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, which is quite robust for some scenarios but fails in some others, 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 the 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：本文中，作者开发了基于深度学习的回归方法，用于平纹帆布分析中的纱线密度估计。以往的方法包括傅里叶分析（在某些场景下较为稳健，但在其他场景下失效）、机器学习工具（需对画作进行预标注）以及纱线交叉点分割（无需预标注即可在所有场景下提供良好估计）。分割方法因需在定位交叉点后才能估计密度而耗时。在此新方案中，我们通过回归深度学习模型直接从图像计算纱线密度，避免了这一步骤。我们还对输入图像的初始预处理进行了改进，从而影响最终误差。我们提出并分析了多种模型以保留最优者，进一步通过引入半监督方法降低了密度估计误差。通过Ribera、Velázquez和Poussin的作品分析了我们新算法的性能，并将结果与先前方法进行了比较。最后，该方法被应用于支持普拉多博物馆一件杰作的作者归属变更。