The Grey Level Distance Zone Matrix (GLDZM) indicates the number of times each grey level's zones occur within a distance from the zone to the ROI border.
A zone is a continuous set of pixels of same intensity (or "grey level").
The continuity is meant as a 4-connected neighbourhood. For example, the following intensity image matrix I of 2 non-zero intensities 1 and 3 contains 4 zones of intensity 3 -- 1 single-pixel, 1 2-pixel, 1 3-pixel, and 2 4-pixel zones.
I = \begin{bmatrix}
0 & 0 & 0 & 1 & 1 & 1 & 0\\
0 & 0 & \fbox{3} & 1 & \fbox{3} & 1 & 0\\
1 & \fbox{3} & 1 & 1 & \fbox{3} & 1 & 0\\
1 & \fbox{3} & 1 & 1 & \fbox{3} & \fbox{3} & 1\\
\fbox{3} & 1 & \fbox{3} & \fbox{3} & 1 & 1 & 1\\
\fbox{3} & 1 & \fbox{3} & \fbox{3} & 1 & 1 & 0\\
\fbox{3} & 1 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
The zone's distance is the minimum of its each pixel's distance to the ROI or image border measured as the number of pixel boudaries to the first off-ROI or off-image pixel.
Considering the following ROI image
R = \begin{bmatrix}
0 & 0 & 0 & 1 & 1 & 1 & 0\\
0 & 0 & 1 & 1 & 1 & 1 & 0\\
1 & 1 & 1 & 1 & 1 & 1 & 0\\
1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 0\\
1 & 1 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
the distances of zons of intensity 3, ignoring pixels of other non-zero intensities (shown as *), in the masked image (whose off-ROI pixels are shown as \times) are
D = \begin{bmatrix}
\times & \times & \times & * & * & * & \times \\
\times & \times & \fbox{2} & * & \fbox{2} & * & \times \\
* & \fbox{2} & * & * & \fbox{2} & * & \times \\
* & \fbox{2} & * & * & \fbox{2} & \fbox{2} & * \\
\fbox{1} & * & \fbox{2} & \fbox{2} & * & * & * \\
\fbox{1} & * & \fbox{2} & \fbox{2} & * & * & \times \\
\fbox{1} & * & \times & \times & \times & * & \times
\end{bmatrix}
The following example is an image having 5 discrete grey values masked with the above ROI mask R :
I_2 = \begin{bmatrix}
\times & \times & \times & 4 & 4 & 4 & \times \\
\times & \times & 3 & 1 & 3 & 4 & \times \\
2 & 1 & 1 & 1 & 3 & 2 & \times \\
4 & 4 & 2 & 2 & 3 & 3 & 1 \\
3 & 5 & 3 & 3 & 2 & 1 & 1 \\
3 & 5 & 3 & 3 & 2 & 4 & \times \\
3 & 1 & \times & \times & \times & 4 & \times
\end{bmatrix}
Its distance map D_2 is:
D_2 = \begin{bmatrix}
\times & \times & \times & 1 & 1 & 1 & \times \\
\times & \times & 1 & 2 & 2 & 1 & \times \\
1 & 1 & 2 & 3 & 2 & 1 & \times \\
1 & 2 & 3 & 3 & 3 & 2 & 1 \\
1 & 2 & 2 & 2 & 2 & 2 & 1 \\
1 & 2 & 1 & 1 & 1 & 1 & \times \\
1 & 1 & \times & \times & \times & 1 & \times
\end{bmatrix}
In a grey level distance zone matrix (GLDZM) M, the element (x,d) describes the number of zones in an image with grey level x located at distance d from the edge of the ROI or image border.
Applied to the example, the GLDZM M(I_2) of image I_2 having distance matrix D_2, is:
M(I_2)=\begin{bmatrix}
3 & 0 & 0\\
3 & 0 & 1\\
3 & 1 & 0\\
2 & 0 & 0\\
1 & 1 & 0\end{bmatrix}
Let m(x,d) be an element ofthe distance zone matrix corresponding to grey level x and zone distance d ,
N_g -- the number of grey levels ,
N_d -- the maximum zone distance, and
N_s -- the number of zones of any non-zero intensity.
p(x,d) be an element of the normalized distance zone matrix expressing the relative probability of element (x,d), defined as
p_{x,d} = \frac{m_{x,d}}{N_s} .
N_v is the number of ROI image pixels.
In addition, the marginal totals
m_{x,\cdot} = m_x = \sum_d m_{x,d}
represent the total of all zones with a given intensity x, and
m_{\cdot, d} = m_d = \sum_x m_{x,d}
represent the total of all zones with a given distance d.
The following features are then defined:
\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDE}} {\textup{Small Distance Emphasis}} = \frac{1}{N_s} \sum_d \frac{m_d}{d^2}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDE}} {\textup{Large Distance Emphasis}} = \frac{1}{N_s} \sum_d d^2 m_d
\underset{\mathrm{Nyxus \, code: \, GLDZM\_LGLE}} {\textup{Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \frac{m_x}{x^2}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_HGLE}} {\textup{High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x x^2 m_x
\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDLGLE}} {\textup{Small Distance Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{ m_{x,d}}{x^2 d^2}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDHGLE}} {\textup{Small Distance High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{x^2 m_{x,d}}{d^2}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDLGLE}} {\textup{Large Distance Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{d^2 m_{x,d}}{x^2}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDHGLE}} {\textup{Large Distance High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \x^2 d^2 m_{x,d}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLNU}} {\textup{Grey Level Non-Uniformity}} = \frac{1}{N_s} \sum_x m_x^2
\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLNUN}} {\textup{Grey Level Non-Uniformity Normalized}} = \frac{1}{N_s^2} \sum_x m_x^2
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDNU}} {\textup{Zone Distance Non-Uniformity}} = \frac{1}{N_s} \sum_d m_d^2
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDNUN}} {\textup{Zone Distance Non-Uniformity Normalized}} = \frac{1}{N_s^2} \sum_d m_d^2
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZP}} {\textup{Zone Percentage}} = \frac{N_s}{N_v}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLM}} {\textup{Grey Level Mean}} = \mu_x = \sum_x \sum_d x p_{x,d}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLV}} {\textup{Grey Level Variance}} = \sum_x \sum_d \left(x - \mu_x \right)^2 p_{x,d}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDM}} {\textup{Zone Distance Mean}} = \mu_d = \sum_x \sum_d d p_{x,d}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDV}} {\textup{Zone Distance Variance}} = \sum_x \sum_d \left(d - \mu_d \right)^2 p_{x,d}
\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDE}} {\textup{Zone Distance Entropy}} = - \sum_x \sum_d p_{x,d} \textup{log}_2 ( p_{x,d} )
\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLE}} {\textup{Grey Level Entropy}} = - \sum_x \sum_d p_{x,d} \textup{log}_2 ( p_{x,d} )
Thibault, G., Angulo, J., and Meyer, F. (2014); Advanced statistical matrices for texture characterization: application to cell classification; IEEE transactions on bio-medical engineering, 61(3):630-7.