论文标题
改进的恢复保证和采样策略，以最大程度地减少压缩成像
Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging
论文作者
论文摘要
在本文中，我们考虑使用总变化（TV）最小化来压缩成像。也就是说，从亚次采样测量中重建图像。专注于两种重要的成像方式 - 即通过Walsh-Hadamard变换，傅立叶成像和结构化的二进制成像 - 我们得出统一的恢复保证，可以保证对任意随机抽样策略的稳定恢复稳定而强大的恢复。然后，我们得出一类理论上最佳的采样策略。对于傅立叶采样，我们显示了从$ M \ gtrsim_d s \ cdot \ log^2（s）\ cdot \ log^4（n）$测量值的图像的恢复，该图像的恢复大约$ s -s $ sparse梯度，in $ d \ geq 1 $尺寸。当$ d = 2 $时，这会改善当前最新结果，而$ \ log（s）\ cdot \ log（n）$。它还将其扩展到任意维度$ d \ geq 2 $。对于WALSH采样，我们证明$ M \ gtrsim_d s \ cdot \ log^2（s）\ cdot \ log^2（n/s）\ cdot \ log^3（n）$测量足以在$ d \ geq 2 $ dimensions中进行。据我们所知，这是对电视最小化结构化二进制采样的首次恢复保证。
In this paper, we consider the use of Total Variation (TV) minimization for compressive imaging; that is, image reconstruction from subsampled measurements. Focusing on two important imaging modalities -- namely, Fourier imaging and structured binary imaging via the Walsh--Hadamard transform -- we derive uniform recovery guarantees asserting stable and robust recovery for arbitrary random sampling strategies. Using this, we then derive a class of theoretically-optimal sampling strategies. For Fourier sampling, we show recovery of an image with approximately $s$-sparse gradient from $m \gtrsim_d s \cdot \log^2(s) \cdot \log^4(N)$ measurements, in $d \geq 1$ dimensions. When $d = 2$, this improves the current state-of-the-art result by a factor of $\log(s) \cdot \log(N)$. It also extends it to arbitrary dimensions $d \geq 2$. For Walsh sampling, we prove that $m \gtrsim_d s \cdot \log^2(s) \cdot \log^2(N/s) \cdot \log^3(N) $ measurements suffice in $d \geq 2$ dimensions. To the best of our knowledge, this is the first recovery guarantee for structured binary sampling with TV minimization.
