Reconstruction of the scattering coefficient is typically more challenging than inverting for µa because the image depends on µa to first order, whereas this is not the case for µs. A consequence of this, as observed in Chapter 5, is that the functional gradient with respect to scattering is significantly smaller than the gradient with respect to absorption.
This also has the effect that the error functional is much more sensitive to MC noise in H(i) which impacts the accuracy and efficiency of the linesearch. Recalling the Wolfe conditions, which require a sufficient decrease in the error function for a step length α(i) and that the gradient at this step length is less than that at the current position, in the presence of noise in the error functional many values of α(i)would have to be tested to satisfy simultaneously the two Wolfe conditions.
The inversion was tested in the same domain in which the gradients were validated in Section 5.3.4.2. The distribution of µs is reproduced in Fig. 6.5(a) and the true absorption coefficient was set to a homogeneous value of 0.03mm-1. The data was simulated using 108photons, and simulation of the forward and adjoint radiance fields was also achieved using 108 photons. The GD optimisation was once again used. It was found that the optimisation spent several hours within the linesearch on the first iteration testing candidate values of α(i). Recalling that inversions for the absorption coefficient completed one iteration every 20 minutes on average, this indicates that the HZ-linesearch performs highly inefficiently in the presence of noise in the error functional and the small magnitude of the functional gradient. Since requiring that both Wolfe conditions are satisfied negatively impacts the performance of the optimisation, it may be worthwhile considering a backtracking linesearch [100], which requires only the sufficient decrease condition to be satisfied for a given α(i). The backtracking linesearch starts with a large candidate step length; a suitable choice is α(i,j=0) = 1/ max(∇(i)).
The sufficient decrease condition (Eq. (6.5)) is then tested for α(i,j)τ whilst iterating over j, with τ ∈ [0, 1].
The gradient descent optimisation using the backtracking linesearch was used to invert the image for the scattering coefficient shown in Fig. 6.5(a). The value of ν1 and τ were set to 0.2 and 0.5 as this yielded positive results. The true absorption coefficient
was homogeneous and was known when inverting for the scattering coefficient. The measured data was computed using a line source across all x positions at z=0mm. The forward and adjoint RMC simulations used 107photons and 10 Fourier harmonics. The initial step, α(1), was chosen to be 1012 because this produced an update that would make a substantial change to the scattering coefficient. Subsequent steps were obtained using the backtracking linesearch with a starting α of 1012, iteratively decreased by an order of magnitude. However, to prevent the linesearch from iterating for an extended period of time and producing an update that was negligble in magnitude, a minimum step length was set to 106. Note that positivity of the scattering coefficient was enforced by setting all values of µs < 0 to zero, which is consistent with the definition of the scattering coefficient. The optimisation was run on a Dell 2U R820 32-core server on Legion and took about 13.6 hours. The value of the error functional is plotted as a function of iteration number in Fig. 6.5(c) and the final estimate of the scattering coefficient is shown in Fig. 6.5(b), with profiles through the true and final estimate of the scattering coefficent shown in Fig. 6.5(d) at x=2mm for all z positions, indicated by the vertical black and grey lines in (a) and (b).
x [mm]
FI G U R E 6 . 5 : (a) True scattering coefficient with a background absorption coefficient of 5mm-1and inclusions where the scattering coefficient is equal to 10mm-1and 15mm-1; (b) Estimate of scattering coefficient estimated with known absorption coefficient and GD optimisation using 107photons; (c) Value of error functional plotted as a function of iteration during GD optimisation for estimating µs; (d) Profiles through the true and reconstructed scattering coefficient at x=2mm for all z-positions (positions shown by
the black and grey lines in (a) and (b)).
From the estimate in Fig. 6.5(b), it is clear that edges are not well reconstructed;
this is in part due to the diffusive nature of scattering, but may also be due to early termination of the optimisation. Reconstruction of the scattering coefficient may benefit from some regularisation, which has been demonstrated in the literature [107, 109, 113, 188] to improve the quality of the scattering estimate. Specifically, given the
piecewise constant distribution of µa and µs used here, total variation regularisation (TVR) [189] may help stabilise the solution. TVR may also be able to suppress the overestimation of µson the left- and right-hand sides of the square inclusion as well as prevent the underestimation of µs in the deeper region, behind the inclusion, through TVR’s edge preserving properties. Nevertheless, the overall accruacy of the estimate of µs in the inclusion was quite good and improved with each iteration as demonstrated by Fig. 6.6 which shows a plot of the maximum and average estimate of the scattering coefficient in the inclusion converging to their respective true values. Thus, perhaps with regularisation and if the optimisation were allowed to iterate for longer, a more accurate reconstruction would have been obtained.
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15
Iteration µ s [mm−1 ]
Maximum estimate of µs in inclusion Average estimate of µs in inclusion Global average estimate of µs Global average of true µs
FI G U R E 6 . 6 : Plot of maximum and average estimate of µs in the inclusion in Fig.
6.5(a) as a function of iteration, and global estimate of µs as a function of iteration.
Plots obtained from GD optimisation used to reconstruct µsusing 107photons in the forward and adjoint simulations. The true value of µsin the inclusion was 15mm-1and
the true global average of µs, 5.63mm-1, is plotted in black.
The limited accuracy of the inversions for the scattering coefficient may compromise
the accuracy of reconstruction of the absorption coefficient when these are estimated