[Paper Review] Optimal Sparse Recovery for Multi-Sensor Measurements

Il Yong Chun, Ben Adcock|arXiv (Cornell University)|Mar 22, 2016

Sparse and Compressive Sensing Techniques14 references3 citations

TL;DR

This paper establishes optimal sparse recovery guarantees for multi-sensor compressed sensing systems with diagonal sensor profile matrices, introducing improved theoretical bounds for both distinct and identical sampling scenarios. It shows that with appropriate sensor profiles, the total number of measurements scales linearly with sparsity and independently of the number of sensors, enabling efficient, scalable signal recovery across diverse applications like MRI and multi-view imaging.

ABSTRACT

Many practical sensing applications involve multiple sensors simultaneously acquiring measurements of a single object. Conversely, most existing sparse recovery guarantees in compressed sensing concern only single-sensor acquisition scenarios. In this paper, we address the optimal recovery of compressible signals from multi-sensor measurements using compressed sensing techniques, thereby confirming the benefits of multi- over single-sensor environments. Throughout the paper, we consider a broad class of sensing matrices, and two fundamentally different sampling scenarios (distinct and identical respectively), both of which are relevant to applications. For the case of diagonal sensor profile matrices (which characterize environmental conditions between a source and the sensors), this paper presents two key improvements over existing results. First, a simpler optimal recovery guarantee for distinct sampling, and second, an improved recovery guarantee for identical sampling, based on the so-called sparsity in levels signal model.

Motivation & Objective

• To address the lack of theoretical recovery guarantees in multi-sensor compressed sensing, particularly for systems with multiple sensors acquiring measurements of a single signal.
• To analyze and improve recovery performance in two fundamental sampling scenarios: distinct sampling (independent sensor matrices) and identical sampling (shared measurement matrices with different profiles).
• To derive new sufficient conditions on sensor profile matrices that ensure optimal recovery with minimal measurements, independent of the number of sensors.
• To provide computable recovery thresholds that enable practical sensor design and system optimization in real-world applications.
• To validate theoretical findings with numerical phase transition experiments across different sensor configurations and signal models.

Proposed method

• Proposes a generalized parallel acquisition model where each sensor uses a measurement matrix formed by a standard CS matrix multiplied by a deterministic sensor profile matrix.
• Introduces a novel recovery condition based on the sparsity in levels model, which allows for improved bounds in the identical sampling case.
• Defines two key computable quantities, $\Upsilon_{\mathrm{distinct}}$ and $\Upsilon_{\mathrm{identical}}$, that determine the recovery performance and are computable in $\mathcal{O}(CN)$ and $\mathcal{O}(CN^2)$ time respectively.
• Applies the basis pursuit algorithm with a noise-constrained $\ell^1$-minimization to reconstruct signals from multi-sensor measurements.
• Derives theoretical recovery guarantees by analyzing the coherence and structure of sensor profile matrices, particularly for piecewise constant and banded profiles.
• Validates results through empirical phase transition curves using random Fourier sensing matrices and banded diagonal profiles across varying numbers of sensors.

Experimental results

Research questions

• RQ1 Can optimal sparse recovery be guaranteed in multi-sensor compressed sensing systems with general sensor profile matrices?
• RQ2 How does the number of required measurements scale with sparsity and the number of sensors in distinct and identical sampling scenarios?
• RQ3 What structural properties of sensor profile matrices ensure optimal recovery performance independent of the number of sensors?
• RQ4 Can the sparsity in levels model improve recovery guarantees in identical sampling compared to standard sparsity models?
• RQ5 How do specific sensor profile designs—such as piecewise constant or banded profiles—affect the theoretical and empirical recovery performance?

Key findings

• For distinct sampling, optimal recovery is guaranteed when the sensor profile matrix $V$ is incoherent with coherence $\mu(V) \lesssim C^{-1}$, ensuring that the total number of measurements scales linearly with sparsity and independently of $C$.
• In the identical sampling case, optimal recovery is achieved when the number of partitions $D$ satisfies $C/D \lesssim 1$, which allows recovery of signals sparse in levels with minimal measurement overhead.
• The banded sensor profile example with $r_1=1$, $r_2=0$ yields $\Upsilon_{\mathrm{distinct}}, \Upsilon_{\mathrm{identical}} \leq 2$ for any $C$, confirming optimal recovery regardless of the number of sensors.
• Numerical phase transition results show that empirical recovery success probability increases with the number of sensors $C$, validating the theoretical predictions for both distinct and identical sampling.
• The recovery thresholds $\Upsilon_{\mathrm{distinct}}$ and $\Upsilon_{\mathrm{identical}}$ are computable in $\mathcal{O}(CN)$ and $\mathcal{O}(CN^2)$ time, respectively, enabling practical design of optimal sensor profiles.
• The results confirm that multi-sensor systems can achieve optimal recovery with average measurements per sensor scaling as $s/C$, demonstrating the theoretical advantage of multi-sensor over single-sensor compressed sensing.