Texture features / NGLDM

The neighbouring grey level dependence matrix (NGLDM) features quantify coarseness of the texture in a ROI in a rotationally invariant way.

NGLDM is based on the concept of a neighbourhood around a pixel determined by position and a concept of dependence between two pixels determined by grey level values. All pixels within Chebyshev distance \(\delta\) are considered to belong to the pixel’s neighbourhood. All pixels whose grey level differenece is within \(\alpha\) are considered to be dependent.

Let \(I_u\) be intensity of a pixel at location \(u\), \(d_{cheb}(u,v) = max(|u-v|)\) be the Chebyshev distance between locations \(u\) and \(v\), \([ b ]\) be the Iverson bracket over expression \(b\).

A pixel at location \(m\) is said to belong to the neighborhood of a “central” pixel at location \(k\) if \(d_{cheb} (k-m) \leq \delta\). Additionally, the neighboring pixel is said to be dependent on the central pixel if \(|I_k - I_m| \leq \alpha\) with respect to coarseness parameter \(\alpha \geq 0\). The number of dependent pixels \(j_k\) within the neighborhood of pixel at location \(k\) is defined as

\[j_k = \sum_{m_y{=}-\delta}^\delta \sum_{m_x{=}-\delta}^\delta \big[|X_{d}(\mathbf{k})-X_{d}(\mathbf{k}+\mathbf{m})| \leq \alpha\big]\]

Let \(N_g\) be the number of unique grey level values of the pixels within the ROI, \(N_r=\text{max}(j_k)\) be the maximum dependence count across all the neighborhood pixels with respect to chosen \(\delta\).

Let \(\mathbf{M}\) be the \(N_g \times N_r\) neighbouring grey level dependence matrix (NGLDM). Element \(s_{ij}\) of \(\mathbf{M}\) is then the number of neighbourhoods with a center pixel with discretised grey level \(i\) and a neighbouring dependence \(j\).

Let \(N_v\) be the number of pixels in the ROI and \(N_s = \sum_{i=1}^{N_g}\sum_{j=1}^{N_n} s_{ij}\) the number of neighbourhoods.

The following marginal totals can be defined. Let \(s_{i.}=\sum_{j=1}^{N_r}\) be the number of neighbourhood pixels having grey level \(i\), and let \(s_{j.}=\sum_{i=1}^{N_g}s_{ij}\) be the number of neighbourhood pixels having dependence \(j\), regardless of grey level.

Note

that Nyxus presets the coarseness parameter \(\alpha=0\) and the neighbourhood radius \(\delta=1\).

The following features are then defined:

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_LDE}} {\textup{Low dependence emphasis}} = \frac{1}{N_s} \sum_{j=1}^{N_r} \frac{s_{.j}}{j^2}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_HDE}} {\textup{High dependence emphasis}} = \frac{1}{N_s} \sum_{j=1}^{N_r} j^2 s_{.j}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_LGLCE}} {\textup{Low grey level count emphasis}}= \frac{1}{N_s} \sum_{i=1}^{N_g} \frac{s_{i.}}{i^2}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_HGLCE}} {\textup{ High grey level count emphasis }} = \frac{1}{N_s} \sum_{i=1}^{N_g} i^2 s_{i.}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_LDLGLE }} {\textup{ Low dependence low grey level emphasis }} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} \frac{s_{ij}}{i^2 j^2}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_LDHGLE }} {\textup{ Low dependence high grey level emphasis }} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} \frac{i^2 s_{ij}}{j^2}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_HDLGLE }} {\textup{ High dependence low grey level emphasis }} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} \frac{j^2 s_{ij}}{i^2}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_HDHGLE }} {\textup{ High dependence high grey level emphasis }} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} i^2 j^2 s_{ij}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_GLNU }} {\textup{ Grey level non-uniformity }} = \frac{1}{N_s} \sum_{i=1}^{N_g} s_{i.}^2\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_GLNUN }} {\textup{ Normalised grey level non-uniformity }} = \frac{1}{N_s^2} \sum_{i=1}^{N_g} s_{i.}^2\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCNU }} {\textup{ Dependence count non-uniformity }} = \frac{1}{N_s} \sum_{j=1}^{N_r} s_{.j}^2\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCNUN }} {\textup{ Normalised dependence count non-uniformity }} = \frac{1}{N_s^2} \sum_{i=1}^{N_r} s_{.j}^2\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCP }} {\textup{ Dependence count percentage }} = \frac{N_s}{N_v}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_GLV }} {\textup{ Grey level variance }} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} (i-\mu)^2 p_{ij}\]

where the mean intensity is defined as \(\mu = \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} i\,p_{ij}\).

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCV }} {\textup{ Dependence count variance }} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} (j-\mu)^2 p_{ij}\]

where the mean dependence count is defined as \(\mu = \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} j\,p_{ij}\).

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCE }} {\textup{ Dependence count entropy }} = - \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} p_{ij} \log_2 p_{ij}\]

\[\underset{\mathrm{Nyxus \, code: \, NGLDM\_DCENE }} {\textup{ Dependence count energy }} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_r} p_{ij}^2\]

References

Chengjun Sun; William G Wee (1983). “Neighboring gray level dependence matrix for texture classification”. Computer Vision, Graphics, and Image Processing, Volume 23, Issue 3, 1983, Pages 341-352, ISSN 0734-189X.