脉冲图像去噪的稀疏模型双阶段优化方法

A Two-Stage Optimization Method Based on Impulse Image Denoising Sparse Model

摘要: 脉冲图像去噪是图像处理领域的关键问题，其核心挑战在于噪声的稀疏分布特性与模型的非凸优化困境。本文针对上述问题，提出一种针对脉冲图像去噪模型的双阶段优化方法。首先，通过复合凹函数与绝对值函数构造$l_0$ 惩罚的替代函数，建立脉冲噪声去噪模型。该模型在保留稀疏表征能力的同时，利用非凸连续函数规避NP难问题。进一步，基于Fenchel变换，将原非凸问题等价为双变量优化模型，并提出外循环–内循环架构的双阶段优化方法：外循环通过闭式求解不断调整内循环目标模型，内循环采用对偶型交替方向乘子法(ADMM)高效求解非光滑核心子问题。该方法通过设计交替优化策略，生成凸优化序列，确保解列逐步逼近原始模型的最优解。实验证明，相比传统去噪模型，提出的双阶段方法对稀疏函数的具体构造形式依赖性低，在噪声抑制与细节保留方面具有显著优势。同时，提出的算法在工程层面易于实现，为大规模图像处理提供了保障与支撑。

Abstract: Impulse image denoising is a critical challenge in image processing, with its core difficulties lying in the sparse distribution characteristics of noise and the non-convex optimization dilemma of traditional models. To address these issues, this paper proposes a two-stage optimization method for impulse image denoising model. First, by constructing a surrogate function for the $l_0$ penalty through the combination of continuous concave functions and absolute value functions, we establish a novel impulse noise denoising model. This model retains the ability to represent sparsity while avoiding the NP-hard problem by using non-convex continuous function. Furthermore, based on Fenchel transformation, the original non-convex problem is equivalently reformulated into a bi-variable optimization model. A new two-stage optimization framework with an outer-inner loop architecture is developed: the outer loop adjusts the model of the inner loop via closed-form solutions, while the inner loop employs a dual Alternating Direction Method of Multipliers (ADMM) to efficiently solve the non-smooth core subproblem. This design generates a sequence of convex optimization problems through an alternating optimization strategy, ensuring that the solution sequence progressively converges to the optimal solution of the original model. Experimental results demonstrate that, compared to traditional denoising models, the proposed two-stage method exhibits lower dependency on specific sparse function constructions and achieves superior performance in noise suppression and detail preservation. Additionally, the proposed algorithm is easy to realize at the engineering level, providing guarantee and support for large-scale image processing.