Ongie Greg, Biswas Sampurna, Jacob Mathews
Department of EECS, University of Michigan, Ann Arbor, MI 48108 USA.
Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA, 52245 USA.
IEEE Trans Signal Process. 2018 Jan;66(1):236-250. doi: 10.1109/TSP.2017.2750111. Epub 2017 Sep 7.
We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero level-set of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low-rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.
我们考虑使用凸矩阵完备算法从非均匀傅里叶样本中恢复连续域分段常数图像。我们假设图像的不连续性/边缘位于一个带限函数的零水平集上。这一假设在图像的傅里叶系数之间引入了线性相关性，从而导致由傅里叶系数构成的两倍块Toeplitz矩阵是低秩的。所提出的算法将未知傅里叶系数的恢复重新表述为一个结构化低秩矩阵完备问题，其中在结构和数据约束下使矩阵的核范数最小化。我们表明，当图像的边缘集满足不相干性质时，以高概率实现精确恢复是可能的。我们还表明，不相干性质取决于边缘集曲线的几何形状，这意味着对于较小的曲线采样负担更大。本文将最近关于孤立狄拉克函数或具有有限创新率信号的超分辨率恢复的工作推广到分段常数图像的恢复。
