On the convergence of the Born series in optical tomography with diffuse light

Abstract

We provide a simple sufficient condition for convergence of Born series in the forward problem of optical diffusion tomography. The condition does not depend on the shape or spatial extent of the inhomogeneity but only on its amplitude.

Speckle contrast optical tomography: A new method for deep tissue three-dimensional tomography of blood flow

“…6(d) suggest to adopt an iterative Born inversion instead of the first Born approximation. As discussed above, the estimation of errors associated with the Born's approximation was previously carried out in the context of DOT [35] and was experimentally demonstrated [39]. A more accurate nonlinear-iterative reconstruction algorithm can be adopted to improve the accuracy of the reconstruction as was already presented for DCT and DOT using numerical methods such as FEM [13,30].…”

“…The sensitivity relation in Eq.11 can be employed for living tissue by setting V = 0 and (V 2 ) δ = 0 and considering D 0 B as the baseline flow with D δ B as the perturbation in flow from the baseline value. The estimation of errors associated with the Born's approximation in the context of DOT is addressed in [35] which can be used to estimate the errors for the results presented here, but it is beyond the scope of this paper. .…”

Convergence and stability of the inverse scattering series for diffuse waves

“…which converges if µ p η L p (Ba) < 1. When the Born series converges, we may estimate the remainder as follows: We note that L ∞ convergence of the Born series for diffuse waves has also been considered in [11]. Corresponding results for the L ∞ convergence of the Born series for acoustic waves have been described by Colton and Kress [4].…”

Diffuse optical tomography by simulated annealing via a spin Hamiltonian

“…When the perturbation δµ a is small, the Born series converges and u = lim n→∞ u n . In particular for nonnegative δµ a , a concise proof of the convergence is known under the condition 0 ≤ δµ a (x) ≤ μa for any x ∈ Ω [13]. Let us define the first and second Rytov approximations u R , u R2 as…”