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Improved Recovery Guarantees and Sampling Strategies for TV Minimization in Compressive Imaging
SIAM Journal on Imaging Sciences ( IF 2.1 ) Pub Date : 2021-08-04 , DOI: 10.1137/20m136788x
Ben Adcock , Nick Dexter , Qinghong Xu

SIAM Journal on Imaging Sciences, Volume 14, Issue 3, Page 1149-1183, January 2021.
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 sampling strategies which are theoretically near-optimal for recovery of approximately gradient-sparse images. For Fourier sampling, we show recovery of such an image 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.


中文翻译:

压缩成像中电视最小化的改进恢复保证和采样策略

SIAM Journal on Imaging Sciences,第 14 卷,第 3 期,第 1149-1183 页,2021 年 1 月。
在本文中,我们考虑将总变差 (TV) 最小化用于压缩成像,即从子采样测量中重建图像。专注于两种重要的成像方式——即傅立叶成像和通过 Walsh--Hadamard 变换的结构化二元成像——我们推导出均匀恢复保证,保证任意随机采样策略的稳定和鲁棒恢复。使用这个,我们然后推导出一类采样策略,这些策略在理论上对于近似梯度稀疏图像的恢复来说是接近最佳的。对于傅立叶采样,我们展示了从 $m \gtrsim_d s \cdot \log^2(s) \cdot \log^4(N)$ 测量中恢复这样的图像,在 $d \geq 1$ 维度。当 $d = 2$ 时,这将当前最先进的结果提高了 $\log(s) \cdot \log(N)$ 的因子。它还将其扩展到任意维度 $d \geq 2$。对于沃尔什采样,我们证明 $m \gtrsim_d s \cdot \log^2(s) \cdot \log^2(N/s) \cdot \log^3(N) $ 测量值在 $d \geq 2 中就足够了$ 尺寸。据我们所知,这是具有 TV 最小化的结构化二进制采样的第一个恢复保证。
更新日期:2021-08-05
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