Ongie Greg, Biswas Sampurna, Jacob Mathews
Department of Mathematics, University of Iowa, IA, USA.
Department of Electrical and Computer Engineering, University of Iowa, IA, USA.
Proc Int Conf Image Proc. 2016 Sep;2016:963-967. doi: 10.1109/icip.2016.7532500. Epub 2016 Aug 19.
We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.
我们使用一种凸矩阵补全算法，从最少数量的非均匀傅里叶样本中，推导出了用于精确恢复分段常数二维图像的理论保证。我们假设图像的不连续性局限于一个带限函数的零水平集，这在傅里叶域中会引起某些线性相关性，使得由傅里叶数据构建的多重托普利兹矩阵已知是低秩的。恢复算法将已知的傅里叶样本排列成结构化矩阵，然后通过在结构和数据约束下最小化核范数来尝试恢复缺失的傅里叶数据。这项工作采用了Chen和Chi通过对类似多重汉克尔结构进行核范数最小化来恢复孤立狄拉克函数的结果。我们表明，当描述边缘集的带限函数满足不相干性质时，以高概率实现精确恢复是可能的。最后，我们展示了该算法在欠采样MRI数据恢复上的应用。
