Ongie Greg, Jacob Mathews
Department of Mathematics, University of Iowa, Iowa City, Iowa.
Department of Electrical and Computer Engineering, University of Iowa, Iowa City, Iowa.
SIAM J Imaging Sci. 2016;9(3):1004-1041. doi: 10.1137/15M1042280. Epub 2016 Jul 21.
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.
我们介绍了一种从少量低通傅里叶样本中恢复分段常数二维图像连续域表示的方法。假设图像的边缘集位于三角多项式的零集上，我们证明图像偏导数的傅里叶系数满足线性消去关系。我们给出了使用消去关系从有限低通傅里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里叶域中进行外推，从而恢复基础图像的精确傅里叶系数。我们在从低通傅里叶样本进行MRI体模和真实MRI数据的超分辨率恢复上演示了我们的算法，这显示出优于单图像超分辨率MRI的标准方法的优势。
