[Paper Review] L1-norm minimization for quaternion signals

Jiasong Wu, Xu Zhang+3 more|arXiv (Cornell University)|Feb 24, 2012

Sparse and Compressive Sensing Techniques17 references3 citations

TL;DR

This paper proposes an algorithm for L1-norm minimization in quaternion signals by reformulating the problem as second-order cone programming (SOCP), enabling efficient and stable recovery in compressed sensing. The method ensures perfect signal reconstruction under suitable conditions, demonstrating potential for applications in quaternion-based signal processing systems.

ABSTRACT

The l1-norm minimization problem plays an important role in the compressed sensing (CS) theory. We present in this letter an algorithm for solving the problem of l1-norm minimization for quaternion signals by converting it to second-order cone programming. An application example of the proposed algorithm is also given for practical guidelines of perfect recovery of quaternion signals. The proposed algorithm may find its potential application when CS theory meets the quaternion signal processing.

Motivation & Objective

• To address the challenge of sparse recovery in quaternion-valued signals using L1-norm minimization.
• To develop a computationally efficient and numerically stable algorithm for solving L1-minimization problems in the quaternion domain.
• To extend compressed sensing theory to quaternion signals by enabling perfect recovery under appropriate conditions.
• To provide a practical framework for implementing L1-minimization in quaternion signal processing applications.

Proposed method

• The L1-norm minimization problem for quaternion signals is reformulated as a second-order cone program (SOCP), enabling use of standard convex optimization solvers.
• The quaternion structure is preserved throughout the transformation by exploiting the algebraic properties of quaternions and their norm equivalence to vector norms.
• The reformulation allows the use of interior-point methods for efficient and accurate solution of the optimization problem.
• The algorithm is validated through a numerical example demonstrating perfect signal recovery under ideal conditions.
• The method leverages existing SOCP solvers, ensuring compatibility with standard optimization toolboxes.

Experimental results

Research questions

• RQ1 Can L1-minimization for quaternion signals be effectively reformulated as a convex optimization problem?
• RQ2 What is the computational efficiency and numerical stability of the proposed SOCP-based approach?
• RQ3 Under what conditions can perfect recovery of quaternion signals be guaranteed using this method?
• RQ4 How does the proposed algorithm compare to existing methods in terms of reconstruction accuracy and computational cost?

Key findings

• The L1-minimization problem for quaternion signals is successfully reformulated as a second-order cone program, enabling efficient solution via standard convex optimization techniques.
• The proposed algorithm achieves perfect recovery of sparse quaternion signals when the sensing matrix satisfies the restricted isometry property (RIP) in the quaternion domain.
• Numerical results confirm that the algorithm converges reliably and reconstructs signals with high accuracy under ideal conditions.
• The method provides a practical and scalable framework for compressed sensing in quaternion signal processing applications.