Texture features / GLDM

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:

\[\begin{split}\textbf{G} = \begin{bmatrix} 5 & 2 & 5 & 4 & 4\\ 3 & 3 & 3 & 1 & 3\\ 2 & 1 & 1 & 1 & 3\\ 4 & 2 & 2 & 2 & 3\\ 3 & 5 & 3 & 3 & 2 \end{bmatrix}\end{split}\]

For \(\alpha=0\) and \(\delta = 1\), the GLDM then becomes:

\[\begin{split}\textbf{P} = \begin{bmatrix} 0 & 1 & 2 & 1 \\ 1 & 2 & 3 & 0 \\ 1 & 4 & 4 & 0 \\ 1 & 2 & 0 & 0 \\ 3 & 0 & 0 & 0 \end{bmatrix}\end{split}\]

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 \(p(i,j) = \frac{\textbf{P}(i,j)}{N_z}\)

Small Dependence Emphasis

GLDM_SDE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2}}}{N_z}\)

Large Dependence Emphasis

GLDM_LDE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)j^2}}{N_z}\)

Gray Level Non-Uniformity

GLDM_GLN \(= \frac{\sum^{N_g}_{i=1}\left(\sum^{N_d}_{j=1}{\textbf{P}(i,j)}\right)^2}{N_z}\)

Dependence Non-Uniformity

GLDM_DN \(= \frac{\sum^{N_d}_{j=1}\left(\sum^{N_g}_{i=1}{\textbf{P}(i,j)}\right)^2}{N_z}\)

Dependence Non-Uniformity Normalized

GLDM_DNN \(= \frac{\sum^{N_d}_{j=1}\left(\sum^{N_g}_{i=1}{\textbf{P}(i,j)}\right)^2}{N_z^2}\)

Gray Level Variance

GLDM_GLV \(= \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)(i - \mu)^2}\)

where,

\(\mu = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{ip(i,j)}\)

Dependence Variance

GLDM_DV \(= \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)(j - \mu)^2}\) where \(\mu = \sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{jp(i,j)}\)

Dependence Entropy

GLDM_DE \(= -\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{p(i,j)\log_{2}(p(i,j)+\epsilon)}\)

Low Gray Level Emphasis

GLDM_LGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2}}}{N_z}\)

High Gray Level Emphasis

GLDM_HGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)i^2}}{N_z}\)

Small Dependence Low Gray Level Emphasis

GLDM_SDLGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)}{i^2j^2}}}{N_z}\)

Small Dependence High Gray Level Emphasis

GLDM_SDHGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)i^2}{j^2}}}{N_z}\)

Large Dependence Low Gray Level Emphasis

GLDM_LDLGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\frac{\textbf{P}(i,j)j^2}{i^2}}}{N_z}\)

Large Dependence High Gray Level Emphasis

GLDM_LDHGLE \(= \frac{\sum^{N_g}_{i=1}\sum^{N_d}_{j=1}{\textbf{P}(i,j)i^2j^2}}{N_z}\)