Generic multivariate model for color texture classification in RGB color space

Abstract

This paper presents a new method for modeling magnitudes of dual-tree complex wavelet coefficients, in the context of color texture classification. Based on the characterization of dependency between RGB color components, Gaussian copula associated with Generalized Gamma marginal function is proposed to design the multivariate generalized Gamma density (MG$\Gamma$D) modeling. MG$\Gamma$D has the advantages of genericity in terms of fitting over a variety of existing joint models. On the one hand, the generalized Gamma density function offers free-shape parameters to characterize a wide range of heavy-tailed densities, i.e., the genericity. On the other hand, the inter-component, inter-band dependency is captured by the Gaussian Copula which offers adapted flexibility. Moreover, this model leads to a closed form for the probabilistic similarity measure in terms of parameters, i.e., Kullback–Leibler divergence. By exploiting the separability between the copula and the marginal spaces, the closed form enables us to minimize the computational time needed to measure the discrepancy between two Multivariate Generalized Gamma densities in comparison to other models which imply using a Monte Carlo method characterized by an expensive time computing. For evaluating the performance of our proposal, a K-nearest neighbor (KNN) classifier is then used to test the classification accuracy. Experiments on different benchmarks using color texture databases are conducted to highlight the effectiveness of the proposed model associated to the Kullback–Leibler divergence.

Appendices

Appendix A

Supposing $y=(y_1,y_2,\ldots ,y_M)$, a set of M independent coefficients, the maximum likelihood function of the sample is defined as:

$$\begin{aligned} y:L=\log \prod _{i=1}^M f(y;\alpha ,\tau ,\lambda ) \end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial L}{\partial \alpha }= -M(\tau \log \lambda -\psi (\alpha ))+{\sum _{i=1}^{M}}\tau \log y_i=0. \end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial L}{\partial \tau }&= M\left( \frac{1}{\tau }-\alpha \log \lambda \right) \nonumber \\&\quad +\,{\sum _{i=1}^{M}}\alpha \log y_i-\left( \frac{y_i}{\lambda }\right) ^\tau \log \frac{y_i}{\lambda }=0. \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial L(y;\alpha ,\tau ,\lambda )}{\partial \lambda }=-\frac{M\alpha \tau }{\lambda }+\frac{\tau \lambda ^{-\tau }}{\lambda }{\sum _{i=1}^{M}}y_i=0. \end{aligned}$$

(22)

Thus, the parameters are deduced by solving a system of three equations:

$$\begin{aligned}&\hat{\lambda }=\left[ \frac{1}{M\hat{\alpha }}{\sum _{i=1}^{M}}y_i\hat{^\tau }\right] ^{\frac{1}{\hat{\tau }}}.\end{aligned}$$

(23)

$$\begin{aligned}&\hat{\alpha }=\frac{1}{\hat{\tau }}\left[ \frac{{\sum \nolimits _{i=1}^{M}}y_i\hat{^\tau }\log y_i}{{\displaystyle \sum \nolimits _{i=1}^{M}}y_i\hat{^\tau }}-\log y_i\right] ^{-1}.\end{aligned}$$

(24)

$$\begin{aligned}&\log \frac{M\hat{\alpha }(\prod \nolimits _{i=1}^M y_i)^\frac{\hat{\tau }}{M}}{{\displaystyle \sum \nolimits _{i=1}^{M}}y_i\hat{^\tau }}- \psi (\hat{\alpha })=0. \end{aligned}$$

(25)

where $\psi$ denotes the digamma function. In [5], we tackled the high nonlinearity of the ML equations using a numerical approximation based on the algorithm of Cohen et al. [25]. However, a faster algorithm was proposed in [6], in which a scale-independent shape estimation (SISE) method is used to find roots of the ML equations.

Appendix B

The proposition presented in [17] shows one attractive feature of the Copula representation of dependence, namely that the dependence structure when modeled by a Copula is invariant under increasing and continuous transformations of the marginals.

If $(x_1,\ldots ,x_n)^t$ has copula C and $T_1, \ldots ,T_n$ are increasing continuous functions, then $(T_1(x_1), \ldots , T_n(x_n))^t$ also has copula C.