With the dominance of digital imaging systems, we are often dealing with discrete-domain samples of an analog image. Due to physical limitations, all imaging devices apply a blurring kernel on the input image before taking samples to form the output pixels. In this paper, we focus on the reconstruction of binary shape images from few blurred samples. This problem has applications in medical imaging, shape processing, and image segmentation. Our method relies on representing the analog shape image in a discrete grid much finer than the sampling grid. We formulate the problem as the recovery of a rank r matrix that is formed by a Hankel structure on the pixels. We further propose efficient ADMM-based algorithms to recover the low-rank matrix in both noiseless and noisy settings. We also analytically investigate the number of required samples for successful recovery in the noiseless case. For this purpose, we study the problem in the random sampling framework, and show that with O(r log4(n1n2)) random samples (where the size of the image is assumed to be n1 x n2) we can guarantee the perfect reconstruction with high probability under mild conditions. We further prove the robustness of the proposed recovery in the noisy setting by showing that the reconstruction error in the noisy case is bounded when the input noise is bounded. Simulation results confirm that our proposed method outperform the conventional total variation minimization in the noiseless settings.

中文翻译：

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通过 Hankel 结构的低阶矩阵恢复从模糊图像中重建二值形状。


随着数字成像系统的主导地位，我们经常处理模拟图像的离散域样本。由于物理限制，所有成像设备在采样形成输出像素之前都会在输入图像上应用模糊内核。在本文中，我们专注于从少量模糊样本中重建二值形状图像。这个问题在医学成像、形状处理和图像分割中都有应用。我们的方法依赖于在比采样网格更精细的离散网格中表示模拟形状图像。我们将问题表述为恢复由像素上的汉克尔结构形成的秩 r 矩阵。我们进一步提出了基于 ADMM 的高效算法来恢复无噪声和噪声设置中的低秩矩阵。我们还分析研究了在无噪声情况下成功恢复所需的样本数量。为此，我们研究了随机采样框架中的问题，并表明使用 O(r log4(n1n2)) 随机样本（假设图像的大小为 n1 x n2），我们可以保证完美的重建温和条件下概率较高。我们进一步证明了所提出的恢复在噪声环境中的鲁棒性，表明当输入噪声有界时，噪声情况下的重建误差是有界的。仿真结果证实，我们提出的方法在无噪声设置中优于传统的总变差最小化。