# 3D Texture Feature Extraction and Classification Using GLCM and LBP-Based Descriptors
## 3 The Proposed Method
### The Improved 3D Gray-Level Co-Occurrence Matrix (IGLCM_3D)  
#### 3.2.1 Definition and Method Overview  
We proposed in the preliminary work [29] an improved version of the 3D GLCM, the IGLCM_3D. The functional diagram is given in Figure 6.  
As shown in Figure 6, the feature extraction process is based on the computation of 4 types of co-occurrence matrices. The first one is the 3D standard GLCM defined in (12), which is computed directly from the input image intensities. The other three types of 3D co-occurrence matrices are based on the gradient image information. This can depict the distribution of strong or weak edges and the degree of uniformity of the input image. The matrix based on the gradient magnitude information is labelled Gradient Magnitude Co-Occurrence Matrix (GMCM) and is given in (13) for an input image I of size m \(\times\) n \(\times\) p.  
\[\text{GMCM}_{\text{d}_{\text{x}},\text{d}_{\text{y}},\text{d}_{\text{z}}}( \text{i},\text{j})=\sum_{\text{x}=1}^{\text{m}}\sum_{\text{y}=1}^{\text{n}} \sum_{\text{z}=1}^{\text{p}}\left\{\begin{array}{ll}1,&\text{if }\text{M}(\text{x},\text{y},\text{z})= \text{i}&\text{and }\text{M}\Big{(}\text{x}+\text{d}_{\text{x}},\text{y}+\text{d}_{\text{y}}, \text{z}+\text{d}_{\text{z}}\Big{)}=\text{j}\\ &0,&\text{otherwise}\end{array}\right. \tag{13}\]  
where \(\text{M}=\sqrt{\text{G}_{\text{x}}^{2}+\text{G}_{\text{y}}^{2}+\text{G}_{ \text{Z}}{}^{2}}\) is the gradient magnitude and \(\text{G}_{\text{x}}\), \(\text{G}_{\text{y}}\) and \(\text{G}_{\text{Z}}\) are the components of the gradient vector in the three directions.  
Figure 5: The multi-resolution strategy for BM3 DELBP_3D © [2020] IEEE. Reprinted [adapted], with permission, from [28]).  
The other two 3D co-occurrence matrices are based on the gradient orientation, which offers additional information about the local anisotropy for the considered textured images. The two matrices are labelled Azimuth Angle Co-Occurrence Matrix (AACM) and Elevation Angle Co-Occurrence Matrix (EACM), given in (14) and (15), respectively:  
\[\text{AACM}_{\text{d}_{\text{x}},\text{d}_{\text{y}},\text{d}_{\text{z}}}( \text{i},\text{j})=\sum_{\text{x}=1}^{\text{m}}\sum_{\text{y}=1}^{n}\sum_{\text{ z}=1}^{\text{p}}\left\{\begin{array}{cc}1,&\text{if}\ \theta(\text{x},\text{y},\text{z})=\ \text{i}\ \text{and}\ \theta(\text{x}+\text{d}_{\text{x}},\text{y}+\text{d}_{ \text{y}},\text{z}+\text{d}_{\text{z}})=\ \text{j}\\ &0,\ \text{otherwise}\end{array}\right. \tag{14}\]  
\[\text{where}\ \theta\ =\ \tan^{-1}\Big{(}\frac{\text{G}_{\text{x}}}{\text{G}_{ \text{x}}}\Big{)}\ \text{is the azimuth angle.} \tag{15}\]  
\[\text{EACM}_{\text{d}_{\text{x}},\text{d}_{\text{y}},\text{d}_{\text{z}}}( \text{i},\text{j})=\sum_{\text{x}=1}^{\text{m}}\sum_{\text{y}=1}^{n}\sum_{ \text{z}=1}^{\text{p}}\left\{\begin{array}{cc}1,&\text{if}\ \Phi(\text{x},\text{y},\text{z})=\ \text{i}\ \text{and}\ \Phi(\text{x}+\text{d}_{\text{x}},\text{y}+\text{d}_{ \text{y}},\text{z}+\text{d}_{\text{z}})=\ \text{j}\\ &0,\ \text{otherwise}\end{array}\right. \tag{16}\]  
\[\text{where}\ \Phi\ =\ \tan^{-1}\Big{(}\frac{\text{G}_{\text{Z}}}{\sqrt{\text{G}_{ \text{x}}{{}^{2}}+\text{G}_{\text{Y}}{{}^{2}}}}\Big{)}\ \text{is the elevation angle.} \tag{17}\]