Table 4 The detailed information of the features

X3D-venous-wavelet.LHH

Glcm–cluster shade \(\mathop \sum \limits_{i = 1}^{{N_{g} }} \mathop \sum \limits_{j = 1}^{{N_{g} }} \left( {i + j - \mu_{x} - \mu_{y} } \right)^{3} p\left( {i,j} \right)\)

Cluster Shade is a measure of the skewness and uniformity of the GLCM. A higher cluster shade implies greater asymmetry about the mean

X3D-venous-wavelet.HHH

Firstorder –mean absolute deviation (MAD)\(\frac{1}{{N_{p} }}\mathop \sum \limits_{i = 1}^{{N_{p} }} \left| {X\left( i \right) - X } \right|\)

Mean Absolute Deviation is the mean distance of all intensity values from the Mean Value of the image array

X2D-venous-wavelet.HLH X2D-delay-wavelet.LHHX2D-Artery-wavelet.HHH

Firstorder—skewness \(\frac{{\mu_{3} }}{{\sigma^{3} }} = \frac{{\frac{1}{{N_{p} }}\mathop \sum \nolimits_{i = 1}^{{N_{p} }} \left( {X\left( i \right) - X} \right)^{3} }}{{\left( {\sqrt {\frac{1}{{N_{p} }}} \sqrt {\mathop \sum \nolimits_{i = 1}^{{N_{p} }} \left( {X\left( i \right) - X} \right)^{2} } } \right)^{3} }}\)

Skewness measures the asymmetry of the distribution of values about the Mean value. Depending on where the tail is elongated and the mass of the distribution is concentrated, this value can be positive or negative

X3D-delay-wavelet.LHH

Glcm—IDMN \(\mathop \sum \limits_{k = 0}^{{N_{g} - 1}} \frac{{P_{x - y} \left( k \right)}}{{1 + \left( {\frac{{k^{2} }}{{N_{g}^{2} }}} \right)}}\)

IDMN (inverse difference moment normalized) is a measure of the local homogeneity of an image. IDMN weights are the inverse of the Contrast weights (decreasing exponentially from the diagonal i = ji = j in the GLCM). Unlike Homogeneity2, IDMN normalizes the square of the difference between neighboring intensity values by dividing over the square of the total number of discrete intensity values

X2D-venous

Original shape elongnation \(\sqrt {\frac{{\lambda_{minor} }}{{\lambda_{major} }}}\)

Elongation shows the relationship between the two largest principal components in the ROI shape. For computational reasons, this feature is defined as the inverse of true elongation

X2D-venous-wavelet.HHH

Glszm–Zone entropy (ZE)\(- \mathop \sum \limits_{i = 1}^{{N_{g} }} \mathop \sum \limits_{j = 1}^{{N_{s} }} p\left( {i,j} \right)\log_{2} \left( {p\left( {i,j} \right) + \in } \right)\)

ZE measures the uncertainty/randomness in the distribution of zone sizes and gray levels. A higher value indicates more heterogeneneity in the texture patterns

X2D-venous-wavelet.HHH

Glszm–gray level non-uniformity (GLN)\(\frac{{\mathop \sum \nolimits_{i = 1}^{{N_{g} }} \left( {\mathop \sum \nolimits_{j = 1}^{{N_{s} }} P\left( {i,j} \right)} \right)^{2} }}{{N_{z} }}\)

GLN measures the variability of gray-level intensity values in the image, with a lower value indicating more homogeneity in intensity values

X3D-artery-wavelet.HLL

Firstorder—Kurtosis \(\frac{{\mu _{4} }}{{\sigma ^{4} }} = \frac{{\frac{1}{{{\text{N}}_{{\text{p}}} }}\sum\nolimits_{{{\text{i}} = 1}}^{{{\text{N}}_{{\text{p}}} }} {\left( {{\text{X}}\left( {\text{i}} \right) - {\text{X}}} \right)^{4} } }}{{\left( {\frac{1}{{{\text{N}}_{{\text{p}}} }}\sum\nolimits_{{{\text{i}} = 1}}^{{{\text{N}}_{{\text{p}}} }} {\left( {{\text{X}}\left( {\text{i}} \right) - {\text{X}}} \right)^{2} } } \right)^{2} }}\)

Kurtosis is a measure of the ‘peakedness’ of the distribution of values in the image ROI. A higher kurtosis implies that the mass of the distribution is concentrated towards the tail(s) rather than towards the mean. A lower kurtosis implies the reverse: that the mass of the distribution is concentrated towards a spike near the Mean value

X2D-delay-wavelet.HHH

Glszm–small area high gray level emphasis (SAHGLE)\(\frac{{\mathop \sum \nolimits_{i = 1}^{{N_{g} }} \mathop \sum \nolimits_{j = 1}^{{N_{s} }} \frac{{P\left( {i,j} \right)i^{2} }}{{j^{2} }}}}{{N_{z} }}\)

SAHGLE measures the proportion in the image of the joint distribution of smaller size zones with higher gray-level values

X2D-venous-wavelet.LLL

Firstorder–range max(X)-min(X)

The range of gray values in the ROI