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结构的低秩矩阵恢复从模糊图像中重建二进制形状。

在数字成像系统的主导下，我们经常处理模拟图像的离散域样本。由于物理限制，所有成像设备在对样本进行采样以形成输出像素之前，会对输入图像应用模糊核。在本文中，我们着重于从少量模糊样本中重建二值形状图像。此问题已应用于医学成像，形状处理和图像分割。我们的方法依赖于在比采样网格精细得多的离散网格中表示模拟形状图像。我们将问题表述为由汉克尔结构在像素上形成的秩r矩阵的恢复。我们进一步提出了有效的基于ADMM的算法，以在无噪声和高噪声设置下恢复低秩矩阵。我们还分析性地研究了在无噪声情况下成功回收所需样品的数量。为此，我们在随机采样框架中研究该问题，并表明，使用O（r log4（n1n2））个随机样本（假设图像大小为n1 x n2），我们可以保证使用在温和条件下的可能性很高。我们通过证明当输入噪声为有界时有噪情况下的重构误差为有界，从而进一步证明了在有噪环境下所提出的恢复的鲁棒性。仿真结果证实了我们提出的方法在无噪声设置下优于常规的总变化最小化。并表明，使用O（r log4（n1n2））个随机样本（假设图像大小为n1 x n2），我们可以保证在温和条件下以高概率进行完美重建。我们通过证明当输入噪声为有界时有噪情况下的重构误差为有界，从而进一步证明了在有噪环境下所提出的恢复的鲁棒性。仿真结果证实了我们提出的方法在无噪声设置下优于常规的总变化最小化。并表明，使用O（r log4（n1n2））个随机样本（假设图像大小为n1 x n2），我们可以保证在温和条件下以高概率进行完美重建。我们通过证明当输入噪声为有界时有噪情况下的重构误差为有界，从而进一步证明了在有噪环境下所提出的恢复的鲁棒性。仿真结果证实了我们提出的方法在无噪声设置下优于常规的总变化最小化。