Multiwave tomography with reflectors: Landweber's iteration

Figure 12.  The unstable case, reconstructions with the Landweber method with noise. Left: unfiltered data. Right: filtered data.

Figure 1.  $\log_{10}$ error at each step vs. $\gamma$ with exact data. This is the graph of $\log_{10}\| K\|$ as a function of $\gamma$ for $\mu^2=1$, $\|\mathcal{L}\|^2=20$. The optimal $\gamma^*$ is $\gamma^*=2/21\approx0.0952$. The error increases fast on the right of $\gamma^*$.

Figure 3.  Error vs. $\gamma$ for the Landweber iterations in the stable example. Plotted are errors after $10$ iterations (boxes), after $30$ iterations (diamonds) and after $50$ ones (dots). Right: Error vs. the number of the iterations. The bottom curve with the circles are the ATR errors. The other curves correspond to $\gamma$ ranging from $0.1$ to $0.06$, as on the left, starting form the top to the bottom in the top left corner.

Figure 2.  The functions $g_N(\lambda)$ with $\gamma=1$ and $N=5, 20, 40, 80$. As the number of iterations $N$ increases, the maximum increases as $C_1\sqrt N$ and its location shifts to the left to $\lambda_N=C_2/\sqrt N$.

Figure 4.  A stable case. The smooth increasing curve represents the eigenvalues of $\mathcal{L}^*\mathcal{L}$ in the discrete realization on an $101\!\times\!101$ grid restricted to a $94\!\times\!94$ grid. There are small imaginary parts along the horizontal line in the middle. $\mathcal{L}^*$ is computed as the adjoint to the matrix representation of $\mathcal{L}$. The rough curve represents the squares of the Fourier coefficients of the SL phantom.

Figure 5.  The eigenfunctions of $L^*L$ corresponding to $\lambda_1$, $\lambda_{1,000}$ and $\lambda_{7,500}$

Figure 6.  Error plots in the stable case with data not in the range. Approximate convergence.

Figure 7.  Reconstruction and error plots in the stable case with noise with $200$ iterations; $\gamma=0.01,0.02,0.03,0.04$. A slow divergence. For $\gamma=0.05$ we get a fast divergence.

Figure 8.  Speed $1$ with $T=1.8$ with invisible singularities on the top. From left to right: (a) ATR with 50 steps, a cut near the border, 34\% error. (b) Landweber with a cut near the borders, error $31\%$; (c) Landweber without a cut near the borders, error $36\%$.

Figure 9.  Error vs. $\gamma$ in the unstable case. Left: errors after $10$ iterations (boxes), after $30$ iterations (diamonds) and after $50$ ones (dots). The vertical axis is on a $\log_{10}$ scale and the horizontal axis represents $\gamma$. The lower curve, corresponding to 50 iterations, is flatter than in the stable case. On the right, the curve with the diamond marks is the error with the ATR method.

Figure 10.  The unstable case. The smooth increasing curve represents the eigenvalues of $\mathcal{L}^*\mathcal{L}$ in the discrete realization on an $101\!\times\!101$ grid restricted to a $94\!\times\!94$ grid. $\mathcal{L}^*$ is computed as the adjoint matrix rather than with the wave equation solver. The rough curve represents the squares of the Fourier coefficients of the SL phantom. The arrows indicate the effective lower bound of the power spectrum of the SL phantom, and the first positive eigenvalue modulo small errors, respectively.

Figure 11.  A unstable case with Gaussian noise. Left: error curves with $\gamma$ ranging from $0.03$ to $0.17$ (the diverging curve). The critical value of $\gamma$ looks close to that in the zero noise case in Figure 9. The iterations for $\gamma\le 0.16$ diverge slowly in contrast with the noise free case in Figure 9. Right: Filtered data, $\gamma\le 0.15$. The errors look like in Figure 9.

Figure 13.  Reconstructions with a discontinuous speed, plotted on the left. The original is black and white disks on a gray background. Data on the marked part of the boundary, $T=4$. Center: ATR reconstruction, error $1\%$. Right: Landweber reconstruction, error $1.5\%$.

Figure 14.  Left: the original SL phantom. Right: The reconstructed SL phantom with $c=1$ and $T=2$. The phantom was originally on a $201\times 201$ grid with $\Delta t=\Delta x /\sqrt{2}$. The data was computed on a grid rescaled by a factor of $5.7$ in the spatial variables and $7.41$ times in the time variable; and then rescaled to the the original one on the boundary before inversion. The ripple artifacts are due to high frequency waves in the time reversal moving slower than the speed $c=1$.

Figure 15.  Same situation as in Figure 14 but the phantom contains Gaussians with predominately low frequency content. Left: original. Right: reconstruction with 50 iterations. The relative $L^2$ error is $1.8\%$, the $L^\infty$ error is $3.5\%$.