Stable and robust sampling strategies for compressive imaging (1210.2380v3)

Abstract: In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence -- the so-called local coherence -- measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell_1$-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded \emph{average} coherence from sensing basis to sparsity basis -- as opposed to bounded maximal coherence -- as long as the sampling strategy is adapted accordingly.

Overview of Stable and Robust Sampling Strategies for Compressive Imaging

The paper authored by Felix Krahmer and Rachel Ward proposes novel sampling methodologies in the field of compressive imaging, focusing specifically on image acquisition strategies that exploit sparsity in transform domains such as wavelets or spatial finite differences using frequency domain samples. The authors present empirical evidence and theoretical insight suggesting that variable density sampling strategies, which concentrate on lower frequencies, can lead to superior image reconstruction compared to uniform sampling methods.

Key Contributions and Theoretical Insights

The paper's core contribution lies in addressing the coherence issues between the Fourier and wavelet domains by introducing the concept of local coherence, which quantifies the correlation of each sensing vector with the sparsity basis individually. This nuanced notion of coherence allows the researchers to provide a robust mathematical framework that underpins their sampling strategy.

1. Local Coherence and Sampling Density: The authors define local coherence and illustrate its utility in designing sampling schemes. By controlling and bounding local coherence for Fourier measurements and Haar wavelet sparsity, they prove the restricted isometry property (RIP) for matrices composed of frequencies sampled from an inverse square power-law density. This results in an optimized dimensional embedding for compressive imaging applications.
2. Stable Recovery Guarantees: The paper establishes conditions under which stable recovery of images is achievable by $\ell_1$-minimization and total variation minimization, demonstrating resilience to sparsity defects and measurement noise. These guarantees are contingent upon employing a suitable variable-density sampling strategy, thereby highlighting the practical implications of the theoretical results.
3. Numerical Simulations: To validate the theoretical claims, the authors include numerical examples and simulations that underscore the efficacy of the proposed sampling strategies in compressive imaging scenarios.

Implications for AI and Future Work

The proposed framework has several implications for advancing practical and theoretical developments in AI and signal processing:

• Enhanced Sparse Recovery: By leveraging local coherence, researchers and engineers can design compressive sensing systems that optimize sparse recovery, potentially improving data acquisition in medical imaging, astronomy, and other domains where high-quality reconstructions from minimal measurements are vital.
• Refinements to Compressive Imaging: The insights from this paper could inform improvements in the speed and quality of imaging techniques, particularly in areas like MRI, where reducing measurement numbers can lower costs and exposure times.
• Continued Exploration of Sparsity Structures: Future research may build upon these findings by examining other sparsity structures that could benefit from variable-density sampling or investigating extensions to the framework that could accommodate more complex image models or iterative reconstruction techniques.

Methodological Strengths and Analytical Rigor

The paper stands out for its methodical approach in tackling a nuanced aspect of signal processing—the reconciliation of incoherent bases through local coherence measurements. This approach exemplifies a significant step forward in understanding how different transform domains interact and how these interactions can be modeled to improve compressive sensing outcomes.

Despite the strong theoretical foundations laid in this paper, practical implementation constraints—such as hardware limitations in MRI systems—are outside its scope and remain a running challenge in the field. It is also noted that further reduction in the number of necessary measurements or computational complexity could make these strategies even more advantageous in real-world scenarios.

Overall, "Stable and Robust Sampling Strategies for Compressive Imaging" advances the dialogue in compressive sensing by providing a mathematically rigorous yet practically relevant approach to image recovery through innovative sampling strategies, thereby contributing to the broader quest for efficient data acquisition methods in the digital age.