[论文解读] Reconstruction of Nonnegative Sparse Signals Using Accelerated Proximal-Gradient Algorithms

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

Sparse and Compressive Sensing Techniques参考文献 42被引用 3

一句话总结

该论文提出了一种基于Nesterov方法的加速近似梯度算法，结合函数重启与基于ADMM的近似映射，用于从欠定测量中重建非负稀疏信号。通过在小波域中联合施加信号非负性与稀疏性（利用l1-范数和指示函数），该方法在压缩感知与泊松和高斯噪声模型下的断层扫描中，实现了优于现有方法的重建性能。

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.

研究动机与目标

• 解决在断层扫描与成像应用中，从欠定线性测量中重建非负稀疏信号的挑战。
• 通过在变换域中同时利用信号非负性与稀疏性，提升重建精度。
• 开发一种快速、稳定且收敛的优化框架，专用于非负稀疏信号恢复。
• 在泊松与高斯测量模型下，优于现有的压缩感知与断层扫描重建方法。

提出的方法

• 该方法采用无约束正则化框架，结合凸数据保真项（负对数似然）与正则化项，以强制实现非负性与稀疏性。
• 采用带函数重启的Nesterov近似梯度（NPG）方法，加速优化过程的收敛。
• 通过交替方向乘子法（ADMM）计算近似映射，以实现高效且稳定的子问题求解。
• 采用步长选择方案，考虑负对数似然函数的局部Lipschitz常数变化，以提升收敛性。
• 采用自适应延续策略，增强数值稳定性并进一步加速收敛。
• 该方法在压缩感知与断层扫描重建任务中，针对高斯线性与泊松广义线性测量模型进行了评估。

实验结果

研究问题

• RQ1 基于函数重启与自适应延续的加速近似梯度方法，能否提升非负稀疏信号的收敛性与重建质量？
• RQ2 在小波域中联合强制非负性与稀疏性，对欠定系统中的重建性能有何影响？
• RQ3 在泊松与高斯噪声下的断层扫描与压缩感知应用中，该方法是否优于现有最先进算法？
• RQ4 基于局部Lipschitz常数的步长选择方案，对收敛速度与稳定性有何影响？
• RQ5 ADMM在NPG框架内实现高效且稳定的近似映射中起到何种作用？

主要发现

• 通过在小波域中联合利用信号非负性与稀疏性，该方法显著优于现有方法，实现了更优的重建性能。
• 函数重启与自适应延续策略增强了NPG迭代的数值稳定性，并加速了收敛。
• 考虑负对数似然函数局部Lipschitz常数的步长选择方案，提升了收敛效率。
• 基于ADMM的近似映射实现了精确且稳定的近似步计算，对整体算法性能至关重要。
• 在泊松与高斯测量模型下，该方法在压缩感知与断层扫描重建任务中均表现出更优的重建精度。
• 非负性约束与变换系数上l1-范数正则化的结合，显著提升了信号恢复性能，尤其在低信噪比条件下。