[Paper Review] Using Correlated Subset Structure For Compressive Sensing Recovery

Atul Divekar, Deanna Needell|arXiv (Cornell University)|Feb 16, 2013

Sparse and Compressive Sensing Techniques15 references3 citations

TL;DR

This paper proposes a novel compressive sensing recovery algorithm that exploits correlated column structures in sampling matrices arising from physical imaging processes like super-resolution. By grouping wavelet coefficients into tree-structured subsets and using a sum-based selection criterion, the method improves reconstruction accuracy over standard $β$-minimization, especially when nonzero coefficients are concentrated in spatially coherent wavelet trees.

ABSTRACT

Compressive sensing is a methodology for the reconstruction of sparse or compressible signals using far fewer samples than required by the Nyquist criterion. However, many of the results in compressive sensing concern random sampling matrices such as Gaussian and Bernoulli matrices. In common physically feasible signal acquisition and reconstruction scenarios such as super-resolution of images, the sensing matrix has a non-random structure with highly correlated columns. Here we present a compressive sensing recovery algorithm that exploits this correlation structure. We provide algorithmic justification as well as empirical comparisons.

Motivation & Objective

• Address the challenge of compressive sensing recovery when sampling matrices exhibit strong intra-column correlation due to physical acquisition constraints, such as in image super-resolution.
• Overcome limitations of standard $β$-minimization and CoSaMP in scenarios with non-random, correlated sensing matrices that violate the Restricted Isometry Property (RIP).
• Improve signal reconstruction accuracy by exploiting the inherent tree-structured correlation of wavelet coefficients in image signals.
• Develop and validate a practical recovery algorithm that leverages subset-wise correlation patterns to enhance selection of significant coefficients.
• Demonstrate the effectiveness of a sum-based estimator for identifying wavelet trees containing nonzero coefficients in sparse signal recovery.

Proposed method

• Introduce the Partial Inversion (PartInv) algorithm, a modified version of CoSaMP, to reduce noise in the initial estimator by inverting only a subset of the sensing matrix columns.
• Group columns of the sampling matrix $Φ = SH\Psi$ into spatially coherent wavelet tree subsets, where columns within each subset are highly correlated due to overlapping spatial support.
• Define a strength estimator $s_I = \sum_{j \in I} |\hat{c}_j|$ for each wavelet tree $I$, used to rank and select the most promising subsets for signal recovery.
• Iteratively refine the support estimate by solving a least-squares problem on the selected subset $I^{(k)}$, then update the residual and re-estimate the support using the new correlation-aware selection.
• Use a stopping condition based on residual decrease and convergence of the support estimate to terminate the iterative process.
• Apply the method to 2D image patches using a Daubechies-5 wavelet basis and a blurring filter to simulate realistic optical sampling, with subsampling patterns generated via replication of base patterns.

Experimental results

Research questions

• RQ1 Can compressive sensing recovery be significantly improved by exploiting the inherent tree-structured correlation in wavelet-based representations of images?
• RQ2 How does the performance of $β$-minimization compare to a modified CoSaMP algorithm (PartInv) when the sensing matrix has correlated column subsets?
• RQ3 To what extent does a sum-based estimator of wavelet tree energy improve the selection of nonzero coefficient sets compared to standard thresholding?
• RQ4 Does the proposed method maintain high recovery accuracy under realistic sampling conditions such as downsampling and blurring in image super-resolution?
• RQ5 Can the algorithm achieve stable and accurate reconstruction when the sampling matrix fails to satisfy the Restricted Isometry Property (RIP) due to strong column correlations?

Key findings

• The proposed PartInv algorithm with the sum-based estimator significantly improves recovery performance compared to standard $β$-minimization, particularly in scenarios with correlated column subsets.
• For sampling rates $\delta = \frac{2}{16}$ to $\frac{14}{16}$, the method achieves a success rate exceeding 90% in reconstructing signals with nonzero coefficients concentrated on wavelet trees, as measured by $\frac{1}{N}\|c - \hat{c}\|^2 < 10^{-5}$.
• The sum estimator $s_I = \sum_{j \in I} |\hat{c}_j|$ effectively identifies wavelet trees containing significant coefficients, outperforming simple thresholding in selecting relevant support sets.
• The algorithm demonstrates robustness to noise and structural correlation in the sensing matrix, even when $\Phi = SH\Psi$ violates the Restricted Isometry Property.
• Empirical results show that the correlation structure in $\Phi$, arising from spatially overlapping wavelet atoms filtered by a low-pass kernel, can be effectively exploited to enhance recovery accuracy.
• The method achieves high reconstruction fidelity with only $M/N = 14/16$ sampling rate, indicating strong performance even at high compression rates in image super-resolution settings.