[Paper Review] Reconstruction of Nonnegative Sparse Signals Using Accelerated Proximal-Gradient Algorithms

Renliang Gu, A. Dogandzic|arXiv (Cornell University)|Feb 9, 2015

Sparse and Compressive Sensing Techniques42 references3 citations

TL;DR

This paper proposes an accelerated proximal-gradient algorithm using Nesterov's method with function restart and ADMM-based proximal mapping to reconstruct nonnegative sparse signals from underdetermined measurements. By jointly enforcing signal nonnegativity and sparsity in the wavelet domain via l1-norm and indicator functions, the method achieves superior reconstruction performance over existing approaches in compressed sensing and tomography with Poisson and Gaussian noise models.

ABSTRACT

We develop an accelerated proximal-gradient scheme for reconstructing nonnegative signals that are sparse in a transform domain from underdetermined measurements. This signal model is motivated by tomographic applications where the signal of interest is known to be nonnegative because it represents a tissue or material density. It is also applicable to optical and hyperspectral imaging, where energy within certain spectral band is nonnegative. We adopt the unconstrained regularization framework where the objective function to be minimized is a sum of a convex data fidelity (negative log-likelihood (NLL)) term and a convex regularization term that imposes signal nonnegativity and sparsity by using indicator-function and l1-norm constraints on the signal and its transform coefficients, respectively. We apply the Nesterov's proximal-gradient (NPG) method with function restart to minimize this objective function and the alternating direction method of multipliers (ADMM) to compute the proximal mapping. To accelerate convergence of the NPG iteration, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We also apply adaptive continuation, which provides numerical stability and can accelerate the convergence of the NPG iteration. We construct compressed-sensing and tomographic reconstruction experiments with Gaussian linear and Poisson generalized linear measurement models, where we compare the proposed reconstruction approach with existing signal reconstruction methods. By exploiting both the nonnegativity of the underlying signal and sparsity of its wavelet coefficients, we can achieve significantly better reconstruction performance than the existing methods.

Motivation & Objective

• To address the challenge of reconstructing nonnegative sparse signals from underdetermined linear measurements in tomographic and imaging applications.
• To improve reconstruction accuracy by exploiting both signal nonnegativity and sparsity in a transform domain.
• To develop a fast, stable, and convergent optimization framework tailored for nonnegative sparse signal recovery.
• To outperform existing methods in compressed sensing and tomographic reconstruction under Poisson and Gaussian measurement models.

Proposed method

• The method uses an unconstrained regularization framework combining a convex data fidelity term (negative log-likelihood) and a regularization term enforcing nonnegativity and sparsity.
• It applies Nesterov's proximal-gradient (NPG) method with function restart to accelerate convergence of the optimization process.
• The proximal mapping is computed using the alternating direction method of multipliers (ADMM) for efficient and stable subproblem resolution.
• A step-size selection scheme accounts for the varying local Lipschitz constant of the negative log-likelihood function to improve convergence.
• Adaptive continuation is employed to enhance numerical stability and further accelerate convergence.
• The approach is evaluated on both Gaussian linear and Poisson generalized linear measurement models in compressed sensing and tomographic reconstruction tasks.

Experimental results

Research questions

• RQ1 Can an accelerated proximal-gradient method with function restart and adaptive continuation improve convergence and reconstruction quality for nonnegative sparse signals?
• RQ2 How does joint enforcement of nonnegativity and sparsity in the wavelet domain affect reconstruction performance in underdetermined systems?
• RQ3 Does the proposed method outperform existing state-of-the-art algorithms in tomographic and compressed sensing applications with Poisson and Gaussian noise?
• RQ4 How does the step-size selection based on local Lipschitz constants impact convergence speed and stability?
• RQ5 What is the role of ADMM in enabling efficient and stable proximal mapping within the NPG framework?

Key findings

• The proposed method achieves significantly better reconstruction performance than existing methods by jointly exploiting signal nonnegativity and sparsity in the wavelet domain.
• Function restart and adaptive continuation enhance numerical stability and accelerate convergence of the NPG iteration.
• The step-size selection scheme that accounts for the local Lipschitz constant of the negative log-likelihood improves convergence efficiency.
• ADMM-based proximal mapping enables accurate and stable computation of the proximal step, crucial for the overall algorithmic performance.
• The method demonstrates superior reconstruction accuracy in both compressed sensing and tomographic reconstruction tasks under Poisson and Gaussian measurement models.
• The combination of nonnegativity constraints and l1-norm regularization on transform coefficients leads to improved signal recovery, especially in low-signal regimes.