Solution of the nonlinear inverse scattering problem by $T$-matrix completion. II. Simulations

This is Part II of the paper series on data-compatible $T$-matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043317 (2016)] contains theory and here we present simulations for inverse scattering of scalar waves. The underlying mathematical model is the scalar wave equation and the object function that is reconstructed is the medium susceptibility. The simulations are relevant to ultrasound tomographic imaging and seismic tomography. It is shown that DCTMC is a viable method for solving strongly nonlinear inverse problems with large data sets. It provides not only the overall shape of the object, but the quantitative contrast, which can correspond, for instance, to the variable speed of sound in the imaged medium.

Figure 1

The shapes of all three targets and the color scheme used in all figures below. Even though all model shape functions satisfy (by definition) the condition $0\le \mathrm{\Theta }\le 1$, the color scale can be used to represent any quantity in the range $[-0.3,1.3]$. The allowances are made because the reconstructed values of ${\chi }_n/{\chi }_0$ can become negative or larger than unity. Reconstructed values that are smaller than $-0.3$ will be shown by the uniform black color and values that are larger than 1.3 will be shown as white.

Figure 2

Linear (top and middle rows, marked FB and FR) and nonlinear (bottom row, marked NL) reconstructions of the small target for different levels of contrast ${\chi }_0$. The quantity shown by the color scale in each plot is ${\chi }_n/{\chi }_0$, where ${\chi }_n$ is the reconstructed susceptibility of the $n\mathrm{th}$ voxel (real under the assumptions used) and ${\chi }_0$ is the amplitude of the shape function. For the linearized reconstructions, FB denotes first Born approximation and FR denotes first Rytov approximation.

Figure 3

Convergence data for the small target. Errors ${\eta }_{\chi }$ and ${\eta }_{\mathrm{\Phi }}$ are plotted vs the iteration number $i$. Different line types correspond to different contrast ${\chi }_0$ as follows: ${\chi }_0=0.00175$ (1), ${\chi }_0=0.0175$ (2), ${\chi }_0=0.175$ (3), ${\chi }_0=0.875$ (4), and ${\chi }_0=1.75$ (5).

Figure 4

Same as in Fig. 2, but for the large target and a somewhat different set of contrasts ${\chi }_0$. Utilization of first Rytov approximation for linearized reconstruction does not provide any improvements over first Born approximation, and the corresponding results are not shown in this figure.

Figure 5

Convergence data for the large target. The different line types correspond to different contrast ${\chi }_0$ as follows: ${\chi }_0=0.002$ (1), ${\chi }_0=0.02$ (2), ${\chi }_0=0.2$ (3), ${\chi }_0=1$ (4), and ${\chi }_0=2$ (5).

Figure 6

Two different initial guesses (DCTMC iteration starting points) for the small target with ${\chi }_0=0.175$. The left panel is the initial guess for the $T$ matrix, $T_1=T_{\mathrm{expt}}$; the image is then computed by $V_1=\mathcal{D}\left[{(I+T_1\mathrm{\Gamma })}^{-1}{T}_1\right]$. The right panel is $V_1=V_{\mathrm{lin}}$, where the linearized reconstruction $V_{\mathrm{lin}}$ was obtained by using the first Born approximation.

Figure 7

Convergence rate for the small target with ${\chi }_0=0.175$. Different curves correspond to using the initial guess shown in Fig. 6, as labeled.

Figure 8

Convergence data for the small target with ${\chi }_0=0.0175$ comparing the original DCTMC algorithm and the one using reciprocity of sources and detectors.

Figure 9

DCTMC reconstruction of the small target after 150 iterations with varying degrees of $R$, where the row summing is done over voxels separated by no more than $R$.

Figure 10

Convergence data ${\eta }_{\chi }$ for the reconstruction shown in Fig. 9.

Figure 11

Reconstructions of the small target by the algorithm combining all three improvements discussed in this section after 75 iterations for the five levels of contrast previously considered.

Figure 12

Convergence data of ${\eta }_{\chi }$ for the algorithm combining all improvements discussed in this section. The horizontal lines show the respective error attained after 900 iterations of DCTMC without any of the improvements. Different line types correspond to different contrast ${\chi }_0$ as follows: ${\chi }_0=0.00175$ (1), ${\chi }_0=0.0175$ (2), ${\chi }_0=0.175$ (3), ${\chi }_0=0.875$ (4), and ${\chi }_0=1.75$ (5).

Figure 13

A typical Gauss-Newton (with Levenburg-Marquardt damping technique) reconstruction of the small target with ${\alpha }_0=0.00175$.

Figure 14

Convergence data for the Gauss-Newton (with Levenberg-Marquardt damping technique) reconstruction shown in Fig. 13. Note that for $i>5$, the error ${\eta }_{\mathrm{\Phi }}\left(i\right)$ continues to decrease, as is guaranteed by the algorithm, although this decrease is not visible due to the logarithmic vertical scale used in the figure.

Figure 15

Gauss-Newton (with Levenburg-Marquardt damping technique) reconstructions of the tiny target for different levels of contrast as labeled.