Rao-Geodesic distance on the generalized gamma manifold: Study of three sub-manifolds and application in the Texture Retrieval domain

Texture retrieval is a very challenging issue that combines two major tasks, feature extraction (FE) and similarity measurement (SM). The FE step consists of computing some texture characteristics that, even small in size, describe perfectly the texture content by using stochastic models like Gaussian or non-Gaussian (Gamma, Weibull,...) distributions. The estimated parameters will form the vector of characteristics of the texture. Secondly, the SM step consists of deriving a distance on the chosen model manifold, in order to nd the closest textures of a query using their characteristic vectors. In this context, the commonly used similarity measure is the Kullback-leibler divergence (KLD). Nevertheless, KLD is not a distance since it does not satisfy symmetry and triangular inequality properties. In this paper we propose Geodesic distance (GD) as a similarity measure on the Generalized Gamma (GG) manifold, in order to illustrate the importance of geometric reasoning in the image retrieval field. The principle idea is the use of the distances between the probability distributions in precise manner through the GD, as an application in the SM between the texture images which are represented by the parameters of the probability distributions. And that can be a good illustration of the value of the Riemannian geometry through statistical manifold in an applied field such as the texture retrieval. Generalized Gamma is a three parameters distribution that covers Gamma, Weibull and Exponential models as special cases, which allowed the modeling of a wide range of texture families. We take advantage of this property in order to make a prior study of the GD for the Gamma, Weibull and Exponential sub-manifolds due to the cumbersomeness of deriving GD for the generalized gamma directly. Experiments are carried out considering texture retrieval in the domains of dual tree complex wavelet transform and steerable pyramid transform, using the Vistex texture database. Results show that GD achieves performances that are close or higher to KLD for the three sub-manifolds, which is of a great interest since GD is a Riemannian metric contrary to KLD.