Feature Family  Specific
Features  Parameter
Name  Range  Default  Description, Formula and Comments 
Intensity Features
(FirstOrder Statistics) 

Minimum

Maximum

Mean

Standard Deviation

Variance

Skewness

Kurtosis
 N.A.  N.A.  N.A. 

Minimum Intensity = \( Min (I_{k}). \) where \( I_{k} \) is the intensity of pixel or voxel at index k.

Maximum Intensity = \( Max (I_{k}). \) where \( I_{k} \) is the intensity of pixel or voxel at index k.

Mean= \( \frac{\sum(X_{i})}{N} \) where N is the number of voxels/pixels.

Standard Deviation = \( \sqrt{\frac{\sum(X\mu)^{2}}{N}}\) where \(\mu\) is the mean of the data.

Variance = \( \frac{\sum(X\mu)^{2}}{N} \) where \(\mu\) is the mean intensity.

Skewness = \( \frac{\sum_{i=1}^{N}(X_{i}  \bar{X})^{3}/N} {s^{3}} \) where \(\bar{X}\) is the mean, s is the standard deviation and N is the number of pixels/voxels.

Kurtosis = \( \frac{\sum_{i=1}^{N}(X_{i}  \bar{X})^{4}/N}{s^{4}} \) where \(\bar{X}\) is the mean, s is the standard deviation and N is the number of pixels/voxels.
All features in this family are extracted from the raw intensities. 
Histogram
based 
 Bins  N.A.  10 

Uses number of bins as input and the number of pixels in each bin would be the output.
All features in this family are extracted from the discretized intensities. 
Volumetric 
 Dimensions
Axis  2D:3D
x,y,z  3D
z 

Volume/Area (depending on image dimension) and number of voxels/pixels in the ROI.

Morphologic 

Elongation

Perimeter

Roundness

Eccentricity

Ellipse Diameter

Equivalent Spherical Radius
 Dimensions
Axis  2D:3D
x,y,z  3D
z 

Elongation = \( \sqrt{\frac{i_{2}}{i_{1}}} \) where \(i_{n}\) are the second moments of particle around its principal axes.

Perimeter = \( 2 \pi r \) where r is the radius of the circle enclosing the shape.

Roundness = \( As/Ac = (Area of a shape)/(Area of circle) \) where circle has the same perimeter.

Eccentricity = \( \sqrt{1  \frac{a*b}{c^{2}}} \) where c is the longest semiprincipal axis of an ellipsoid fitted on an ROI, and a and b are the 2nd and 3rd longest semiprincipal axes of the ellipsoid.

Edge
Enhancing
Index   Normalizing factor  \( \eta \)  5  The edgeenhancing index of an image I is defined as:

E(I) = \( (\frac{\lambda_{1}\lambda_{2}}{\lambda_{1}+\lambda_{2}+\eta})^2 \)
where \( \lambda_{1} \) and \( \lambda_{2} (\lambda_{1}>\lambda_{2}) \) are eigenvalues of the diffusion tension matrix of Image (I) and \( \eta \) is a normalizing factor. Available only on 2D. 
Local Binary
Pattern (LBP)   Radius
Neighborhood  1
2:4:8  1
8  The pixelwise LBP codes are computed using N number of neighbors on a circle of radius R around each pixel and using a rotation invariant implementation. The output value corresponds to the mean of the LBP map. 
Grey Level
Cooccurrence
Matrix
(GLCM) 

Energy (Angular Second Moment)

Contrast (Inertia)

Joint Entropy

Homogeneity (Inverse Difference Moment)

Correlation

Variance

SumAverage

Auto
Correlation
 Bins
Radius
Dimensions
Offset
Axis  N.A.
N.A.
2D:3D
Individual/Average/Combined
x,y,z  10
13
2
3D
Average
z  For a given image, a Grey Level Cooccurrence Matrix is created and \( g(i,j) \) represents an element in matrix

Energy = \( \sum_{i,j}g(i, j)^2 \)

Contrast = \( \sum_{i,j}(i  j)^2g(i, j) \)

Joint Entropy = \( \sum_{i,j}g(i, j) \log_2 g(i, j) \)

Homogeneity = \( \sum_{i,j}\frac{1}{1 + (i  j)^2}g(i, j) \)

Correlation = \( \sum_{i,j}\frac{(i  \mu)(j  \mu)g(i, j)}{\sigma^2} \)

Sum Average = \( \sum_{i,j}i \cdot g(i, j) = \sum_{i,j}j \cdot g(i, j)\)(due to matrix symmetry)

Variance = \( \sum_{i,j}(i  \mu)^2 \cdot g(i, j) = \sum_{i,j}(j  \mu)^2 \cdot g(i, j)\) (due to matrix symmetry)

AutoCorrelation = \(\frac{\sum_{i,j}(i, j) g(i, j)\mu_t^2}{\sigma_t^2}\) where \(\mu_t\) and \(\sigma_t\) are the mean and standard deviation of the row (or column, due to symmetry) sums.
All features are estimated within the ROI in an image, considering 26connected neighboring voxels in the 3D volume. Note that the creation of the GLCM and its corresponding aforementioned features for all offsets are calculated using an existing ITK filter. The Individual option gives features for each individual offset, Average estimates the average across all offsets and assigns a single value for each feature and Combined combines the GLCM matrices generated across offsets and calculates a single set of features from this matrix. 
Grey Level
RunLength
Matrix
(GLRLM) 

SRE

LRE

GLN

RLN

LGRE

HGRE

SRLGE

SRHGE

LRLGE

LRHGE
 Bins
Radius
Dimensions
Axis
Offset  N.A.
N.A.
2D:3D
x,y,z
Individual/Average/Combined  10
13
2
3D
z
Average
1  For a given image, a runlength matrix \( P(i; j)\) is defined as the number of runs with pixels of gray level i and run length j. Please note that some features are only extracted in DebugMode (by using the "d" parameter from the command line); these defines features that are mathematically formulated in previous published material but not completely aligned with The Image Biomarker Standardisation Initiative.

[COMPLETE MODE] Short Run Emphasis (SRE) = \( \frac{1}{n_r}\sum_{i,j}^{N}\frac{p(i,j)}{j^2} \)

[COMPLETE MODE] Long Run Emphasis (LRE) = \( \frac{1}{n_r}\sum_{j}^{N}p(i,j) \cdot j^2\)

[COMPLETE MODE] Grey Level Nonuniformity (GLN) = \( \frac{1}{n_r}\sum_{i}^{M}\Big(\sum_{j}^{N}p(i,j) \Big)^2 \)

[COMPLETE MODE] Run Length Nonuniformity (RLN) = \( \frac{1}{n_r}\sum_{j}^{N}\Big(\sum_{i}^{M}p(i,j) \Big)^2 \)

Low GreyLevel Run Emphasis (LGRE)= \( \frac{1}{n_r}\sum_{i}^{M}\frac{p_g(i)}{i^2} \)

High GreyLevel Run Emphasis (HGRE)= \( \frac{1}{n_r}\sum_{i}^{M}p_g(i) \cdot i^2 \)

Short Run Low GreyLevel Emphasis (SRLGE)= \(\frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j)}{i^2 \cdot j^2} \)

Short Run High GreyLevel Emphasis (SRLGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j) \cdot i^2 }{j^2}\)

[COMPLETE MODE] Long Run Low GreyLevel Emphasis (LRLGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j) \cdot j^2 }{i^2} \)

[COMPLETE MODE] Long Run High GreyLevel Emphasis (LRHGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}p(i,j) \cdot i^2 \cdot j^2 \)
All features are estimated within the ROI in an image, considering 26connected neighboring voxels in the 3D volume. Note that the creation of the GLRLM and its corresponding aforementioned features for all offsets are calculated using an existing ITK filter. The Individual option gives features for each individual offset, Average estimates the average across all offsets and assigns a single value for each feature and Combined combines the GLRLM matrices generated across offsets and calculates a single set of features from this matrix. 
Neighborhood
GreyTone
Difference
Matrix
(NGTDM) 

Coarseness

Contrast

Busyness

Complexity

Strength
 Bins
Dimensions
Axis  N.A.
2D:3D
x,y,z  10
13
3D
N.A.
1 

Coarseness = \( \Big[ \epsilon + \sum_{i=0}^{G_{k}} p_{i}s(i) \Big]\)

Contrast = \( \Big[\frac{1}{N_{s}(N_{s}1)}\sum_{i}^{G_{k}}\sum_{j}^{G_{k}}p_{i}p_{j}(ij)^2\Big]\Big[\frac{1}{n^2}\sum_{i}^{G_{k}}s(i)\Big] \)

Busyness = \( \Big[\sum_{i}^{G_{k}}p_{i}s(i)\Big]\Big/ \Big[\sum_{i}^{G_{k}}\sum_{j}^{G_{k}}i p_{i}  j p_{j}\Big] \)

Complexity = \( \sum_{i}^{G_{k}}\sum_{j}^{G_{k}} \Big[ \frac{(ij)}{(n^{2}(p_{i}+p_{j}))} \Big] \Big[ p_{i}s(i)+p_{j}s(j) \Big]\)

Strength = \( \Big[\sum_{i}^{G_{k}}\sum_{j}^{G_{k}}(p_{i}+p_{j})(ij)^{2}\Big]/\Big[\epsilon + \sum_{i}^{G_{k}} s(i)\Big]\)
Where \(p_{i}\) is the probability of occurrence of a voxel of intensity i and \(s(i)\) represents the NGTDM value of intensity i calculated as: \( \sum │i  Ai│\). Ai indicates the average intensity of the surrounding voxels without including the central voxel.

Grey Level
SizeZone
Matrix
(GLSZM) 

SZE

LZE

GLN

ZSN

ZP

LGZE

HGZE

SZLGE

SZHGE

LZLGE

LZHGE

GLV

ZLV
 Bins
Radius
Dimensions
Axis  N.A.
N.A.
2D:3D
x,y,z  10
13
2
3D
z
4  For a given image, a runlength matrix \( P(i; j)\) is defined as the number of runs with pixels of gray level i and run length j.

Small Zone Emphasis (SZE) = \( \frac{1}{n_r}\sum_{i,j}^{N}\frac{p(i,j)}{j^2} \)

Large Zone Emphasis(LZE) = \( \frac{1}{n_r}\sum_{j}^{N}p(i,j) \cdot j^2\)

GrayLevel Nonuniformity (GLN) = \( \frac{1}{n_r}\sum_{i}^{M}\Big(\sum_{j}^{N}p(i,j) \Big)^2 \)

ZoneSize Nonuniformity (ZSN) = \( \frac{1}{n_r}\sum_{j}^{N}\Big(\sum_{i}^{M}p(i,j) \Big)^2 \)

Zone Percentage (ZP) = \( \frac{n_{r}}{n_p} \) where \( n_r \) is the total number of runs and \( n_p \) is the number of pixels in the image.

Low GreyLevel Zone Emphasis (LGZE)= \( \frac{1}{n_r}\sum_{i}^{M}\frac{p_g(i)}{i^2} \)

High GreyLevel Zone Emphasis (HGZE)= \( \frac{1}{n_r}\sum_{i}^{M}p_g(i) \cdot i^2 \)

Short Zone Low GreyLevel Emphasis (SZLGE)= \(\frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j)}{i^2 \cdot j^2} \)

Short Zone High GreyLevel Emphasis (SZLGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j) \cdot i^2 }{j^2}\)

Long Zone Low GreyLevel Emphasis (LZLGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}\frac{p(i,j) \cdot j^2 }{i^2} \)

Long Zone High GreyLevel Emphasis (LZHGE) = \( \frac{1}{n_r}\sum_{i}^{M}\sum_{j}^{N}p(i,j) \cdot i^2 \cdot j^2 \)
All features are estimated within the ROI in an image, considering 26connected neighboring voxels in the 3D volume. 
Gabor Wavelets 

Mean

Standard deviation

Variance

Maximum

Sum
 Radius
Direction
Level
Gamma
Fmax  N.A.  1
N.A.
4
\( \sqrt{2} \)
0.25  For a given image, gabor filters are created for the number of directions specificed, with maximum sinusoid frequency determined by F_max. The \( \gamma \) parameter determines the spatial aspect ratio, and the levels parameter determines the number of levels to perform wavelet decomposition. Features are extracted for each resulting image generated for each direction and level. 
Power Spectrum  Beta  Center  N.A.  center of image  A discrete fourier transform is applied to each image to decompose the information in the image into signals with varying frequencies. The power spectrum is computed for each signal. Plotting the relation between spatial frequency and average power, the slope of a line fit to this relation is calculated.

Lattice
based 

Selected features

Feature Maps


FullImage

Window

Step

Boundary

PatchBoundary


01

(mm)0:ImageSize

(mm)0:ImageSize

NoPadding:ZeroPadding

Full:ROI:None

 When activated, this option performs feature calculation on multiple local square (for 2D images) or cubic (for 3D images) regions defined by a lattice virtually overlaid on the image.
Please see ${CaPTk_InstallDir}/data/features/2_params_default_lattice.csv for detailed descriptions. 