Features & Mathematical Formulations

fastrad implements seven foundational radiomics feature classes completely compliant with the IBSI (Image Biomarker Standardisation Initiative). While our backend circumvents procedural looping via PyTorch vectorization, the output mathematical equations are identical to legacy packages like PyRadiomics.

First Order Statistics

(Note: Prior to feature extraction, `fastrad` supports native PyTorch GPU Mathematical filters mirroring PyRadiomics including `LoG`, `Square`, `SquareRoot`, `Logarithm`, and `Exponential`.)

First-order statistics describe the distribution of voxel intensities within the image region $X$ defined by the mask through basic and commonly used metrics. Let $N_p$ be the number of voxels, and $X(i)$ be the intensity of the $i$-th voxel.

\[\text{Mean} = \frac{1}{N_p} \sum_{i=1}^{N_p} X(i)\]
\[\text{Variance} = \frac{1}{N_p - 1} \sum_{i=1}^{N_p} (X(i) - \text{Mean})^2\]
\[\text{Entropy} = -\sum_{i=1}^{N_g} p(i) \log_2(p(i) + \epsilon)\]

Where $N_g$ is the number of discrete intensity bins and $p(i)$ is the probability of a voxel having intensity $i$.

  • Available Features: Mean, Median, Variance, Skewness, Kurtosis, Maximum, Minimum, Range, Interquartile Range, Root Mean Squared (RMS), Energy, Total Energy, Entropy, Uniformity, 90th/10th Percentiles, Robust Mean Absolute Deviation.

Shape (2D and 3D)

Shape descriptors quantify the morphological and geometric properties of the Region of Interest (ROI), completely independent of the actual pixel intensities inside the tumor or organ.

\[\text{Sphericity} = \frac{\pi^{\frac{1}{3}} (6V)^{\frac{2}{3}}}{A}\]

Where $V$ is the Volume (number of voxels $times$ physical voxel volume) and $A$ is the Surface Area computed via marching-cubes or voxel-faces matching.

  • Available Features: Maximum 3D Diameter, Volume, Surface Area, Sphericity, Surface to Volume Ratio, Elongation, Flatness, Major/Minor Axis Length (via PCA constraints).

Gray Level Co-occurrence Matrix (GLCM)

A GLCM of size $N_g times N_g$ describes the second-order joint probability function of an image region. Let $P(i, j)$ be the probability that a voxel with intensity $i$ is separated from a voxel with intensity $j$ by a fixed distance $delta$ in a defined angle.

\[\text{Contrast} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_g} (i - j)^2 P(i,j)\]
\[\text{Correlation} = \frac{\sum_{i=1}^{N_g} \sum_{j=1}^{N_g} p(i,j) i j - \mu_{x} \mu_{y}}{\sigma_{x}(i) \sigma_{y}(j)}\]

fastrad computes all 13 symmetric 3D angles simultaneously using volumetric shifted tensor concatenations (torch.roll), eliminating nested pixel lookups entirely.

  • Available Features: Autocorrelation, Joint Average, Cluster Prominence, Cluster Shade, Cluster Tendency, Contrast, Correlation, Inverse Difference Moment (IDM), Maximum Probability, Sum Average, Sum Variance.

Gray Level Size Zone Matrix (GLSZM)

A connected component $c$ is defined as a sequence of logically connected voxels exhibiting identical discretized gray-levels. The GLSZM $P(i, j)$ represents the number of zones with gray-level $i$ and voxel size $j$.

\[\text{Small Area Emphasis (SAE)} = \frac{1}{N_z} \sum_{i=1}^{N_g}\sum_{j=1}^{N_s} \frac{P(i,j)}{j^2}\]

Where $N_z$ is the total zone count in the volume, and $N_s$ is the maximum zone size limits.

Note: fastrad uniquely implements this algorithm utilizing a high-performance CuPy / cuCIM topological labeling subroutine when executed on GPUs, avoiding the severe performance trap common to zone labeling.

  • Available Features: Small/Large Area Emphasis, Gray Level Non-Uniformity, Zone Percentage, Zone Variance, Size Zone Non-Uniformity.

Gray Level Dependence Matrix (GLDM)

A GLDM $P(i, j)$ tracks occurrences where a center voxel defined as gray level $i$ has precisely $j$ surrounding neighbor voxels within a radius $delta$ that share an identical gray level (or fall within a strict alpha tolerance boundary).

\[\text{Dependence Entropy} = -\sum_{i=1}^{N_g}\sum_{j=1}^{N_d} p(i,j) \log_2(p(i,j) + \epsilon)\]

fastrad accelerates 3D dependency aggregations via dynamic spherical boolean masks multiplied across massive 4D pre-allocated tensors.

  • Available Features: Small/Large Dependence Emphasis, Gray Level/Dependence Non-Uniformity, Dependence Variance, Dependence Entropy.

Neighbourhood Gray Tone Difference Matrix (NGTDM)

Quantifies the difference between a specific gray value $i$ and the average gray value of its immediate spatial neighbors $bar{A}_i$ extending up to distance $delta$.

\[s_i = \sum^{N_p}_{k=1} |i - \bar{A}_i| \text{ for } X(k)=i\]
\[\text{Coarseness} = \frac{1}{\sum_{i=1}^{N_g} p_i \cdot s_i}\]

  • Available Features: Coarseness, Contrast, Busyness, Complexity, Strength.