结构化随机过程的去噪
Denoising of structured random processes
摘要：
平稳过程$(X_i)_{i \in Z}$的去噪是信息论和统计信号处理中的一个经典和基本问题。尽管在设计有效的去噪算法方面取得了相当大的进展，但对于一般的模拟源，尚未找到理论上建立的计算有效的方法。例如，在将被噪声$Z^n$破坏的$X^n$去噪为$Y^n=X^n+Z^n$时，给定$X^n$的全分布，最小均方误差（MMSE）去噪器需要计算$E[X^n|Y^n]$。然而，对于一般源，计算$E[X^n|Y^n]$在计算上是非常具有挑战性的，如果不是不可行的话。在本文中，开始由贝叶斯设置，其中的源分布是完全已知的，一种新的去噪方法，即量化最大后验（Q-MAP）去噪，提出并分析其渐近性能在高信噪比制度。无论是无记忆源，和结构化的一阶马尔可夫源，它表明，渐近$σ$收敛到零，${1\over σ^2}E[(X_i-\hat{X}^{\rm Q-MAP}_i)^2]$实现的Q-MAP去噪收敛到源的信息维。对于所研究的无记忆源，该限制已知是最佳的。Q-MAP去噪器的一个主要优点是，与MMSE去噪器不同，它突出了要在其去噪中使用的源分布的关键属性。该特性显著地降低了近似Q-MAP去噪器的解的计算复杂性。此外，它自然会导致基于学习的降噪。使用ImageNet数据库进行训练，初步仿真结果探索了这种基于学习的去噪器在图像去噪中的性能。
英文摘要：
Denoising stationary process $(X_i)_{i \in Z}$ corrupted by additive white Gaussian noise is a classic and fundamental problem in information theory and statistical signal processing. Despite considerable progress in designing efficient denoising algorithms, for general analog sources, theoretically-founded computationally-efficient methods are yet to be found. For instance in denoising $X^n$ corrupted by noise $Z^n$ as $Y^n=X^n+Z^n$, given the full distribution of $X^n$, a minimum mean square error (MMSE) denoiser needs to compute $E[X^n|Y^n]$. However, for general sources, computing $E[X^n|Y^n]$ is computationally very challenging, if not infeasible. In this paper, starting by a Bayesian setup, where the source distribution is fully known, a novel denoising method, namely, quantized maximum a posteriori (Q-MAP) denoiser, is proposed and its asymptotic performance in the high signal to noise ratio regime is analyzed. Both for memoryless sources, and for structured first-order Markov sources, it is shown that, asymptotically, as $σ$ converges to zero, ${1\over σ^2}E[(X_i-\hat{X}^{\rm Q-MAP}_i)^2]$ achieved by Q-MAP denoiser converges to the information dimension of the source. For the studied memoryless sources, this limit is known to be optimal. A key advantage of the Q-MAP denoiser is that, unlike an MMSE denoiser, it highlights the key properties of the source distribution that are to be used in its denoising. This property dramatically reduces the computational complexity of approximating the solution of the Q-MAP denoiser. Additionally, it naturally leads to a learning-based denoiser. Using ImageNet database for training, initial simulation results exploring the performance of such a learning-based denoiser in image denoising are presented.
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