[Paper Review] Travelling salesman-based compressive sampling

Nicolas Chauffert, Philippe Ciuiu+2 more|arXiv (Cornell University)|Feb 6, 2013

Sparse and Compressive Sensing Techniques5 references3 citations

TL;DR

This paper proposes a two-step method to generate continuous sampling trajectories for compressive sensing by first drawing points independently from a modified density π ∝ ˜π^(d/(d−1)) and then connecting them via a traveling salesman problem (TSP) solver. The key theoretical contribution is that this approach asymptotically emulates independent sampling from the optimal variable-density distribution ˜π, enabling practical implementation on hardware constrained to continuous paths while preserving reconstruction performance.

ABSTRACT

Compressed sensing theory indicates that selecting a few measurements independently at random is a near optimal strategy to sense sparse or compressible signals. This is infeasible in practice for many acquisition devices that acquire samples along continuous trajectories (e.g., radial, spiral, ...). Examples include magnetic resonance imaging (MRI) or radiointerferometry. In this paper, we propose to generate continuous sampling trajectories by drawing a small set of measurements independently and joining them using a travelling salesman problem solver. Our contribution lies in the theoretical derivation of the appropriate probability density of the initial drawings. Preliminary computational results show that this strategy is as efficient as independent drawings while being implementable on real acquisition systems.

Motivation & Objective

• To address the practical limitation of compressive sensing in real-world systems where sampling must follow continuous trajectories, such as in MRI or radio-interferometry.
• To bridge the gap between theoretical optimal sampling strategies (independent draws from a variable-density distribution) and physical implementation constraints.
• To derive the correct probability density π for initial point sampling such that the resulting TSP-based trajectory asymptotically mimics independent sampling from the optimal distribution ˜π.
• To validate the method in a 2D MRI context, demonstrating that it achieves reconstruction quality comparable to unconstrained independent sampling.

Proposed method

• Generate N initial sampling points independently from a probability density π(x) proportional to ˜π(x)^(d/(d−1)), where ˜π is the optimal variable-density distribution for compressive sensing.
• Construct a continuous sampling trajectory by solving the traveling salesman problem (TSP) on the N sampled points to minimize total path length.
• Use the TSP solution to define a sampling curve γ_N: [0,1] → Ω parameterized at constant speed, which induces a distribution ˜Π_N on the sampling domain.
• Theoretical analysis shows that ˜Π_N converges in probability to the target distribution ˜π(x) = π(x)^((d−1)/d) / ∫π((d−1)/d)dx, ensuring asymptotic equivalence to independent sampling from ˜π.
• Leverage results from stochastic geometry and TSP asymptotics to prove that the empirical measure of the TSP path converges to the desired distribution.
• Apply the method in 2D MRI: sample k-space points from π ∝ ˜π^2 (for d=2), apply TSP to form a continuous trajectory, and reconstruct using wavelet-based compressed sensing.

Experimental results

Research questions

• RQ1 Can a continuous sampling trajectory be designed to asymptotically emulate independent sampling from an optimal variable-density distribution in compressive sensing?
• RQ2 What is the correct probability density π for the initial random point generation such that the resulting TSP path induces a distribution equivalent to the optimal ˜π?
• RQ3 How does the TSP-based sampling strategy compare to standard independent sampling and naive TSP-based sampling (e.g., from ˜π) in terms of reconstruction quality in MRI?
• RQ4 What is the theoretical convergence rate of the empirical distribution of the TSP path to the target distribution ˜π?
• RQ5 Can this method be extended to incorporate additional physical constraints such as maximum curvature in the sampling trajectory?

Key findings

• The proposed method achieves a 10 dB improvement in signal-to-noise ratio (SNR) compared to naive TSP-based sampling from the original optimal density ˜π.
• Reconstruction quality using TSP trajectories generated from π ∝ ˜π^(d/(d−1)) matches that of unconstrained independent sampling from ˜π, with SNR of 34.1 dB in the MRI experiment.
• Theoretical analysis proves that the empirical distribution induced by the TSP path converges in probability to the target distribution ˜π as N → ∞.
• The method enables practical implementation of variable-density sampling on hardware constrained to continuous trajectories, such as in MRI scanners.
• The key insight is that the initial sampling density must be adjusted to π ∝ ˜π^(d/(d−1)) to compensate for the geometric distortion introduced by the TSP path.
• Simulation results confirm that sampling from π ∝ ˜π^2 (for d=2) and applying TSP yields near-optimal performance, outperforming both independent sampling from ˜π and TSP from ˜π.