A Gray Level Dependence Matrix (GLDM) quantifies gray level dependencies in an image. A gray level dependency is defined as a the number of connected voxels within distance \delta that are dependent on the center voxel. A neighbouring voxel with gray level j is considered dependent on center voxel with gray level i if |i-j|\le\alpha. In a gray level dependence matrix \textbf{P}(i,j) the (i,j)-th element describes the number of times a voxel with gray level i with j dependent voxels in its neighbourhood appears in image.
As an example, consider the following 5x5 ROI image having 5 gray levels:
\textbf{G} = \begin{bmatrix} 5 & 2 & 5 & 4 & 4\\ 3 & 3 & 3 & 1 & 3\\ 2 & 1 & 1 & 1 & 3\\ 4 & 2 & 2 & 2 & 3\\ 3 & 5 & 3 & 3 & 2 \end{bmatrix}
For \alpha=0 and \delta = 1, the GLDM then becomes:
\textbf{P} = \begin{bmatrix} 0 & 1 & 2 & 1 \\ 1 & 2 & 3 & 0 \\ 1 & 4 & 4 & 0 \\ 1 & 2 & 0 & 0 \\ 3 & 0 & 0 & 0 \end{bmatrix}
Let:
- N_g be the number of discrete intensity values in the image
- N_d be the number of discrete dependency sizes in the image
- N_z be the number of dependency zones in the image, which is equal to \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)}
- \textbf{P}(i,j) be the dependence matrix
- p(i,j) be the normalized dependence matrix, defined as p(i,j) = \frac{\textbf{P}(i,j)}{N_z}
GLDM_SDE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2}}}{N_z}
GLDM_LDE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)j^2}}{N_z}
GLDM_GLN = \frac{\sum^{N_g}_{i=1}\left(\sum^{N_d}_{j=1}{\textbf{P}(i,j)}\right)^2}{N_z}
GLDM_DN = \frac{\sum^{N_d}_{j=1}\left(\sum^{N_g}_{i=1}{\textbf{P}(i,j)}\right)^2}{N_z}
GLDM_DNN = \frac{\sum^{N_d}_{j=1}\left(\sum^{N_g}_{i=1}{\textbf{P}(i,j)}\right)^2}{N_z^2}
GLDM_GLV = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)(i - \mu)^2}
where,
\mu = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{ip(i,j)}
GLDM_DV = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)(j - \mu)^2} where \mu = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{jp(i,j)}
GLDM_DE = -\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)\log_{2}(p(i,j)+\epsilon)}
GLDM_LGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2}}}{N_z}
GLDM_HGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)i^2}}{N_z}
GLDM_SDLGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2j^2}}}{N_z}
GLDM_SDHGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)i^2}{j^2}}}{N_z}
GLDM_LDLGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)j^2}{i^2}}}{N_z}
GLDM_LDHGLE = \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)i^2j^2}}{N_z}