.
The inverse problem involves the solution of a linear perturbation equation,
which is often severely underdetermined. To overcome this problem, we propose
to represent the unknown image of optical properties by a set of linearly
independent basis functions, with the number of basis functions being equal
to or less than the number of independent detector readings. The accuracy of the
solution depends on the choice of the basis. We have explored the use of the
weight functions associated with different source and detector pairs (i.e. the
rows in the weight matrix of the perturbation equation) as the basis functions.
By choosing those source and detector pairs which have uncorrelated weight functions,
the inverse problem is transformed into a well-posed, uniquely determined problem.
The system matrix in the transformed representation has a dimension significantly
smaller than the original matrix, so that it is feasible to perform the inversion
using singular value decomposition (SVD). This new method has been integrated with
a previously reported forward solver, and applied to data generated from numerical
simulations using diffusion approximation. Compared to the Conjugate Gradient
Descent (CGD) method used in previously reported studies, the new
method takes substantially less computation time, while providing equal, if not better,
image reconstruction quality at similar noise levels.