Deep denoisers have shown excellent performance in solving inverse problems in signal and image processing. In order to guarantee the convergence, the denoiser needs to satisfy some Lipschitz conditions like non-expansiveness. However, enforcing such constraints inevitably compromises recovery performance. This paper introduces a novel training strategy that enforces a weaker constraint on the deep denoiser called pseudo-contractiveness. By studying the spectrum of the Jacobian matrix, relationships between different denoiser assumptions are revealed. Effective algorithms based on gradient descent and Ishikawa process are derived, and further assumptions of strict pseudo-contractiveness yield efficient algorithms using half-quadratic splitting and forward-backward splitting. The proposed algorithms theoretically converge strongly to a fixed point. A training strategy based on holomorphic transformation and functional calculi is proposed to enforce the pseudo-contractive denoiser assumption. Extensive experiments demonstrate superior performance of the pseudo-contractive denoiser compared to related denoisers. The proposed methods are competitive in terms of visual effects and quantitative values.

翻译：深度去噪器在解决信号与图像处理中的逆问题方面展现出卓越性能。为保证收敛性，去噪器需满足非扩张性等利普希茨条件。然而，施加此类约束不可避免地会损害恢复性能。本文提出一种新颖的训练策略，对深度去噪器施加称为伪压缩性的较弱约束。通过研究雅可比矩阵的谱，揭示了不同去噪器假设之间的关联。推导出基于梯度下降和Ishikawa过程的有效算法，进一步利用严格伪压缩性假设得到基于半二次分裂和前向后向分裂的高效算法。所提算法理论上强收敛到不动点。提出基于全纯变换和泛函演算的训练策略来实施伪压缩去噪器假设。大量实验表明，与相关去噪器相比，伪压缩去噪器具有更优越的性能。所提方法在视觉效果和量化指标上均具有竞争力。