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

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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\(\Gamma \)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\(\Gamma \)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.

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Authors and Affiliations

  1. Engineering Sciences Laboratory, National School of Applied Sciences, Ibn Tofail University, Kenitra, Morocco

    Zakariae Abbad & Said Ouatik El Alaoui

  2. LRIT Rabat IT Center, Faculty of sciences, Mohammed V University, Rabat, Morocco

    Ahmed Drissi El Maliani

  3. FLSH, Mohammed V University in Rabat, Rabat, Morocco

    Mohammed El Hassouni

  4. Laboratory of Mathematical Sciences and Applications, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Fez, Morocco

    Mohamed Tahar Kadaoui Abbassi

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  5. Mohamed Tahar Kadaoui Abbassi
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Appendix

Appendix

The Christoffel symbols \(\Gamma _{ij}^{k}\) associated with the G\(\Gamma \)D manifold throughout the Fisher-Rao metric are highly nonlinear and cannot be simplified, so we will limit ourselves to the seventeen Christoffel symbols that are used in the geodesic equations. For the simplicity of the expressions of the Christoffel symbols, we will define them by:

$$\begin{aligned}&\Gamma _{\alpha \alpha }^{\alpha } = -\frac{A}{2\,\alpha \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\alpha \beta }^{\alpha } = \frac{C}{2\,\beta \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\alpha \lambda }^{\alpha } = -\frac{\psi (\lambda )}{2}, \Gamma _{\alpha \lambda }^{\beta } = - \frac{\beta ^2}{2\alpha }\\&\Gamma _{\beta \lambda }^{\alpha } = \frac{D}{2\,\beta ^2\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\alpha \alpha }^{\beta } = -\frac{\beta ^3\,\left( 2\,\lambda -2\,\lambda ^2\,\psi '\left( \lambda \right) +1\right) }{2\,\alpha ^2\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\beta \beta }^{\beta } = -\frac{E}{2\,\beta \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\beta \beta }^{\alpha } = -\frac{F}{2\,\beta ^3\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\lambda \lambda }^{\beta } = \frac{\beta \,\left( 2\,\lambda \,{\psi '\left( \lambda \right) }^2-2\,\psi '\left( \lambda \right) +\psi ''\left( \lambda \right) \right) }{-2\,\lambda ^2\,{\psi '\left( \lambda \right) }^2+2\,\psi '\left( \lambda \right) +2}\\&\Gamma _{\beta \lambda }^{\beta } = \frac{G}{-2\,\lambda ^2\,{\psi '\left( \lambda \right) }^2+2\,\psi '\left( \lambda \right) +2}\\&\Gamma _{\alpha \alpha }^{\lambda }= -\frac{\beta ^2\,\left( \lambda -\lambda ^2\,\psi '\left( \lambda \right) +1\right) }{2\,\alpha ^2\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\beta \beta }^{\lambda } = -\frac{H}{2\,\beta ^2\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\lambda \lambda }^{\lambda } = -\frac{\left( \lambda -1\right) \,\psi ''\left( \lambda \right) +2\,\psi '\left( \lambda \right) +\lambda ^2\,\psi '\left( \lambda \right) \,\psi ''\left( \lambda \right) }{-2\,\lambda ^2\,{\psi '\left( \lambda \right) }^2+2\,\psi '\left( \lambda \right) +2}\\&\Gamma _{\alpha \beta }^{\lambda } = -\frac{\left( \lambda \,\left( \lambda \,\psi '\left( \lambda \right) -1\right) -1\right) \,\left( \psi \left( \lambda \right) +\lambda \,\psi '\left( \lambda \right) +1\right) }{2\,\alpha \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\alpha \beta }^{\beta } = \frac{\beta \,\left( \psi \left( \lambda \right) \,\left( 2\,\lambda +1\right) -\lambda \,\psi '\left( \lambda \right) -2\,\lambda ^2\,\psi \left( \lambda \right) \,\psi '\left( \lambda \right) +1\right) }{2\,\alpha \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }\\&\Gamma _{\beta \lambda }^{\lambda } = \beta \,\left( \frac{\lambda \,\psi ''\left( \lambda \right) +2\,\psi '\left( \lambda \right) }{2\,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) }+\frac{1}{2}\right) \\&\Gamma _{\lambda \lambda }^{\alpha } = \frac{J}{2\,\beta \,\left( -\lambda ^2\,{\psi '\left( \lambda \right) }^2+\psi '\left( \lambda \right) +1\right) } \end{aligned}$$

Where:

$$\begin{aligned} A= & {} \left( \beta \,\left( \psi \left( \lambda \right) -\lambda \,\left( \psi '\left( \lambda \right) -2\,\psi \left( \lambda \right) \right. \right. \right. \\&\left. \left. +2\,\lambda \,\psi \left( \lambda \right) \,\psi '\left( \lambda \right) \right) +1\right) \\&\left. -2\,\lambda ^2\,{\psi '\left( \lambda \right) }^2+2\,\psi '\left( \lambda \right) +2 \right) \end{aligned}$$

Where \(\psi ^{(m)}(x) = \frac{d^{(m)}}{dx^{m}} \psi (x)\) is the polygamma function and \(\psi ^{(0)}(x)\) is the digamma function.

$$\begin{aligned} C= & {} \left( {\psi \left( \lambda \right) }^2-2\,\lambda \,\psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) -\psi '\left( \lambda \right) \right) \right. \\&\left. +\lambda ^2\,\left( 2\,{\psi \left( \lambda \right) }^2+\psi '\left( \lambda \right) \right) \,\psi '\left( \lambda \right) +1 \right) \\ D= & {} \left( \alpha \,\left( \left( \psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) +2\right) -1\right) \,\psi '\left( \lambda \right) \right. \right. \\&+{\psi \left( \lambda \right) }^2-3\,{\psi '\left( \lambda \right) }^2-\lambda \,\left( 2\,\psi \left( \lambda \right) \,{\psi '\left( \lambda \right) }^2\right. \\&-\left( \psi \left( \lambda \right) -\psi '\left( \lambda \right) \right) \,\psi ''\left( \lambda \right) \\&+\lambda \,\left( \left( {\psi \left( \lambda \right) }^2-\psi '\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right. \\&\left. \left. \left. \left. +\psi \left( \lambda \right) \,\psi ''\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) \right) \right) \\ E= & {} \left( \psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) +2\right) \right. \\&+3\,\psi '\left( \lambda \right) +2\,\lambda \,\psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) -\psi '\left( \lambda \right) \right. \\&\left. \left. +\psi ''\left( \lambda \right) -2\,\lambda \,\left( {\psi \left( \lambda \right) }^2+\psi '\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) +2 \right) \\ F= & {} \left( \alpha \,\left( \lambda \,\left( 2\,{\psi \left( \lambda \right) }^3+{\psi '\left( \lambda \right) }^2\right. \right. \right. \\&+\left( \psi \left( \lambda \right) +1\right) \,\psi ''\left( \lambda \right) \\&\quad -3\,{\psi \left( \lambda \right) }^2\,\psi '\left( \lambda \right) -\lambda \,\left( 2\,{\psi \left( \lambda \right) }^3\right. \\&\left. \left. -\psi ''\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) -\left( \psi \left( \lambda \right) -1\right) \,\psi '\left( \lambda \right) \\&\left. \left. +{\psi \left( \lambda \right) }^2\,\left( \psi \left( \lambda \right) +3\right) \right) \right) \\ G= & {} \left( \psi \left( \lambda \right) \,\left( \psi '\left( \lambda \right) +1\right) \right. \\&+2\,\psi '\left( \lambda \right) -\lambda \,\left( 2\,{\psi '\left( \lambda \right) }^2-\psi ''\left( \lambda \right) \right. \\&\left. \left. +\lambda \,\left( \psi \left( \lambda \right) \,\psi '\left( \lambda \right) +\psi ''\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) \right) \\ H= & {} \left( \psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) +2\right) -\lambda \,\left( \lambda \,\left( \left( {\psi \left( \lambda \right) }^2\right. \right. \right. \right. \\&\left. +3\,\psi '\left( \lambda \right) \right) \,\psi '\left( \lambda \right) +\psi ''\left( \lambda \right) \\&\quad \left. +\lambda \,\left( 2\,\psi \left( \lambda \right) \,\psi '\left( \lambda \right) +\psi ''\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) \\&\quad -\psi \left( \lambda \right) \,\left( \psi \left( \lambda \right) +2\,\psi '\left( \lambda \right) +2\right) +\psi '\left( \lambda \right) \\&\quad \left. \left. -\psi ''\left( \lambda \right) \right) +3\,\psi '\left( \lambda \right) +2 \right) \\ J= & {} \left( \alpha \,\left( 2\,{\psi '\left( \lambda \right) }^2-\psi \left( \lambda \right) \,\left( 2\,\psi '\left( \lambda \right) -\psi ''\left( \lambda \right) \right) +\psi ''\left( \lambda \right) \right. \right. \\&\left. \left. +\lambda \,\left( 2\,\psi \left( \lambda \right) \,\psi '\left( \lambda \right) +\psi ''\left( \lambda \right) \right) \,\psi '\left( \lambda \right) \right) \right) \end{aligned}$$

The expressions of Christoffel symbols \(\Gamma _{ij}^{k}\) associated with the submanifold defined by fixed scale parameter are defined by:

$$\begin{aligned}&\Gamma _{\beta \beta }^{\beta } = -\frac{K}{M}\quad ,\quad \Gamma _{\beta \beta }^{\lambda } = -\frac{N}{P}\quad ,\quad \Gamma _{\beta \lambda }^{\beta } = \frac{Q}{R}\\&\Gamma _{\lambda \lambda }^{\beta } = -\frac{V}{R}\quad ,\quad \Gamma _{\lambda \lambda }^{\lambda } = \frac{W}{R}\quad ,\quad \Gamma _{\beta \lambda }^{\lambda } = \frac{S}{M} \end{aligned}$$

Where:

$$\begin{aligned} K= & {} \left( \left( (-2\psi (\lambda )^2+\psi (\lambda )^3+2\psi '(\lambda )\right. \right. \\&+7\psi (\lambda )\psi '(\lambda )+4\lambda \psi ^2\psi '(\lambda )\\&\left. \left. +2\lambda \psi '(\lambda )^2+\lambda \psi (\lambda )\psi ''(\lambda ))\right) \right) \\ M= & {} \left( \left( 2\beta (-\psi (\lambda )^2+\psi '(\lambda )\right. \right. \\&+2\psi (\lambda )\psi '(\lambda )+\lambda \psi (\lambda )^2\\&\left. \left. +\psi '(\lambda )+\psi '(\lambda )^2\right) \right) \\ N= & {} \left( \left( (1+2\psi (\lambda )+\lambda \psi (\lambda )^2+\lambda \psi '(\lambda ))(psi(\lambda )^2\right. \right. \\&\left. \left. +3\psi '(\lambda )+2\lambda \psi (\lambda )\psi '(\lambda )+\lambda \psi ''(\lambda )))\right) \right) \\ P= & {} \left( \left( 2\beta ^2(-\psi (\lambda )^2+\psi '(\lambda )\right. \right. \\&+2\psi (\lambda )\psi '(\lambda )+\lambda \psi (\lambda )^2\\&\left. \left. +\psi '(\lambda )+\psi '(\lambda )^2\right) \right) \\ R= & {} \left( \left( 2(-\psi (\lambda )^2+\psi '(\lambda )+2\psi (\lambda )\psi '(\lambda )\right. \right. \\&\left. \left. +\lambda \psi (\lambda )^2\psi '(\lambda )+\lambda \psi '(\lambda )^2)\right) \right) \\ Q= & {} \left( \left( \psi '(\lambda )(\psi (\lambda )^2+3\psi '(\lambda )\right. \right. \\&\left. \left. +2\lambda \psi (\lambda )\psi '(\lambda )+\lambda \psi ''(\lambda ))\right) \right) \\ S= & {} \left( \left( \psi (\lambda )(\psi (\lambda )^2+3\psi '(\lambda )\right. \right. \\&\left. \left. +2\lambda \psi (\lambda )\psi '(\lambda )+\lambda \psi ''(\lambda ))\right) \right) \\ V= & {} \left( \left( \beta (2\psi '(\lambda )^2-\psi (\lambda )\psi ''(\lambda )) \right) \right) \\ W= & {} \left( -2\psi (\lambda )\psi '(\lambda )+\psi ''(\lambda )\right. \\&\left. +2\psi (\lambda )\psi ''(\lambda )+\lambda \psi (\lambda )^2\psi ''(\lambda )+\lambda \psi '(\lambda )\psi ''(\lambda )\right) \end{aligned}$$

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Abbad, Z., Maliani, A.D.E., Alaoui, S.O.E. et al. The Geodesic Distance on the Generalized Gamma Manifold for Texture Image Retrieval. J Math Imaging Vis 64, 243–260 (2022). https://doi.org/10.1007/s10851-021-01063-x

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