Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples.
Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples.
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DOI：
10.1137/15m1042280
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发表时间：
2016
影响因子：
2.1
通讯作者：
Jacob M
中科院分区：
文献类型：
--
作者：
Ongie G;Jacob M
介绍了一种从少量低通傅里叶样本中恢复二维图像连续域表示的方法。假设图像的边缘集定位于三角多项式的零集，我们证明了图像的偏导数的傅里叶系数满足线性湮灭关系。给出了利用湮灭关系从有限低通傅立叶样本中唯一恢复图像的充分必要条件。我们还提出了一种实用的两阶段恢复算法，该算法对模型失配和噪声具有鲁棒性。在第一阶段，我们估计图像边缘集的连续域表示。在第二阶段，我们通过最小二乘二维线性预测在傅里叶域中进行外推，该外推恢复了底层图像的精确傅里叶系数。我们在低通傅立叶样本的MRI幻象和真实MRI数据的超分辨率恢复上展示了我们的算法，这显示了单图像超分辨率MRI的标准方法的优势。
We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm which is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.