Texture features / GLDZM

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.

\[\begin{split}I = \begin{bmatrix} 0 & 0 & 0 & 1 & 1 & 1 & 0\\ 0 & 0 & \fbox{3} & 1 & \fbox{3} & 1 & 0\\ 1 & \fbox{3} & 1 & 1 & \fbox{3} & 1 & 0\\ 1 & \fbox{3} & 1 & 1 & \fbox{3} & \fbox{3} & 1\\ \fbox{3} & 1 & \fbox{3} & \fbox{3} & 1 & 1 & 1\\ \fbox{3} & 1 & \fbox{3} & \fbox{3} & 1 & 1 & 0\\ \fbox{3} & 1 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}\end{split}\]

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 boundaries to the first off-ROI or off-image pixel.

Considering the following ROI image

\[\begin{split}R = \begin{bmatrix} 0 & 0 & 0 & 1 & 1 & 1 & 0\\ 0 & 0 & 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}\end{split}\]

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

\[\begin{split}D = \begin{bmatrix} \times & \times & \times & * & * & * & \times \\ \times & \times & \fbox{2} & * & \fbox{2} & * & \times \\ * & \fbox{2} & * & * & \fbox{2} & * & \times \\ * & \fbox{2} & * & * & \fbox{2} & \fbox{2} & * \\ \fbox{1} & * & \fbox{2} & \fbox{2} & * & * & * \\ \fbox{1} & * & \fbox{2} & \fbox{2} & * & * & \times \\ \fbox{1} & * & \times & \times & \times & * & \times \end{bmatrix}\end{split}\]

The following example is an image having 5 discrete grey values masked with the above ROI mask \(R\) :

\[\begin{split}I_2 = \begin{bmatrix} \times & \times & \times & 4 & 4 & 4 & \times \\ \times & \times & 3 & 1 & 3 & 4 & \times \\ 2 & 1 & 1 & 1 & 3 & 2 & \times \\ 4 & 4 & 2 & 2 & 3 & 3 & 1 \\ 3 & 5 & 3 & 3 & 2 & 1 & 1 \\ 3 & 5 & 3 & 3 & 2 & 4 & \times \\ 3 & 1 & \times & \times & \times & 4 & \times \end{bmatrix}\end{split}\]

Its distance map \(D_2\) is:

\[\begin{split}D_2 = \begin{bmatrix} \times & \times & \times & 1 & 1 & 1 & \times \\ \times & \times & 1 & 2 & 2 & 1 & \times \\ 1 & 1 & 2 & 3 & 2 & 1 & \times \\ 1 & 2 & 3 & 3 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 2 & 1 & 1 & 1 & 1 & \times \\ 1 & 1 & \times & \times & \times & 1 & \times \end{bmatrix}\end{split}\]

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:

\[\begin{split}M(I_2)=\begin{bmatrix} 3 & 0 & 0\\ 3 & 0 & 1\\ 3 & 1 & 0\\ 2 & 0 & 0\\ 1 & 1 & 0\end{bmatrix}\end{split}\]

Let \(m(x,d)\) be an element of the 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:

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDE}} {\textup{Small Distance Emphasis}} = \frac{1}{N_s} \sum_d \frac{m_d}{d^2}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDE}} {\textup{Large Distance Emphasis}} = \frac{1}{N_s} \sum_d d^2 m_d\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_LGLE}} {\textup{Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \frac{m_x}{x^2}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_HGLE}} {\textup{High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x x^2 m_x\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDLGLE}} {\textup{Small Distance Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{ m_{x,d}}{x^2 d^2}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_SDHGLE}} {\textup{Small Distance High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{x^2 m_{x,d}}{d^2}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDLGLE}} {\textup{Large Distance Low Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \frac{d^2 m_{x,d}}{x^2}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_LDHGLE}} {\textup{Large Distance High Grey Level Emphasis}} = \frac{1}{N_s} \sum_x \sum_d \x^2 d^2 m_{x,d}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLNU}} {\textup{Grey Level Non-Uniformity}} = \frac{1}{N_s} \sum_x m_x^2\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLNUN}} {\textup{Grey Level Non-Uniformity Normalized}} = \frac{1}{N_s^2} \sum_x m_x^2\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDNU}} {\textup{Zone Distance Non-Uniformity}} = \frac{1}{N_s} \sum_d m_d^2\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDNUN}} {\textup{Zone Distance Non-Uniformity Normalized}} = \frac{1}{N_s^2} \sum_d m_d^2\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZP}} {\textup{Zone Percentage}} = \frac{N_s}{N_v}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLM}} {\textup{Grey Level Mean}} = \mu_x = \sum_x \sum_d x p_{x,d}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLV}} {\textup{Grey Level Variance}} = \sum_x \sum_d \left(x - \mu_x \right)^2 p_{x,d}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDM}} {\textup{Zone Distance Mean}} = \mu_d = \sum_x \sum_d d p_{x,d}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDV}} {\textup{Zone Distance Variance}} = \sum_x \sum_d \left(d - \mu_d \right)^2 p_{x,d}\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_ZDE}} {\textup{Zone Distance Entropy}} = - \sum_x \sum_d p_{x,d} \textup{log}_2 ( p_{x,d} )\]

\[\underset{\mathrm{Nyxus \, code: \, GLDZM\_GLE}} {\textup{Grey Level Entropy}} = - \sum_x \sum_d p_{x,d} \textup{log}_2 ( p_{x,d} )\]

References

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.