用随机波束成形实现射电干涉的压缩感知，大幅降低数据量。
Compressive radio-interferometric sensing with random beamforming as rank-one signal covariance projections
- 通过随机波束成形将信号协方差投影为秩一形式，实现数据压缩。
- 数据量从O(Q²B)降至O(K)，K为图像稀疏度，与天线数无关。
- 适合大阵列射电望远镜的实时成像，尤其适用于稀疏场景。
射电干涉仪（RI）以极高的角分辨率观测宇宙，可研究星系、黑洞等遥远天体。其阵列天线接收来自天空的信号，通过范·希特-曾尼克定理，得到信号协方差矩阵的不完整且含噪傅里叶采样，即可见度（visibilities）。可见度数量级为O(Q²B)，Q为天线数，B为短时积分区间数。针对大型阵列带来的数据激增问题，本文提出在天线测量层面直接应用压缩感知。首先证明波束成形等价于对信号协方差矩阵进行秩一投影（ROP），基于先前工作（arXiv:2306.12698v3），设计基于随机波束成形的压缩方案，将数据量从依赖Q²改为仅需P个ROP。理论证明稀疏图像重建的恢复保证。其次，通过在每个STI区间对ROP向量施加M个伯努利调制，使数据量独立于B。理论分析和相变图均表明样本复杂度为O(K)，K为图像稀疏度，体现该方法潜力。
原文摘要 · Abstract (English)
Radio-interferometry (RI) observes the sky at unprecedented angular resolutions, enabling the study of several far-away galactic objects such as galaxies and black holes. In RI, an array of antennas probes cosmic signals coming from the observed region of the sky. The covariance matrix of the vector gathering all these antenna measurements offers, by leveraging the Van Cittert-Zernike theorem, an incomplete and noisy Fourier sensing of the image of interest. The number of noisy Fourier measurements -- or visibilities -- scales as $\mathcal O(Q^2B)$ for $Q$ antennas and $B$ short-time integration (STI) intervals. We address the challenges posed by this vast volume of data, which is anticipated to increase significantly with the advent of large antenna arrays, by proposing a compressive sensing technique applied directly at the level of the antenna measurements. First, this paper shows that beamforming -- a common technique of dephasing antenna signals -- usually used to focus some region of the sky, is equivalent to sensing a rank-one projection (ROP) of the signal covariance matrix. We build upon our recent work arXiv:2306.12698v3 [eess.IV] to propose a compressive sensing scheme relying on random beamforming, trading the $Q^2$-dependence of the data size for a smaller number $P$ ROPs. We provide image recovery guarantees for sparse image reconstruction. Secondly, the data size is made independent of $B$ by applying $M$ Bernoulli modulations of the ROP vectors obtained for the STI. The resulting sample complexities, theoretically derived in a simpler case without modulations and numerically obtained in phase transition diagrams, are shown to scale as $\mathcal O(K)$ where $K$ is the image sparsity. This illustrates the potential of the approach.
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