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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.

Single Image Super Resolution (SISR) methods aim to recover the clean images in high resolution from low resolution observations.A family of patch-based approaches have received considerable attention and development. The minimum mean square error (MMSE) methodis a powerful image restoration method that uses a probability model on the patches of images. This paper proposes an algorithm to learn a jointgeneralized Gaussian mixture model (GGMM) from a pair of the low resolution patches and the corresponding high resolution patches fromthe reference data. We then reconstruct the high resolution image based on the MMSE method. Our numerical evaluations indicate that theMMSE-GGMM method competes with other state of the art methods.

Elliptically-contoured distributions (ECD) play a significant role, in computer vision, image processing, radar, and biomedical signal processing. Maximum likelihood. estimation (MLE) of ECD leads to a system of non-linear equations, most-often addressed using fixed-point (FP) methods. Unfortunately, the computation time required for these methods is unacceptably long, for large-scale or high-dimensional datasets. To overcome this difficulty, the present work introduces a Riemannian optimisation method, the information stochastic gradient (ISG). The ISG is an online (recursive) method, which achieves the same performance as MLE, for large-scale datasets, while requiring modest memory and time resources. To develop the ISG method, the Riemannian information gradient is derived taking into account the product manifold associated to the underlying parameter space of the ECD. From this information gradient definition, we define also, the information deterministic gradient (IDG), an offline (batch) method, which is an alternative, for moderate-sized datasets. The present work formulates these two methods, and demonstrates their performance through numerical simulations. Two applications, to image re-colorization, and to texture classification, are also worked out.

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.

A new Riemannian geometry for the Compound Gaussian distribution is proposed. In particular, the Fisher information metric is obtained, along with corresponding geodesics and distance function. This new geometry is applied on a change detection problem on Multivariate Image Times Series: a recursive approach based on Riemannian optimization is developed. As shown on simulated data, it allows to reach optimal performance while being computationally more efficient., Comment: 13 pages

Most of existing superpixel methods are designed to segment standard planar images as pre-processing for computer vision pipelines. Nevertheless, the increasing number of applications based on wide angle capture devices, mainly generating 360{\deg} spherical images, have enforced the need for dedicated superpixel approaches. In this paper, we introduce a new superpixel method for spherical images called SphSPS (for Spherical Shortest Path-based Superpixels). Our approach respects the spherical geometry and generalizes the notion of shortest path between a pixel and a superpixel center on the 3D spherical acquisition space. We show that the feature information on such path can be efficiently integrated into our clustering framework and jointly improves the respect of object contours and the shape regularity. To relevantly evaluate this last aspect in the spherical space, we also generalize a planar global regularity metric. Finally, the proposed SphSPS method obtains significantly better performance than both planar and recent spherical superpixel approaches on the reference 360{\deg} spherical panorama segmentation dataset.

Superpixels are widely used in computer vision applications. Nevertheless, decomposition methods may still fail to efficiently cluster image pixels according to their local texture. In this paper, we propose a new Nearest Neighbor-based Superpixel Clustering (NNSC) method to generate texture-aware superpixels in a limited computational time compared to previous approaches. We introduce a new clustering framework using patch-based nearest neighbor matching, while most existing methods are based on a pixel-wise K-means clustering. Therefore, we directly group pixels in the patch space enabling to capture texture information. We demonstrate the efficiency of our method with favorable comparison in terms of segmentation performances on both standard color and texture datasets. We also show the computational efficiency of NNSC compared to recent texture-aware superpixel methods.

Most superpixel algorithms compute a trade-off between spatial and color features at the pixel level. Hence, they may need fine parameter tuning to balance the two measures, and highly fail to group pixels with similar local texture properties. In this paper, we address these issues with a new Texture-Aware SuperPixel (TASP) method. To accurately segment textured and smooth areas, TASP automatically adjusts its spatial constraint according to the local feature variance. Then, to ensure texture homogeneity within superpixels, a new pixel to superpixel patch-based distance is proposed. TASP outperforms the segmentation accuracy of the state-of-the-art methods on texture and also natural color image datasets.

The present paper shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry. Precisely, the paper features a new theorem, which states that the Rao-Fisher information metric of any location-scale model, defined on a Riemannian manifold, is a warped Riemannian metric, whenever this model is invariant under the action of some Lie group. This theorem is a valuable tool in finding the expression of the Rao-Fisher information metric of location-scale models defined on high-dimensional Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by only two functions of a single variable, irrespective of the dimension of the underlying Riemannian manifold. Starting from this theorem, several original contributions are made. The expression of the Rao-Fisher information metric of the Riemannian Gaussian model is provided, for the first time in the literature. A generalised definition of the Mahalanobis distance is introduced, which is applicable to any location-scale model defined on a Riemannian manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher information metric defined in terms of warped Riemannian metrics. Finally, using a mixture of analytical and numerical computations, it is shown that the parameter space of the von Mises-Fisher model of $n$-dimensional directional data, when equipped with its Rao-Fisher information metric, becomes a Hadamard manifold, a simply-connected complete Riemannian manifold of negative sectional curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be proved for any value of $n$., Comment: first version, before submission

This paper argues that a class of Riemannian metrics, called warped metrics, plays a fundamental role in statistical problems involving location-scale models. The paper reports three new results : i) the Rao-Fisher metric of any location-scale model is a warped metric, provided that this model satisfies a natural invariance condition, ii) the analytic expression of the sectional curvature of this metric, iii) the exact analytic solution of the geodesic equation of this metric. The paper applies these new results to several examples of interest, where it shows that warped metrics turn location-scale models into complete Riemannian manifolds of negative sectional curvature. This is a very suitable situation for developing algorithms which solve problems of classification and on-line estimation. Thus, by revealing the connection between warped metrics and location-scale models, the present paper paves the way to the introduction of new efficient statistical algorithms., Comment: preprint of a submission to GSI 2017 conference

Data which lie in the space $\mathcal{P}_{m\,}$, of $m \times m$ symmetric positive definite matrices, (sometimes called tensor data), play a fundamental role in applications including medical imaging, computer vision, and radar signal processing. An open challenge, for these applications, is to find a class of probability distributions, which is able to capture the statistical properties of data in $\mathcal{P}_{m\,}$, as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on $\mathcal{P}_{m\,}$. Distributions of this kind were first considered by Pennec in $2006$. However, the present paper gives an exact expression of their probability density function for the first time in existing literature. This leads to two original contributions. First, a detailed study of statistical inference for Riemannian Gaussian distributions, uncovering the connection between maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and implementation of an expectation-maximisation algorithm, for the estimation of mixtures of Riemannian Gaussian distributions. The paper applies this new algorithm, to the classification of data in $\mathcal{P}_{m\,}$, (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches., Comment: 21 pages, 1 table; accepted for publication in IEEE Trans Inf Theory

This paper focuses on the denoising and enhancing of 3-D reflection seismic data. We propose a pre-processing step based on a non linear diffusion filtering leading to a better detection of seismic faults. The non linear diffusion approaches are based on the definition of a partial differential equation that allows us to simplify the images without blurring relevant details or discontinuities. Computing the structure tensor which provides information on the local orientation of the geological layers, we propose to drive the diffusion along these layers using a new approach called SFPD (Seismic Fault Preserving Diffusion). In SFPD, the eigenvalues of the tensor are fixed according to a confidence measure that takes into account the regularity of the local seismic structure. Results on both synthesized and real 3-D blocks show the efficiency of the proposed approach., Comment: 10 pages