Texture pattern classification based on probability density function estimation of the image spatial structure feature with symmetrical weibull distribution model

Abstract

Automatic texture pattern classification (ATPC) has long been an essential issue in computer vision. However, ATPC is still a challenging task since texture is a subjective conception, which is difficult to be expressed concisely by the existing computational models. The visual appearance of the imaged texture pattern (TP) visually depends on the random organization of local homogeneous fragments (LHFs) in it. Hence, it is essential to investigate the latent statistical distribution (LSD) behavior of LHFs for the texture image spatial structural feature (TISSF) characterization and expression to achieve a good performance of ATPC. We presented a probability density function estimation (PDFE)-based ATPC scheme, termed PDFE-ATPC. We demonstrated the Weibull distribution (WD) behavior of LHFs by the sequential fragmentation theory to explain the LSD of the TISSFs. To obtain the multiscale and multi-orientation detail expression of the TISSFs, we introduced an oriented Gaussian derivative filter (OGDF)-based TISSF characterization method, including the steerable isotropic Gaussian derivative filters (SIGDFs) and the oriented anisotropic Gaussian Derivative filters (OAGDFs). Successively, the LSDs of the filter responses were characterized omnidirectionally by a symmetrical WD model (SWDM) and the SWDM-based TISSF parameters, demonstrated to be directly related to the human vision perception (HVP) system, were extracted and applied to the ATPC with a spline regression-based classifier. Effectiveness of the proposed PDFE-ATPC method was verified by extensive experiments on three different texture image databases and compared with four commonly-used statistics-based texture classification methods.

Appendices

Appendix 1. Details of the SWDM parameter estimation

The two expression in (7), the first-order derivates of $\log\;\tilde{L}\left(\mathbf{X}|\boldsymbol{\uptheta} \right)$ over the model parameter vector θ and the Hessian matrix H, are given by,

$$ \nabla \log\;\tilde{L}\left(\mathbf{X}|\boldsymbol{\uptheta} \right)=\left(\begin{array}{l}\frac{1}{\beta^{\lambda }}\left(\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda }-\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\right)\\ {}\frac{1}{\lambda}\left[\begin{array}{l}\sum \limits_{i=1}^N\left({\left|\frac{x_i-\mu }{\beta}\right|}^{\lambda}\log |\frac{x_i-\mu }{\beta }|\right)-\\ {}N-\frac{N\log \lambda }{\lambda }-\frac{N}{\lambda}\boldsymbol{\Phi} \left(\frac{1}{\lambda}\right)\end{array}\right]\\ {}\frac{N}{\beta }-\frac{1}{\beta}\sum \limits_{i=1}^N{\left|\frac{x_i-\mu }{\beta}\right|}^{\lambda}\end{array}\right) $$

(32)

$$ \mathbf{H}=\left(\begin{array}{lll}\frac{\lambda }{\beta^{\lambda }}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda -1}+\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda -1}\end{array}\right)& \frac{1}{\beta^{\lambda }}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda}\log \frac{\mu -{x}_i}{\beta }-\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\log \frac{x_i-\mu }{\beta}\end{array}\right)& \frac{-\lambda }{\beta^{\lambda +1}}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda }-\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\end{array}\right)\\ {}\frac{1}{\beta^{\lambda }}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda}\log \frac{\mu -{x}_i}{\beta }-\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\log \frac{x_i-\mu }{\beta}\end{array}\right)& \left(\begin{array}{l}-\frac{\log \mid \frac{x_i-\mu }{\beta}\mid }{\lambda^2}\sum \limits_{i=1}^N{\left|\frac{x_i-\mu }{\beta}\right|}^{\lambda }+\\ {}\frac{\sum \limits_{i=1}^N{\left|\frac{x_i-\mu }{\beta}\right|}^{\lambda }{\log}^2\mid \frac{x_i-\mu }{\beta}\mid }{\lambda }+\\ {}\frac{N}{\lambda^2}\left(\log \lambda -1+\boldsymbol{\Phi} \left(\frac{1}{\lambda}\right)-\frac{1}{\lambda}\boldsymbol{\uppsi} \left(\frac{1}{\lambda}\right)\right)\end{array}\right)& \frac{-\lambda }{\beta^{\lambda +1}}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda }-\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\end{array}\right)\\ {}\frac{-\lambda }{\beta^{\lambda +1}}\left(\begin{array}{l}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i<\mu \end{array}}^N{\left(\mu -{x}_i\right)}^{\lambda }-\\ {}\sum \limits_{\begin{array}{l}i=1\\ {}{x}_i\ge \mu \end{array}}^N{\left({x}_i-\mu \right)}^{\lambda}\end{array}\right)& -\frac{1}{\beta}\sum \limits_{i=1}^N{\left|\frac{x_i-\mu }{\beta}\right|}^{\lambda}\log \mid \frac{x_i-\mu }{\beta}\mid & -\frac{N}{\beta^2}+\frac{\lambda +1}{\beta^{\lambda +2}}\sum \limits_{i=1}^N{\left|{x}_i-\mu \right|}^{\lambda}\end{array}\right) $$

(33)

where ψ(z) is the trigamma function, $\boldsymbol{\uppsi} (z)=\frac{d^2\boldsymbol{\Gamma} (z)}{d{z}^2}$.

Apparently, the convergence rate of the SWDM parameter estimation iteration process is inevitably related to the initial model parameter θ⁽⁰⁾. To enhance the estimation process, the initial model parameter θ⁽⁰⁾ is assigned as the empirical value which is close to the extremum point, namely, θ⁽⁰⁾ = (μ⁽⁰⁾, λ⁽⁰⁾, β⁽⁰⁾)^T, where

$$ \Big\{{\displaystyle \begin{array}{l}{\mu}^{(0)}= median\left(\mathbf{X}\right)\\ {}{\lambda}^{(0)}=2\left(2\frac{\frac{1}{N}\sum \limits_{i=1}^N{\left({x}_i-\overline{x}\right)}^4}{\frac{1}{N}\sum \limits_{i=1}^N{\left({x}_i-\overline{x}\right)}^2}-3\right)/\left(\frac{\frac{1}{N}\sum \limits_{i=1}^N{\left({x}_i-\overline{x}\right)}^4}{\frac{1}{N}\sum \limits_{i=1}^N{\left({x}_i-\overline{x}\right)}^2}-3\right)+1\\ {}{\beta}^{(0)}={\left(\sum \limits_{i=1}^N{x}_i-\frac{{\left({\mu}^{(0)}\right)}^{\lambda^{(0)}}}{N}\right)}^{\lambda^{(0)}}\end{array}} $$

(34)

However, it was revealed that computing the inverse of the Hessian matrix is a time-consuming task. Actually, the location parameter μ is near the mean value of the sampling points and we can find the relation of the shape parameter λ and scale parameter β by the partial derivative of the LLF equation

$$ \frac{\partial }{\mathrm{\partial \upbeta }}\log \mathrm{L}\left(\mathbf{X}|\boldsymbol{\uptheta} \right)=-\frac{N}{\upbeta}+\frac{1}{\upbeta}\sum \limits_{i=1}^N{\left|\frac{x_i-\upmu}{\upbeta}\right|}^{\uplambda}=0 $$

(35)

and we can obtain,

$$ \beta ={\left(\sum \limits_{i=1}^N{\left|{x}_i-\mu \right|}^{\lambda }/N\right)}^{\frac{1}{\lambda }} $$

(36)

Hence, we subtract a mean value from each sampling points and get a new corrected sample set $\tilde{\mathbf{X}}$, which still obeys the SWD model with the same scale parameter and shape parameter but symmetric around the y-axis. So, if we make a rough estimation of the location parameter with the mean value of the samples and take into account the relation equation of the scale parameter and shape parameter, respectively, substituting the formula (36) into formula,

$$ \frac{\partial }{\mathrm{\partial \uplambda }}\log \mathrm{L}\left(\mathbf{X}|\upmu, \upbeta, \uplambda \right)=\frac{1}{\uplambda^2}\left[N\uplambda +N\log \uplambda +N\boldsymbol{\Phi} \left(\frac{1}{\uplambda}\right)\right]-\frac{1}{\uplambda}\sum \limits_{i=1}^N\left[{\left|\frac{x_i-\upmu}{\upbeta}\right|}^{\lambda}\log |\frac{x_i-\upmu}{\upbeta}\right]=0 $$

(37)

We can achieve an uniparameter equation as follows,

$$ 1+\frac{\log \lambda }{\lambda }+\frac{1}{\lambda}\boldsymbol{\Phi} \left(\frac{1}{\lambda}\right)-\frac{1}{\lambda}\left(\sum \limits_{i=1}^N\frac{{\left|{\tilde{x}}_i\right|}^{\lambda }}{\sum \limits_{i=1}^N{\left|{\tilde{x}}_i\right|}^{\lambda }}\log \frac{N\mid {\tilde{x}}_i\mid }{\sum \limits_{i=1}^N{\left|{\tilde{x}}_i\right|}^{\lambda }}\right)=0 $$

(38)

where ${\tilde{x}}_i$ represents the samples in $\tilde{\mathbf{X}}$. We solve this univariable Eq. (38) by the Newton-Raphson scheme and then with formula (36), we can obtain the scale parameter conveniently.

Appendix 2. Derivation of the weighting coefficient α _{m, i} of the SIGDF

As stated in [10, 18, 29, 32], the design of a steerable SIGDF can be explained easily in the Fourier domain. We transform the rotated filter G_{κ, σ}(XR_θ) into the Fourier domain to facilitate computing, and we can then obtain [18]:

$$ {\displaystyle \begin{array}{c}\Im \left({G}_{K,\sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)\right)=\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}{\left(j{\omega}_x\cos \theta +j{\omega}_y\sin \theta \right)}^i{\left(-j{\omega}_x\sin \theta +j{\omega}_y\cos \theta \right)}^{m-i}{G}_{\sigma}\left({\omega}_{x,}{\omega}_y\right)\\ {}=\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}\left\{\sum \limits_{t=0}^i{C}_i^t{\left(j{\omega}_x\cos \theta \right)}^t{\left(j{\omega}_y\sin \theta \right)}^{i-t}\right\}\left\{\sum \limits_{l=0}^{m-i}{C}_{m-i}^l{\left(-j{\omega}_x\sin \theta \right)}^l{\left(j{\omega}_y\cos \theta \right)}^{m-i-l}\right\}{\widehat{G}}_{\sigma}\left({\omega}_x,{\omega}_y\right)\\ {}=\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{l=0}^{m-i}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l{\left(\cos \theta \right)}^{t+m-i-l}{\left(\sin \theta \right)}^{i-t+l}\underset{\Im \left(\frac{\partial^{t+l}}{\partial x}\frac{\partial^{m-\left(t+l\right)}}{\partial y}{G}_{\sigma}\left(x,y\right)\right)}{\underbrace{{\left(j{\omega}_x\right)}^{t+l}{\left(j{\omega}_y\right)}^{m-\left(t+l\right)}{\widehat{G}}_{\sigma}\left({\omega}_x,{\omega}_y\right)}}\end{array}} $$

(39)

where ℑ(f(x))means the Fourier transform of the function f(x);${C}_m^n$ is the binomial coefficient, ${C}_m^n=\frac{m!}{n!\left(m-n\right)!}$; ${\widehat{G}}_{\sigma}\left({\omega}_x,{\omega}_y\right)$ is the Fourier transform of the Gaussian function G_σ(x, y). According to the convolution theorem, we can perform Fourier transform on both sides of Eq. (13), then we can obtain,

$$ {\displaystyle \begin{array}{c}\Im \left\{I\left(\mathbf{X}\right)\ast {G}_{\kappa, \sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)\right\}=\widehat{I}\left({\omega}_x,{\omega}_y\right)\Im \left\{{G}_{\kappa, \sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)\right\}\\ {}=\widehat{I}\left({\omega}_x,{\omega}_y\right)\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{i=0}^{m-i}\left\{\begin{array}{l}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l\times \\ {}{\left(\cos \theta \right)}^{t+m-i-l}\times \\ {}{\left(\sin \theta \right)}^{i-t+l}\end{array}\right\}\Im \left(\frac{\partial^{t+l}}{\partial x}\frac{\partial^{m-\left(t+l\right)}}{\partial y}{G}_{\sigma}\left(x,y\right)\right)\end{array}} $$

(40)

We then perform inverse Fourier transformation on the formula (40) and we can achieve the time domain expression of I(X) ∗ G_{κ, σ}(XR_θ)

$$ {\displaystyle \begin{array}{c}I\left(\mathbf{X}\right)\ast {G}_{\kappa, \sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)={\Im}^{-1}\left\{\Im \left\{I\left(\mathbf{X}\right)\ast {G}_{\kappa, \sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)\right\}\right\}\\ {}=\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{i=0}^{m-i}\left\{\begin{array}{l}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l\times \\ {}{\left(\cos \theta \right)}^{t+m-i-l}\times \\ {}{\left(\sin \theta \right)}^{i-t+l}\end{array}\right\}{\Im}^{-1}\left\{\Im \left\{I\left(x,y\right)\right\}\Im \left(\frac{\partial^{t+l}}{\partial x}\frac{\partial^{m-\left(t+l\right)}}{\partial y}{G}_{\sigma}\left(x,y\right)\right)\right\}\\ {}=\sum \limits_{m=1}^k\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{i=0}^{m-i}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l{\left(\cos \theta \right)}^{t+m-i-l}{\left(\sin \theta \right)}^{i-t+l}{I}_{m,m-\left(t+l\right)}\left(\mathbf{X}\right)\end{array}} $$

(41)

Compare the formula (41) and (13), if we setj = m − (t + l) where 0 ≤ j ≤ m, then

$$ I\left(\mathbf{X}\right)\ast {G}_{\kappa, \sigma}\left(\mathbf{X}{\mathbf{R}}_{\theta}\right)=\sum \limits_{m=1}^k\sum \limits_{j=0}^m{I}_{m,j}\left(\mathbf{X}\right)\underset{\alpha_{m,j}}{\underbrace{\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{l=0}^{m-i}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l{\left(\cos \theta \right)}^{t+m-i-l}{\left(\sin \theta \right)}^{i-t+l}}} $$

(42)

Consequently, we can obtain α_{m, j} as the following expression [18]:

$$ {\alpha}_{m,j}=\sum \limits_{i=0}^m{k}_{m,i}\sum \limits_{t=0}^i\sum \limits_{l=0}^{m-i}{\left(-1\right)}^l{C}_i^t{C}_{m-i}^l{\left(\cos \theta \right)}^{t+m-i-l}{\left(\sin \theta \right)}^{i-t+l} $$

(43)