大数据干涉成像的傅里叶降维模型
A Fourier dimensionality reduction model for big data interferometric imaging
摘要：
无线电干涉测量中的数据降维可以通过减少存储器占用和每次迭代更轻的计算来节省用于图像重建的计算资源，这对于成像方法的可扩展性到下一代望远镜的大数据设置是重要的。本文从压缩感知理论的角度对降维进行了新的阐述，并研究了其与凸优化背景下设计的成像算法的相互作用。我们提出了一个后网格的线性数据嵌入到测量算子的左奇异向量所跨越的空间，提供了一个低于图像大小的降维。这种嵌入保留了测量算子的零空间，因此也保留了其根据压缩感测理论的采样特性。我们表明，这可以近似通过首先计算脏图像，然后应用加权子采样离散傅立叶变换，以获得最终减少的数据向量。该傅立叶降维模型确保了完整测量算子的快速实现，这对于任何迭代图像重建方法都是必不可少的。建议的减少也保留了i.i.d.原始测量噪声的高斯特性。对于基于凸优化的成像算法，这是证明使用标准L2范数作为数据保真度项的关键。我们的模拟证实，这种降维方法可以利用凸优化算法，相对于从完整的可见性数据集重建图像，成像质量没有损失。在模拟环境中的重建结果没有方向相关的影响或校准误差显示出有前途的性能所提出的降维。GitHub上提供了实现所提出的缩减方法的MATLAB代码。
英文摘要：
Data dimensionality reduction in radio interferometry can provide savings of computational resources for image reconstruction through reduced memory footprints and lighter computations per iteration, which is important for the scalability of imaging methods to the big data setting of the next-generation telescopes. This article sheds new light on dimensionality reduction from the perspective of compressed sensing theory and studies its interplay with imaging algorithms designed in the context of convex optimization. We propose a post-gridding linear data embedding to the space spanned by the left singular vectors of the measurement operator, providing a dimensionality reduction below image size. This embedding preserves the null space of the measurement operator and hence also its sampling properties as per compressed sensing theory. We show that this can be approximated by first computing the dirty image and then applying a weighted subsampled discrete Fourier transform to obtain the final reduced data vector. This Fourier dimensionality reduction model ensures a fast implementation of the full measurement operator, essential for any iterative image reconstruction method. The proposed reduction also preserves the i.i.d. Gaussian properties of the original measurement noise. For convex optimization-based imaging algorithms, this is key to justify the use of the standard L2-norm as the data fidelity term. Our simulations confirm that this dimensionality reduction approach can be leveraged by convex optimization algorithms with no loss in imaging quality relative to reconstructing the image from the complete visibility data set. Reconstruction results in simulation settings with no direction dependent effects or calibration errors show promising performance of the proposed dimensionality reduction. MATLAB code implementing the proposed reduction method is available on GitHub.
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