Recently, much progress has been made in unsupervised denoising learning. However, existing methods more or less rely on some assumptions on the signal and/or degradation model, which limits their practical performance. How to construct an optimal criterion for unsupervised denoising learning without any prior knowledge on the degradation model is still an open question. Toward answering this question, this work proposes a criterion for unsupervised denoising learning based on the optimal transport theory. This criterion has favorable properties, e.g., approximately maximal preservation of the information of the signal, whilst achieving perceptual reconstruction. Furthermore, though a relaxed unconstrained formulation is used in practical implementation, we prove that the relaxed formulation in theory has the same solution as the original constrained formulation. Experiments on synthetic and real-world data, including realistic photographic, microscopy, depth, and raw depth images, demonstrate that the proposed method even compares favorably with supervised methods, e.g., approaching the PSNR of supervised methods while having better perceptual quality. Particularly, for spatially correlated noise and realistic microscopy images, the proposed method not only achieves better perceptual quality but also has higher PSNR than supervised methods. Besides, it shows remarkable superiority in harsh practical conditions with complex noise, e.g., raw depth images. Code is available at https://github.com/wangweiSJTU/OTUR .

Recent studies in perceptual lossy compression have highlighted the advantage of stochastic coding with shared randomness between encoder and decoder, over deterministic encoding in the regime of “high perceptual quality”. While the theoretical benefits of stochastic coding have been well-established and demonstrated by analytic examples, its practical realization remains challenging and largely unexplored. In this work, we propose a practical learning framework for stochastic coding that effectively realizes its theoretical advantages. Starting with a theoretically optimal scheme, we develop an implementation closely approximates it through a two-stage training process: learning a stochastic encoder, followed by a stochastic decoder which is modeled as an optimal transport problem conditioned on minimum mean square error (MMSE) decoding. Additionally, for training stochastic coding models, we prove the equivalence between quantized representation and “noisy” representation. Based on this insight, we introduce a quantization-free training method that effectively addresses the non-differentiability challenge posed by quantization. Experiments on a circular distribution example and the MNIST dataset validate our findings and demonstrate the effectiveness of the proposed method.