This paper addresses the automatic colorization problem, which converts a gray-scale image to a colorized one. Recent deep-learning approaches can colorize automatically grayscale images. However, when it comes to different scenes which contain distinct color styles, it is difficult to accurately capture the color characteristics. In this work, we propose a fully automatic colorization approach based on Symmetric Positive Definite (SPD) Manifold Learning with a generative adversarial network (SPDGAN) that improves the quality of the colorization results. Our SPDGAN model establishes an adversarial game between two discriminators and a generator. The latter is based on ResNet architecture with few alterations. Its goal is to generate fake colorized images without losing color information across layers through residual connections. Then, we employ two discriminators from different domains. The first one is devoted to the image pixel domain, while the second one is to the Riemann manifold domain which helps to avoid color misalignment. Extensive experiments are conducted on the Places365 and COCO-stuff databases to test the effect of each component of our SPDGAN. In addition, quantitative and qualitative comparisons with state-of-the-art methods demonstrate the effectiveness of our model by achieving more realistic colorized images with less artifacts visually, and good results of PSNR, SSIM, and FID values.

Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. This paper provides a twofold contribution to the topic. First, we propose a Gaussian Mixture Model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, called PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not depend on the usually large number of samples and can be solved efficiently by an (inertial) proximal alternating linearized minimization algorithm. Second, we apply our PCA-GMM for the superresolution of 2D and 3D material images based on the approach of Sandeep and Jacob. Numerical results confirm the moderate influence of the dimensionality reduction on the overall superresolution result.