| Abstract: |
| Industrial imaging often suffers from highly noised and undersampled data. As such, many state-of-the-art reconstruction algorithms are not applicable and practitioners rely on simple heuristic methods often paired with a simplified physical model. To improve the reconstruction quality, iterative methods are a favorable choice. One such method is the iterative graph Laplacian, which iteratively solves
$$x_n=\arg\min\limits_{x}\|Ax-b\|_2+\lambda\|\Delta_{x_{n-1}}x\|_1$$
where $\Delta_{x_{n-1}}$ is a graph Laplacian matrix based on the previous solution $x_{n-1}$. The initial guess $x_0$ is obtained using any chosen reconstruction algorithm. This method has proven to outperform other iterative approaches since the data adaptive graph Laplacian $\Delta_{x_{n-1}}$ imitates the structure of the given solution. It can differentiate between desired and undesired features within the data.
However, the above approach requires a decent starting guess $x_0$ with structural features close to the real solution. This is not always the case in industrial imaging. We present an updated version of the iterative graph Laplacian by adjusting the construction of $\Delta_{x_{n-1}}$. The new matrix is context-aware and responds to possible artifacts or anomalies in the given solution $x_{n-1}$. We use a null space projection to counter any undesired edges within the solution as well as a directional $\ell_1/\ell_2$ weight to detect blurred edges. We demonstrate the method on non-destructive testing and seismic exploration examples. |
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