In this paper we study the performance of image reconstruction methods from incomplete samples of the 2D discrete Fourier transform. Inspired by requirements in parallel MRI, we focus on a special sampling pattern with a small number of acquired rows of the Fourier transformed image. We show the importance of the low-pass set of acquired rows around zero in the Fourier space for image reconstruction. A suitable choice of the width $L$ of this index set depends on the image data and is crucial to achieve optimal reconstruction results. We prove that non-adaptive reconstruction approaches cannot lead to satisfying recovery results. We propose a new hybrid algorithm which connects the TV minimization technique based on primal-dual optimization with a recovery algorithm which exploits properties of the special sampling pattern for reconstruction. Our method shows very good performance for natural images as well as for cartoon-like images for a data reduction rate up to 8 in the complex setting and even 16 for real images.

翻译：本文研究了从二维离散傅里叶变换的不完全样本进行图像重建的方法性能。受并行磁共振成像需求的启发，我们专注于一种特殊的采样模式，该模式仅获取傅里叶变换后图像的少量行数据。我们证明了在傅里叶空间中，围绕零点获取的低频行索引集对于图像重建的重要性。该索引集宽度$L$的合适选择取决于图像数据，并且对实现最优重建结果至关重要。我们证明了非自适应重建方法无法获得令人满意的恢复结果。我们提出了一种新的混合算法，该算法将基于原始-对偶优化的全变分最小化技术与一种利用特殊采样模式特性进行重建的恢复算法相结合。我们的方法在复数设定下对自然图像和卡通类图像在数据压缩率高达8倍时表现出色，对于实值图像甚至可达16倍。