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：本文作者开发了基于深度学习的回归方法，用于平纹帆布画布的经纬线密度估算。先前方法基于傅里叶分析（在某些场景下相当稳健，但在其他场景中失效）、机器学习工具（涉及对特定画作进行预标注）、或经纬线交叉点分割（无需预标注即可在所有场景中提供良好估算）。交叉点分割法耗时较长，因为需在定位交叉点后估算密度。本项创新提案通过回归深度学习模型直接从图像计算线密度，从而跳过此步骤。我们还对输入图像的初始预处理进行了改进，对最终误差产生了影响。提出并分析了多种模型以保留最优方案。此外，通过引入半监督方法，进一步降低了密度估算误差。通过分析里贝拉、委拉斯开兹和普桑的作品，将本算法性能与先前方法的结果进行比较。最后，该方法实际应用于支持普拉多博物馆一幅杰作的作者身份变更鉴定。