In this work the authors develop regression approaches based on deep learning to perform thread density estimation for plain weave canvas analysis. Previous approaches were based on Fourier analysis, that are quite robust for some scenarios but fail in some other, in machine learning tools, that involve pre-labeling of the painting at hand, or the segmentation of thread crossing points, that provides good estimations in all scenarios with no need of pre-labeling. The segmentation approach is time-consuming as estimation of the densities is performed after locating the crossing points. In this novel proposal, we avoid this step by computing the density of threads directly from the image with a regression deep learning model. We also incorporate some improvements in the initial preprocessing of the input image with an impact on the final error. Several models are proposed and analyzed to retain the best one. Furthermore, we further reduce the density estimation error by introducing a semi-supervised approach. The performance of our novel algorithm is analyzed with works by Ribera, Vel\'azquez, and Poussin where we compare our results to the ones of previous approaches. Finally, the method is put into practice to support the change of authorship or a masterpiece at the Museo del Prado.

翻译：本文作者开发了基于深度学习的回归方法，用于执行平纹织物画布的经纬线密度估计。以往的方法包括：基于傅里叶分析的方法（在部分场景下相当稳健但在其他场景中失效）、基于机器学习工具的方法（需对绘画作品进行预标注）或基于交叉点分割的方法（无需预标注即可在所有场景中获得良好估计）。然而，分割方法在定位交叉点后才能进行密度估计，较为耗时。在本项创新研究中，我们通过采用回归深度学习模型直接从图像计算经纬线密度，从而省去了这一步骤。我们还对输入图像的初始预处理进行了改进，从而降低了最终误差。我们提出并分析了多种模型以确定最优方案。此外，通过引入半监督方法进一步降低了密度估计误差。我们使用里贝拉、委拉斯开兹和普桑的作品分析了新算法的性能，并将结果与以往方法进行了对比。最终，该方法被应用于普拉多博物馆一幅杰作的作者归属变更验证。