The aim of this paper is to describe a novel non-parametric noise reduction technique from the point of view of Bayesian inference that may automatically improve the signal-to-noise ratio of one- and two-dimensional data, such as e.g. astronomical images and spectra. The algorithm iteratively evaluates possible smoothed versions of the data, the smooth models, obtaining an estimation of the underlying signal that is statistically compatible with the noisy measurements. Iterations stop based on the evidence and the $\chi^2$ statistic of the last smooth model, and we compute the expected value of the signal as a weighted average of the whole set of smooth models. In this paper, we explain the mathematical formalism and numerical implementation of the algorithm, and we evaluate its performance in terms of the peak signal to noise ratio, the structural similarity index, and the time payload, using a battery of real astronomical observations. Our Fully Adaptive Bayesian Algorithm for Data Analysis (FABADA) yields results that, without any parameter tuning, are comparable to standard image processing algorithms whose parameters have been optimized based on the true signal to be recovered, something that is impossible in a real application. State-of-the-art non-parametric methods, such as BM3D, offer slightly better performance at high signal-to-noise ratio, while our algorithm is significantly more accurate for extremely noisy data (higher than $20-40\%$ relative errors, a situation of particular interest in the field of astronomy). In this range, the standard deviation of the residuals obtained by our reconstruction may become more than an order of magnitude lower than that of the original measurements. The source code needed to reproduce all the results presented in this report, including the implementation of the method, is publicly available at https://github.com/PabloMSanAla/fabada

翻译：本文旨在从贝叶斯推断的视角描述一种新颖的非参数噪声抑制技术，该技术可自动提升一维和二维数据（如天文图像及光谱）的信噪比。算法通过迭代评估数据的潜在平滑版本（即平滑模型），获得与含噪观测统计兼容的潜在信号估计。迭代过程基于证据和最新平滑模型的χ²统计量终止，并计算信号期望值为所有平滑模型的加权平均值。本文阐释了该算法的数学形式化与数值实现，并利用一系列真实天文观测数据，基于峰值信噪比、结构相似性指数及时间开销对其性能进行评测。我们的完全自适应贝叶斯数据分析算法（FABADA）无需任何参数调优，即可达到与标准图像处理算法（其参数已基于待恢复真实信号优化）相当的结果，这在真实应用中是不可能的。前沿非参数方法（如BM3D）在高信噪比下表现略优，而我们的算法在极端噪声数据（相对误差高于20-40%，这在天文领域尤为常见）中显著更精确。在此范围内，本算法重建残差的标准差可较原始测量值降低一个数量级以上。复现本报告所有结果（含方法实现）的源代码已公开发布于https://github.com/PabloMSanAla/fabada。