First order statistics

These include local intensity-based statistical measures, computed over the neighbourhood around each voxel, in accordance with the IBSI standard.

List of available features

For the set of grey levels Xgl = {Xgl,1, Xgl,2, . . . , Xgl,Nv} of Nv neighbouring voxels:

Min : The lowest grey level in Xgl.

Max : The highest grey level in Xgl.

10th percentile : The 10th percentile of Xgl. This is more robust to grey level outliers than the minimum grey level.

90th percentile : The 90th percentile of Xgl. This is more robust to grey level outliers than the maximum grey level.

Median : The sample median of Xgl.

$Mean, \ \mu =\frac{\sum_{k=1}^{N_v} X_{gl,k}}{N_v}$

$Range = \max(X_{gl}) - \min(X_{gl})$

$Variance = \frac{\sum_{k=1}^{N_v}(X_{gl,k - \mu})^2}{N_v}$

$Standard \ deviation = \sqrt{Variance}$

$Skewness = \frac{\frac{\sum_{k=1}^{N_v}(X_{gl,k - \mu})^3}{N_v}} {(\frac{\sum_{k=1}^{N_v}(X_{gl,k - \mu})^2}{Nv})^{3/2})}$

$Kurtosis = \frac{\frac{\sum_{k=1}^{N_v}(X_{gl,k - \mu})^4}{N_v}} {(\frac{\sum_{k=1}^{N_v}(X_{gl,k - \mu})^2}{Nv})^2)} - 3$

$Total \ energy = \sum_{k=1}^{N_v} X_{gl,k}^2$

$Root \ Mean \ Square = \sqrt{}\frac{\sum_{k=1}^{N_v} X_{gl,k}^2}{N_v}$

$Mean \ absolute \ deviation = \frac{\sum_{k=1}^{N_v}|X_{gl,k}-\mu|}{N_v}$

$Median \ absolute \ deviation = \frac{\sum_{k=1}^{N_v}|X_{gl,k} - M|}{N_v}$

Robust Mean Absolute Deviation : Mean absolute deviation may be influenced by outliers. To increase robustness, the set of grey levels can be restricted to those which lie closer to the center of the distribution.

$Robust \ mean \ absolute \ deviation = \frac{\sum_{k=1}^{N_{v,10-90}}|X_{gl,10-90,k}-\bar{X}_{gl,10-90,k}|}{N_{v,10-90}}\\$ $where \ X_{gl,10-90} = \{x\in X_{gl}|P_{10}(X_{gl})\leq x\leq P_{90}(X_{gl}) \} \ and$ $X_{gl,10-90} \ denotes \ the \ sample \ mean \ of \ X_{gl,10-90}$

$Inter-Quartile \ Range = P_{75} - P_{25}$
where P₂₅ and P₇₅ are the 25^th and 75^th percentiles of X_gl, respectively

$Quartile \ coefficient \ of \ Dispersion = \frac{P_{75}-P_{25}}{P_{75}+P_{25}}$

$Coefficient \ of \ Variation = \frac{\sigma}{\mu}$

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First order statistics

List of available features

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