Convex recovery of continuous domain piecewise constant images from nonuniform Fourier samples.
Convex recovery of continuous domain piecewise constant images from nonuniform Fourier samples.
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DOI：
10.1109/tsp.2017.2750111
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发表时间：
2018-01
期刊：
影响因子：
--
通讯作者：
Jacob M
中科院分区：
文献类型：
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作者：
Ongie G;Biswas S;Jacob M
我们考虑用凸矩阵补全算法从其非均匀傅立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/边缘被定位到带宽限制函数的零电平集。该假设诱导图像的傅里叶系数之间的线性依赖关系，从而导致由傅里叶系数构建的双层块Toeplitz矩阵是低秩的。提出的算法将未知傅里叶系数的恢复重新表述为结构化低秩矩阵补全问题，其中矩阵的核范数在结构和数据约束下最小化。我们证明了当图像的边缘集满足非相干性时，精确恢复是可能的。我们还表明，非相干性取决于边缘集曲线的几何形状，这意味着较小曲线的采样负担更高。本文将近年来关于有限创新率孤立狄拉克信号的超分辨恢复的研究推广到分段常数图像的恢复。
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.