Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a constrained least-squares problem. As the constraints are nonconvex, many approaches for solving it rely on either relaxing the constraints or heuristics. In this paper, we propose a novel convex formulation, based on the Lagrange dual of the constrained least-squares problem. The resulting problem is a generalized least absolute shrinkage and selection operator problem, which can be solved efficiently. It is a relaxation in the sense that it can only be guaranteed to give a feasible solution, not necessarily the optimal one. In exhaustive experiments on small images ($2\times 2$, $3\times 3$, $4\times 4$), we find, however, that if the problem has a unique solution, our dual approach finds it. In the case of multiple solutions, our approach finds the commonalities between the solutions. Further experiments on realistic numerical phantoms and an experiment on the X-ray dataset show that our method compares favorably to Total Variation and DART.

中文翻译：

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二元断层扫描的凸公式

二值断层扫描涉及从它们的几个投影（即沿各个方向的像素值的总和）中恢复二值图像。要从嘈杂的投影数据重建图像，可以将其视为受约束的最小二乘问题。由于约束是非凸的，许多解决它的方法依赖于放松约束或启发式。在本文中，我们基于约束最小二乘问题的拉格朗日对偶提出了一种新的凸公式。由此产生的问题是一个广义的最小绝对收缩和选择算子问题，可以有效解决。从某种意义上说，这是一种放松，只能保证给出可行的解决方案，而不一定是最优的解决方案。在对小图像的详尽实验中（$2\乘以 2$, $3\times 3$, $4\乘以4$)，然而，我们发现如果问题有一个唯一的解决方案，我们的双重方法就会找到它。在多个解决方案的情况下，我们的方法会找到解决方案之间的共性。对真实数值模型的进一步实验和对 X 射线数据集的实验表明，我们的方法与 Total Variation 和 DART 相比具有优势。