Solution of the nonlinear inverse scattering problem by $T$-matrix completion. I. Theory

We propose a conceptually different method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology, and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential $V$ from external scattering data. Although we seek a local (diagonally dominated) $V$ as the solution to the posed problem, we allow $V$ to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between $V$ and the $T$ matrix of the problem. Here it is important to realize that not every $T$ corresponds to a diagonal $V$ and we, therefore, relax the usual condition of strict diagonality (locality) of $V$. An iterative algorithm is proposed in which we seek $T$ that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential $V$ that is as diagonally dominated as possible. We refer to this algorithm as to the data-compatible $T$-matrix completion. This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043318 (2016)]. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.

Figure 1

Illustration of the imaging geometry. The symbols $A, B, \mathrm{\Gamma }$, and $V$ in the rectangular frames denote the matrices obtained by restricting and sampling the kernels $G_0(\mathbf{r},{\mathbf{r}}^{\prime })$ and $V(\mathbf{r},{\mathbf{r}}^{\prime })$. The scattering diagram corresponds to the second-order term $G_{0}V{G}_{0}V{G}_0$ in the formal power-series expansion of the left-hand side in (4). Note that in the local limit $V(\mathbf{r},{\mathbf{r}}^{\prime })=D\left(\mathbf{r}\right)\delta (\mathbf{r}-{\mathbf{r}}^{\prime })$, the two arrows contract to two vertexes at ${\mathbf{r}}_1={\mathbf{r}}_1^{\prime }$ and ${\mathbf{r}}_2={\mathbf{r}}_2^{\prime }$.

Figure 2

Block diagram of Eq. (9) with sizes of all matrices indicated. Here, $N_d$ and $N_s$ are the numbers of detectors and sources and $N_v$ is the number of voxels.

Figure 3

Left panel: elements of $\tilde{T}$ inside the shaded block can be computed from the data by using (17). Elements outside of the shaded block are not known and can not be in any way inferred from the data. Right panel: the initial guess for the $T$ matrix, $T_{\mathrm{expt}}$. In this initial guess, we set the unknown elements of $\tilde{T}$ (in singular-vector representation) to zero.

Figure 4

Assuming that the singular values ${\sigma }_{\mu }^A$ and ${\sigma }_{\mu }^B$ are arranged in the descending order, this sketch shows an example of a more general shape (compared to Fig. 3) of the region in which the elements of $\tilde{T}$ are known. The numbers above the thick line satisfy ${\sigma }_{\mu }^A{\sigma }_{\nu }^B>{\epsilon }^2$. In the general case, the boundary line can only go from left to right and from bottom to top if followed from the leftmost boundary of the matrix.

Figure 5

Basic flowchart of the DCTMC iteration process for the case when the iterations start with an initial guess for $\tilde{T}$. Numbered iteration steps are defined in Sec. 4. Computational shortcuts are described in Sec. 5. Matrix representations are abbreviated by SV (singular vector) and RS (real space). Exit condition can be checked at various steps of the iterations (see [28] for more detail).

Figure 6

Schematics of computing $\mathcal{N}[\mathcal{R}\left[T\right]]$. Matrices $P_A$ and $P_B$ are obtained from $R_A$ and $R_B$ by setting all columns to zero except for the first $M_A$ and $M_B$ columns, respectively.