[Paper Review] Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging

Ben Adcock, Nick Dexter+1 more|arXiv (Cornell University)|Sep 17, 2020

Sparse and Compressive Sensing Techniques25 references12 citations

TL;DR

This paper presents improved theoretical recovery guarantees and optimal sampling strategies for Total Variation (TV) minimization in compressive imaging using Fourier and Walsh–Hadamard transforms. It establishes that $ m hicksim s imes \log^2(s) \times \log^4(N) $ measurements suffice for stable, robust recovery in $ d \geq 1 $ dimensions under random sampling, improving prior bounds by $ \log(s) \times \log(N) $ in 2D and providing the first such guarantee for structured binary Walsh sampling.

ABSTRACT

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.

Motivation & Objective

• To establish uniform recovery guarantees for TV minimization in compressive imaging under arbitrary random sampling strategies for both Fourier and structured binary (Walsh–Hadamard) sampling.
• To derive theoretically optimal sampling strategies that minimize the number of measurements required for stable and robust image reconstruction.
• To improve upon existing recovery bounds for TV minimization in compressive imaging, particularly for 2D and higher-dimensional images.
• To provide the first theoretical recovery guarantee for TV minimization with structured binary sampling using the Walsh–Hadamard transform.
• To quantify the number of measurements $ m $ required for accurate reconstruction based on gradient sparsity $ s $ and image size $ N $

Proposed method

• Leveraged compressed sensing theory to derive uniform recovery guarantees for TV minimization under random sampling, ensuring stability and robustness to noise.
• Used the connection between TV semi-norm and Haar wavelet coefficients to analyze gradient sparsity and recovery error.
• Derived measurement bounds via probabilistic analysis of random sampling matrices for both Fourier and Walsh–Hadamard transforms.
• Formulated the recovery problem as a constrained $ \ell^1 $-minimization of the gradient, with a noise constraint $ \|Az - y\|_{\ell^2} \leq \eta $.
• Introduced a novel analysis framework to handle the $ \ell^{2,1} $-norm structure of the isotropic TV semi-norm in multiple dimensions.
• Applied results from high-dimensional probability and random matrix theory to bound the restricted isometry property (RIP) for gradient-sparse signals

Experimental results

Research questions

• RQ1 What is the minimal number of measurements $ m $ required to stably recover a $ d $-dimensional image with approximately $ s $-sparse gradient via TV minimization under random Fourier sampling?
• RQ2 Can theoretical recovery guarantees be extended to higher-dimensional images ($ d \geq 2 $) for Fourier-based compressive imaging?
• RQ3 What is the optimal sampling strategy for TV minimization in compressive imaging using the Walsh–Hadamard transform?
• RQ4 Can uniform recovery guarantees be established for structured binary sampling with TV minimization, given the lack of prior theoretical results in this setting?
• RQ5 How do the new bounds compare to existing state-of-the-art results in terms of dependence on $ s $, $ N $, and $ d $

Key findings

• For $ d $-dimensional Fourier sampling, $ m \gtrsim s \cdot \log^2(s) \cdot \log^4(N) $ measurements suffice for stable and robust recovery of images with $ s $-sparse gradient.
• In the 2D case, this improves the prior state-of-the-art bound by a factor of $ \log(s) \cdot \log(N) $, representing a significant quantitative improvement.
• The recovery guarantee extends to arbitrary dimensions $ d \geq 1 $, providing a general framework for multi-dimensional compressive imaging.
• For structured binary Walsh–Hadamard sampling, $ m \gtrsim s \cdot \log^2(s) \cdot \log^2(N/s) \cdot \log^3(N) $ measurements are sufficient, marking the first such theoretical guarantee for this setting.
• The results establish uniform recovery guarantees, meaning a single random sampling draw suffices for recovery of all approximately gradient-sparse images with high probability.
• The derived bounds are optimal up to logarithmic factors, and the sampling strategies are theoretically optimal in minimizing measurement count