1077-2626 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVCG.2017.2769050, IEEE Transactions on Visualization and Computer Graphics Transformation of the Multivariate Generalized Gaussian Distribution for Image Editing Hristina Hristova University of Rennes 1 Rennes, France hristina.hristova@irisa.fr Olivier Le Meur University of Rennes 1 Rennes, France olivier.lemeur@irisa.fr Rémi Cozot University of Rennes 1 Rennes, France remi.cozot@irisa.fr Kadi Bouatouch University of Rennes 1 Rennes, France kadi.bouatouch@irisa.fr Abstract—Multivariate generalized Gaussian distributions (MGGDs) have aroused a great interest in the image processing community thanks to their ability to describe accurately var- ious image features, such as image gradient fields. However, so far their applicability has been limited by the lack of a transformation between two of these parametric distributions. In this paper, we propose a novel transformation between MGGDs, consisting of an optimal transportation of the second- order statistics and a stochastic-based shape parameter trans- formation. We employ the proposed transformation between MGGDs for a color transfer and a gradient transfer between images. We also propose a new simultaneous transfer of color and gradient, which we apply for image color correction. 1. Introduction Multivariate Gaussian distribution (MGD) is commonly used in image processing applications to fit the distributions of image features [1]. Thanks to the analytically tractable density function of the MGD, there exists a number of MGD-based statistical transformations, applied to carry out a color transfer between images [1]. Such transformations benefit from the analytical properties of the MGD, but they also depend on how accurately the MGD approximates the given distributions. To this end, such transformations fail when applied to heavy-tailed distributions (e.g. sparse distributions with exponentially unbounded tails [2]) such as the distributions of image gradient fields and wavelet coefficients. To overcome this limitation, the multivariate Laplacian distribution (MLD) is usually adopted to fit sparse distributions. The MLD is often used in speech recognition to characterize discrete Fourier coefficients [3] as well as in image processing to model gradient fields [4]. As the MGD and the MLD are special cases of the multivariate generalized Gaussian distribution (MGGD) with shape parameters equal to 1 and 0.5 respectively, the ac- curacy of the statistical approximation could be improved by relaxing the constraint on the shape parameters. The MGGD and its properties have been introduced in [5], [6]. The property of the MGGD to model accurately sparse distributions (for a shape parameter less than 1) has been exploited in many applications, such as texture discrimi- nation and texture retrieval [7], [8], [9], image and video segmentation [10], image de-noising [11], [12]. Including the MGD and the MLD as special cases, the MGGD with an unconstrained shape parameter is likely to accurately fit a wide class of image feature distributions, including those of color, gradient, wavelet coefficients, etc. Despite the major role of the MGGD in image pro- cessing, no transformation between two MGGDs has so far been tackled. To address this limitation, we propose a novel transformation between input and target sets of sample vectors, following an MGGD with given parameters. The transformation consists of two main steps. First, a linear optimization transforms the input sample vectors so that their distribution approximates the second-order statistics of the target distribution. Second, a stochastic-based transfer modifies the sample vectors, computed during the first step, so that their distribution gets similar to the target distribution (in terms of both scale and shape). We demonstrate the potential of our method for color and gradient transfers between images. The proposed transformation allows to simultaneously transfer a number of image features by tak- ing into account the dependencies between them. Unlike existing image editing techniques, which are limited to only color transformations, we demonstrate the efficiency of our method for a p-dimensional feature transfer between images. To sum up, the main contributions of this paper are twofold: • a novel transformation between MGGDs, consisting of a linear Monge-Kantorovich transformation and a stochastic-based shape parameter transfer; • two novel applications, carried out by the proposed MGGD-based transformation: – a transfer of gradient between images; – a multidimensional transfer of color and gradient between images. The paper is organized as follows. Section 2 presents related work on existing transformations between MGDs as well as applications of these transformations. Section 3 introduces our transformation between MGGDs and pro- vides details about the evaluation of the transformation. Applications and results are presented in section 4. Finally, the last section concludes the paper.