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.

中文翻译：

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从少量傅立叶样本中逐步恢复分段常量图像的网格。

我们介绍了一种从几个低通傅立叶样本中恢复分段恒定二维图像的连续域表示的方法。假设图像的边缘集定位于三角多项式的零集，则表明图像的偏导数的傅立叶系数满足线性an灭关系。我们提出了必要和充分的条件，使用using灭关系从有限的低通傅立叶样本中唯一恢复图像。我们还提出了一种实用的两阶段恢复算法，该算法对模型不匹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立叶域中进行外推，这将恢复基础图像的精确傅立叶系数。我们演示了关于从低通傅立叶样本中MRI体模和真实MRI数据的超分辨率恢复的算法，该算法显示出优于单图像超分辨率MRI的标准方法的优势。