Functional-analytical reconstruction of high contrast inhomogeneities

Abstract

In practice of acoustic tomography, for example, in medical applications and ocean tomography, the relative deviation of sound speed from its background value usually does not exceed 10-30%. At the same time, in electromagnetic applications, the equivalent contrasts can be noticeably higher than 60%. Since the inverse electromagnetic problem can be reduced in some approximation to Helmholtz equation, a formal comparison of reconstruction results obtained for different "acoustic" contrast and corresponding "dielectric" contrast is possible. In this work examples of such reconstructions are presented, which were obtained by using the functional-analytical algorithm described in works of R.G. Novikov. Previously, the advantages of this algorithm for solving practical problems of acoustic tomography were demonstrated. Results obtained in the present work show that functional-analytical algorithm can also be applied to reconstructing inhomogeneities with high "dielectric" contrast. Moreover, the functional algorithm also perfectly reconstructs very small "dielectric" contrast, recovering of which can be difficult for other approaches due to weak backscattering.

Figures (5)

• Figure 1: Tomography area $V_S$ contains the scattering region $\mathfrak{R}$; quasi-point sources and receivers are located on the boundary $S$.
• Figure 2: The case $c \ll c_0$, $\epsilon \gg 1$. General view of true scatterer $v(\mathbf{r})$ (a) of cylindrical shape with relative contrast of sound speed $\Delta c / c_0 \approx - 0.67$ and corresponding "dielectric" contrast $\epsilon = 9$, which provides the additional phase shift $\Delta \psi \approx 0.94 \pi$ and norm of scattering data $\| f(\phi, \phi^\prime) \| \approx 8.6 / (3 \pi)$, relative size of inhomogeneity is $2 R_0 / \lambda_0 \approx 0.25$; - space distribution of reconstruction results $\hat{v}(\mathbf{r})$ obtained by using the functional-analytical algorithm (b); - central cross sections of true scatterer $v(x, y = 0)$ (c, red line) and of estimate $\hat{v}(x, y~=~0)$ (c, blue line); - central cross sections of functions which space spectrums are filtered in a cylinder with radius $2k_0$ (arguments of functions are omitted): filtered true scatterer $v^{\text{filtered}}$ (d, red line), functional-analytical estimate $\hat{v}^{\text{filtered}}$ (d, blue line) and Born approximation result $\hat{v}^{\text{filtered}}_{\text{born}}$ (d, back line).
• Figure 3: The case $c \gg c_0$, $\epsilon \ll 1$. Space distribution of true scatterer $v(\mathbf{r})$ (a) with relative contrast of sound speed $\Delta c / c_0 = 39$ and corresponding "dielectric" contrast $\epsilon \approx 6 \cdot 10^{-4}$, which provides the additional phase shift $\Delta \psi \approx 1.3 \pi$ and norm of scattering data $\| f(\phi, \phi^\prime) \| \approx 7.7 / (3 \pi)$, relative size of scatterer is $2 R_0 / \lambda_0 \approx 0.67$; - general view of functional-analytical estimate $\hat{v}(\mathbf{r})$ (b); - central cross sections of true inhomogeneity $v(x, y = 0)$ (c, red line) and reconstruction result $\hat{v}(x, y~=~0)$ (c, blue line); - central cross sections ($y = 0$) of space filtering results obtained in a cylinder with radius $2k_0$ (arguments of functions are omitted): filtered true scatterer $v^{\text{filtered}}$ (d, red line), functional-analytical estimate $\hat{v}^{\text{filtered}}$ (d, blue line) and Born approximation result $\hat{v}^{\text{filtered}}_{\text{born}}$ (d, back line).
• Figure 4: Space distribution of true scatterer $v(\mathbf{r})$ (a) with relative contrast of sound speed $\Delta c / c_0 \approx 1.24$ and corresponding "dielectric" contrast $\epsilon = 0.2$, which provides the additional phase shift $\Delta \psi \approx 3.8 \pi$ and norm of scattering data $\| f(\phi, \phi^\prime) \| \approx 18.3 / (3 \pi)$, relative size of inhomogeneity is $2 R_0 / \lambda_0 \approx 3.4$; - general view of functional-analytical estimate $\hat{v}(\mathbf{r})$ (b); - central cross sections of true inhomogeneity $v(x, y = 0)$ (c, red line) and reconstruction result $\hat{v}(x, y~=~0)$ (c, blue line); - central cross sections ($y = 0$) of space filtering results obtained in a cylinder with radius $2k_0$ (arguments of functions are omitted): filtered true scatterer $v^{\text{filtered}}$ (d, red line), functional-analytical estimate $\hat{v}^{\text{filtered}}$ (d, blue line) and Born approximation result $\hat{v}^{\text{filtered}}_{\text{born}}$ (d, back line).
• Figure 5: Central cross sections of true inhomogeneity $v_\text{fig2}(x, y = 0)$ (a, red line) and reconstruction result $\hat{v}^{\text{noise}}_\text{fig2}(x, y~=~0)$ (a, blue line) obtained by using noisy data with rms amplitude deviation $\sigma_{\text{ns}} = 0.5 \bar{G}_{\text{sc}}$ (reconstruction of this scatterer with noise-free data is shown in figure \ref{['figure_2']}); - central cross sections of absolute value of true inhomogeneity space spectrum $| \tilde{v}_\text{fig2}(k_x, k_y = 0) |$ normalized on its maximum value (b, red line) and corresponding normalized value of space spectrum modulus of reconstruction result $| \tilde{v}^{\text{noise}}_{\text{fig2}}(k_x, k_y = 0) |$ (b, blue line), arguments of functions are omitted; - (c), (d) shows the same as on (a), (b), respectively, but for scatterer function $v_\text{fig4}(\mathbf{r})$ and noisy data with rms amplitude deviation $\sigma_{\text{ns}} = 0.0015 \bar{G}_{\text{sc}}$ (reconstruction of this inhomogeneity with noise-free data is shown in figure \ref{['figure_4']}).