The Geodesic Distance on the Generalized Gamma Manifold for Texture Image Retrieval

In this paper, the similarity measurement issue, in the context of texture images comparison, is tackled from a geometrical point of view by computing the Rao Geodesic distance on the Generalized Gamma distributions (G Γ D) manifold. This latter permits a generic and flexible characterization thanks...

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Published in:	Journal of mathematical imaging and vision Vol. 64; no. 3; pp. 243 - 260
Main Authors:	Abbad, Zakariae, Maliani, Ahmed Drissi El, Alaoui, Said Ouatik El, Hassouni, Mohammed El, Abbassi, Mohamed Tahar Kadaoui
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.03.2022 Springer Nature B.V
Subjects:	Applications of Mathematics Computation Computer Science Differential geometry Divergence Image management Image Processing and Computer Vision Image retrieval Manifolds (mathematics) Mathematical Methods in Physics Mathematical models Parameter modification Polynomials Signal,Image and Speech Processing Similarity Texture Information geometry Geodesic distance Texture retrieval Generalized gamma manifold
ISSN:	0924-9907, 1573-7683
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Abstract	In this paper, the similarity measurement issue, in the context of texture images comparison, is tackled from a geometrical point of view by computing the Rao Geodesic distance on the Generalized Gamma distributions (G Γ D) manifold. This latter permits a generic and flexible characterization thanks to its three-parameters modeling. We take advantage of information geometry tools to consider the G Γ D as a geometrical manifold and thus to define the geodesic distance (GD) as a real and intuitive similarity measure rather than the purely statistical Kullback-Leibler divergence. However, the three-parameter space turns out to be cumbersome when it comes to solving the geodesic equations. This explains why the main studies that tried to solve this problem have been content to use mappings to some embedded submanifolds to modify and approximate the accurate geodesic model by computing geodesic distances only on integrable submanifolds. Our main contribution is to approximate the GD in a more general manner, i.e considering all cases of the manifold coordinates. We managed to derive a closed-form of the modified geodesic model on sub-manifolds when the shape parameters are fixed. In the case of the fixed scale parameters, the geodesic equations are solved and a numerical approximation is directly applied to derive the geodesic distance. When the three parameters are allowed to vary, we approximate the GD using two different ways. The first is treated by the mean of a polynomial approach, while the second is subject to a graph-based approach. The proposed approaches are evaluated considering the Content-Based Texture Retrieval (CBTR) application on three different databases (VisTex, Brodatz, and USPTex) in order to show their effectiveness over the state-of-the-art methods.
AbstractList	In this paper, the similarity measurement issue, in the context of texture images comparison, is tackled from a geometrical point of view by computing the Rao Geodesic distance on the Generalized Gamma distributions (GΓD) manifold. This latter permits a generic and flexible characterization thanks to its three-parameters modeling. We take advantage of information geometry tools to consider the GΓD as a geometrical manifold and thus to define the geodesic distance (GD) as a real and intuitive similarity measure rather than the purely statistical Kullback-Leibler divergence. However, the three-parameter space turns out to be cumbersome when it comes to solving the geodesic equations. This explains why the main studies that tried to solve this problem have been content to use mappings to some embedded submanifolds to modify and approximate the accurate geodesic model by computing geodesic distances only on integrable submanifolds. Our main contribution is to approximate the GD in a more general manner, i.e considering all cases of the manifold coordinates. We managed to derive a closed-form of the modified geodesic model on sub-manifolds when the shape parameters are fixed. In the case of the fixed scale parameters, the geodesic equations are solved and a numerical approximation is directly applied to derive the geodesic distance. When the three parameters are allowed to vary, we approximate the GD using two different ways. The first is treated by the mean of a polynomial approach, while the second is subject to a graph-based approach. The proposed approaches are evaluated considering the Content-Based Texture Retrieval (CBTR) application on three different databases (VisTex, Brodatz, and USPTex) in order to show their effectiveness over the state-of-the-art methods. In this paper, the similarity measurement issue, in the context of texture images comparison, is tackled from a geometrical point of view by computing the Rao Geodesic distance on the Generalized Gamma distributions (G Γ D) manifold. This latter permits a generic and flexible characterization thanks to its three-parameters modeling. We take advantage of information geometry tools to consider the G Γ D as a geometrical manifold and thus to define the geodesic distance (GD) as a real and intuitive similarity measure rather than the purely statistical Kullback-Leibler divergence. However, the three-parameter space turns out to be cumbersome when it comes to solving the geodesic equations. This explains why the main studies that tried to solve this problem have been content to use mappings to some embedded submanifolds to modify and approximate the accurate geodesic model by computing geodesic distances only on integrable submanifolds. Our main contribution is to approximate the GD in a more general manner, i.e considering all cases of the manifold coordinates. We managed to derive a closed-form of the modified geodesic model on sub-manifolds when the shape parameters are fixed. In the case of the fixed scale parameters, the geodesic equations are solved and a numerical approximation is directly applied to derive the geodesic distance. When the three parameters are allowed to vary, we approximate the GD using two different ways. The first is treated by the mean of a polynomial approach, while the second is subject to a graph-based approach. The proposed approaches are evaluated considering the Content-Based Texture Retrieval (CBTR) application on three different databases (VisTex, Brodatz, and USPTex) in order to show their effectiveness over the state-of-the-art methods.
Author	Abbad, Zakariae Maliani, Ahmed Drissi El Alaoui, Said Ouatik El Abbassi, Mohamed Tahar Kadaoui Hassouni, Mohammed El
Author_xml	– sequence: 1 givenname: Zakariae orcidid: 0000-0003-4911-2022 surname: Abbad fullname: Abbad, Zakariae organization: Engineering Sciences Laboratory, National School of Applied Sciences, Ibn Tofail University – sequence: 2 givenname: Ahmed Drissi El surname: Maliani fullname: Maliani, Ahmed Drissi El organization: LRIT Rabat IT Center, Faculty of sciences, Mohammed V University – sequence: 3 givenname: Said Ouatik El surname: Alaoui fullname: Alaoui, Said Ouatik El organization: Engineering Sciences Laboratory, National School of Applied Sciences, Ibn Tofail University – sequence: 4 givenname: Mohammed El surname: Hassouni fullname: Hassouni, Mohammed El organization: FLSH, Mohammed V University in Rabat – sequence: 5 givenname: Mohamed Tahar Kadaoui surname: Abbassi fullname: Abbassi, Mohamed Tahar Kadaoui organization: Laboratory of Mathematical Sciences and Applications, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah
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Snippet	In this paper, the similarity measurement issue, in the context of texture images comparison, is tackled from a geometrical point of view by computing the Rao...
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SubjectTerms	Applications of Mathematics Computation Computer Science Differential geometry Divergence Image management Image Processing and Computer Vision Image retrieval Manifolds (mathematics) Mathematical Methods in Physics Mathematical models Parameter modification Polynomials Signal,Image and Speech Processing Similarity Texture
Title	The Geodesic Distance on the Generalized Gamma Manifold for Texture Image Retrieval
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