从少量傅立叶样本恢复分段平滑图像
Recovery of Piecewise Smooth Images from Few Fourier Samples
摘要：
我们引入了一个Prony样的方法来恢复连续域的2-D分段光滑图像从几个傅立叶样本。假设图像的不连续集被局部化到三角多项式的零水平集，我们证明了信号的偏导数的傅里叶变换系数满足湮灭关系。我们提出的必要条件和充分条件，唯一恢复分段常数图像使用上述湮没关系。我们提出的恢复的傅立叶系数的信号从测量作为一个凸矩阵完成算法，它依赖于提升的傅立叶数据的结构化低秩矩阵，这种方法联合估计的信号和零化滤波器。最后，我们证明了我们的算法从几个低分辨率傅立叶样本的MRI幻影的恢复。
英文摘要：
We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples. Assuming the discontinuity set of the image is localized to the zero level-set of a trigonometric polynomial, we show the Fourier transform coefficients of partial derivatives of the signal satisfy an annihilation relation. We present necessary and sufficient conditions for unique recovery of piecewise constant images using the above annihilation relation. We pose the recovery of the Fourier coefficients of the signal from the measurements as a convex matrix completion algorithm, which relies on the lifting of the Fourier data to a structured low-rank matrix; this approach jointly estimates the signal and the annihilating filter. Finally, we demonstrate our algorithm on the recovery of MRI phantoms from few low-resolution Fourier samples.
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