Radiomic Features in GlioMap

First-Order Statistical Features

1. Mean

Average intensity:

$$ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i $$

2. Median

The middle value of the sorted intensities.

3. Mode

The most frequent intensity value.

4. Minimum

Smallest intensity: $ \min(x_i) $

5. Maximum

Largest intensity: $ \max(x_i) $

6. Range

Difference between max and min:

$$ \text{Range} = \max(x) - \min(x) $$

7. Interquartile Range (IQR)

Middle 50% spread:

$$ \text{IQR} = Q_3 - Q_1 $$

8. Variance

Spread of intensity values:

$$ \sigma^2 = \frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2 $$

9. Standard Deviation

Root of variance:

$$ \sigma = \sqrt{ \frac{1}{N-1} \sum (x_i - \mu)^2 } $$

10. Skewness

Asymmetry of distribution:

$$ \text{Skewness} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i - \mu}{\sigma} \right)^3 $$

11. Kurtosis

Tailedness (excess):

$$ \text{Kurtosis} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i - \mu}{\sigma} \right)^4 - 3 $$

12. Energy

Sum of squared intensities:

$$ \text{Energy} = \sum_{i=1}^{N} x_i^2 $$

13. Entropy

Measure of randomness:

$$ \text{Entropy} = -\sum_{j=1}^{B} p_j \log_2(p_j) $$

14. Uniformity

Histogram smoothness:

$$ \text{Uniformity} = \sum_{j=1}^{B} p_j^2 $$

15. Root Mean Square (RMS)

Magnitude of signal:

$$ \text{RMS} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} x_i^2 } $$

16. Mean Absolute Deviation (MAD)

Average deviation from mean:

$$ \text{MAD} = \frac{1}{N} \sum_{i=1}^{N} |x_i - \mu| $$

17. Robust MAD

Mean absolute deviation from the trimmed mean (10–90% data).

18. Median Absolute Deviation

Median of absolute deviations from median:

$$ \text{MAD}_{\text{median}} = \text{median}(|x_i - \text{median}(x)|) $$

19. Coefficient of Variation (CoV)

Standard deviation normalized by mean:

$$ \text{CoV} = \frac{\sigma}{\mu} $$

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GLCM Texture Features

Notation

$ P(i,j) $: Normalized GLCM matrix
$ \mu_x = \sum_i i \sum_j P(i,j) $: Mean of reference pixel
$ \mu_y = \sum_j j \sum_i P(i,j) $: Mean of neighbor pixel
$ \sigma_x, \sigma_y $: Standard deviations along rows and columns
$ p_{x+y}(k) = \sum_{i+j=k} P(i,j) $: Sum distribution
$ p_{|x-y|}(k) = \sum_{|i-j|=k} P(i,j) $: Difference distribution

1. Autocorrelation

Measures repetition of pixel pairs:

$$ \text{Autocorrelation} = \sum_{i,j} i \cdot j \cdot P(i,j) $$

2. Contrast

Local gray-level variations:

$$ \text{Contrast} = \sum_{i,j} (i - j)^2 \cdot P(i,j) $$

3. Correlation

Linear dependency between pixel pairs:

$$ \text{Correlation} = \frac{\sum_{i,j} (i - \mu_x)(j - \mu_y)P(i,j)}{\sigma_x \sigma_y} $$

4. Cluster Prominence

Sharpness or asymmetry in distribution:

$$ \text{Cluster Prominence} = \sum_{i,j} (i + j - \mu_x - \mu_y)^4 \cdot P(i,j) $$

5. Cluster Shade

Skewness of texture:

$$ \text{Cluster Shade} = \sum_{i,j} (i + j - \mu_x - \mu_y)^3 \cdot P(i,j) $$

6. Dissimilarity

Linear gray-level difference:

$$ \text{Dissimilarity} = \sum_{i,j} |i - j| \cdot P(i,j) $$

7. Energy

Angular second moment (uniformity):

$$ \text{Energy} = \sum_{i,j} P(i,j)^2 $$

8. Entropy

Texture randomness:

$$ \text{Entropy} = -\sum_{i,j} P(i,j) \log_2(P(i,j) + \epsilon) $$

9. Homogeneity 1 (IDM)

Inverse difference moment:

$$ \text{Homogeneity}_1 = \sum_{i,j} \frac{P(i,j)}{1 + (i - j)^2} $$

10. Homogeneity 2 (IDN)

Normalized homogeneity:

$$ \text{Homogeneity}_2 = \sum_{i,j} \frac{P(i,j)}{1 + \frac{|i - j|}{N}} $$

11. Maximum Probability

Maximum probability in GLCM:

$$ \text{Max Probability} = \max(P(i,j)) $$

12. Sum Average

Average of sum distribution:

$$ \text{Sum Average} = \sum_k k \cdot p_{x+y}(k) $$

13. Sum Entropy

Entropy of sum distribution:

$$ \text{Sum Entropy} = -\sum_k p_{x+y}(k) \log_2(p_{x+y}(k) + \epsilon) $$

14. Sum Variance

Spread around sum average:

$$ \text{Sum Variance} = \sum_k (k - \text{SumAvg})^2 \cdot p_{x+y}(k) $$

15. Difference Entropy

Entropy of difference distribution:

$$ \text{Diff Entropy} = -\sum_k p_{|x-y|}(k) \log_2(p_{|x-y|}(k) + \epsilon) $$

16. Difference Variance

Variance of difference distribution:

$$ \text{Diff Variance} = \sum_k (k - \mu)^2 \cdot p_{|x-y|}(k) $$

17. Information Measure of Correlation 1

Mutual dependence:

$$ \text{IMC}_1 = \frac{H_{XY} - H_{XY1}}{\max(H_X, H_Y)} $$

18. Information Measure of Correlation 2

Alternative mutual info measure:

$$ \text{IMC}_2 = \sqrt{1 - \exp(-2(H_{XY2} - H_{XY}))} $$

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Gray Level Run Length Matrix (GLRLM) Features

Notation

$ \text{GLRLM}(i,j) $: Number of runs with gray level $ i $ and run length $ j $
$ i $: Gray level index, $ i = 1, 2, \dots, G $
$ j $: Run length index, $ j = 1, 2, \dots, R $
$ N $: Total number of runs, $ N = \sum_{i=1}^{G} \sum_{j=1}^{R} \text{GLRLM}(i,j) $
$ G $: Number of gray levels
$ R $: Maximum run length
$ \text{Total Voxels in ROI} $: Number of voxels in the region of interest (tumor region)

1. Short Run Emphasis (SRE)

Measures the distribution of short runs, emphasizing fine textures.

$$ \text{SRE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} \frac{\text{GLRLM}(i,j)}{j^2} $$

2. Long Run Emphasis (LRE)

Measures the distribution of long runs, emphasizing coarse textures.

$$ \text{LRE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} j^2 \cdot \text{GLRLM}(i,j) $$

3. Gray Level Non-Uniformity (GLNU)

Measures variability of gray level values throughout the image.

$$ \text{GLNU} = \frac{1}{N} \sum_{i=1}^{G} \left( \sum_{j=1}^{R} \text{GLRLM}(i,j) \right)^2 $$

4. Run Length Non-Uniformity (RLNU)

Measures variability of run lengths throughout the image.

$$ \text{RLNU} = \frac{1}{N} \sum_{j=1}^{R} \left( \sum_{i=1}^{G} \text{GLRLM}(i,j) \right)^2 $$

5. Run Percentage (RP)

Measures the density of runs relative to the total voxels in the ROI.

$$ \text{RP} = \frac{N}{\text{Total Voxels in ROI}} $$

6. Low Gray Level Run Emphasis (LGRE)

Measures the distribution of low gray-level runs, highlighting fine low-intensity textures.

$$ \text{LGRE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} \frac{\text{GLRLM}(i,j)}{i^2} $$

7. High Gray Level Run Emphasis (HGRE)

Measures the distribution of high gray-level runs, highlighting coarse high-intensity textures.

$$ \text{HGRE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} i^2 \cdot \text{GLRLM}(i,j) $$

8. Short Run Low Gray Level Emphasis (SRLGE)

Measures the joint distribution of short runs with low gray levels.

$$ \text{SRLGE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} \frac{\text{GLRLM}(i,j)}{i^2 \cdot j^2} $$

9. Short Run High Gray Level Emphasis (SRHGE)

Measures the joint distribution of short runs with high gray levels.

$$ \text{SRHGE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} \frac{i^2 \cdot \text{GLRLM}(i,j)}{j^2} $$

10. Long Run Low Gray Level Emphasis (LRLGE)

Measures the joint distribution of long runs with low gray levels.

$$ \text{LRLGE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} \frac{j^2 \cdot \text{GLRLM}(i,j)}{i^2} $$

11. Long Run High Gray Level Emphasis (LRHGE)

Measures the joint distribution of long runs with high gray levels.

$$ \text{LRHGE} = \frac{1}{N} \sum_{i=1}^{G} \sum_{j=1}^{R} i^2 \cdot j^2 \cdot \text{GLRLM}(i,j) $$

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GLSZM Features

Notation

$ \text{GLSZM}(i, j) $: Number of zones with gray level $ i $ and size $ j $
$ N_g $: Number of gray levels
$ N_s $: Number of zone sizes
$ N_z $: Total number of zones
$ N_p $: Total number of voxels in ROI
$ s_g(i) = \sum_{j=1}^{N_s} \text{GLSZM}(i, j) $: Zones with gray level $ i $
$ s_z(j) = \sum_{i=1}^{N_g} \text{GLSZM}(i, j) $: Zones with size $ j $

1. Small Area Emphasis (SAE)

\[ \text{SAE} = \frac{1}{N_z} \sum_{j=1}^{N_s} \frac{s_z(j)}{j^2} \]

Measures the prevalence of small zones. High SAE suggests finer textures or many small homogeneous areas.

2. Large Area Emphasis (LAE)

\[ \text{LAE} = \frac{1}{N_z} \sum_{j=1}^{N_s} s_z(j) \cdot j^2 \]

Highlights larger homogeneous zones. High LAE implies coarser textures with large uniform areas.

3. Gray Level Non-Uniformity (GLNU)

\[ \text{GLNU} = \frac{1}{N_z} \sum_{i=1}^{N_g} \left( s_g(i) \right)^2 \]

Measures how gray levels are distributed. Low values suggest uniform gray levels in zones.

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Gray Level Size Zone Matrix (GLSZM) Features

Notation

$ \text{GLSZM}(i, j) $: Number of zones with gray level $ i $ and size $ j $
$ N_g $: Number of gray levels
$ N_s $: Number of zone sizes
$ N_z $: Total number of zones
$ N_p $: Total number of voxels in ROI
$ s_g(i) = \sum_{j=1}^{N_s} \text{GLSZM}(i, j) $: Gray level distribution
$ s_z(j) = \sum_{i=1}^{N_g} \text{GLSZM}(i, j) $: Zone size distribution

4. Zone Size Non-Uniformity (ZSNU)

Quantifies how zone sizes vary. Higher values imply dominance by certain zone sizes:

$$ \text{ZSNU} = \frac{1}{N_z} \sum_{j=1}^{N_s} \left( s_z(j) \right)^2 $$

5. Zone Percentage (ZP)

Ratio of total zones to total voxels in ROI. Indicates texture fragmentation:

$$ \text{ZP} = \frac{N_z}{N_p} $$

6. Low Gray Level Zone Emphasis (LGZE)

Emphasizes zones with low gray-level intensities:

$$ \text{LGZE} = \frac{1}{N_z} \sum_{i=1}^{N_g} \frac{s_g(i)}{i^2} $$

7. High Gray Level Zone Emphasis (HGZE)

Gives importance to zones with higher intensity values:

$$ \text{HGZE} = \frac{1}{N_z} \sum_{i=1}^{N_g} s_g(i) \cdot i^2 $$

8. Small Area Low Gray Level Emphasis (SALGLE)

Combines small zone sizes and low gray-levels, emphasizing fine dark textures:

$$ \text{SALGLE} = \frac{1}{N_z} \sum_{i=1}^{N_g} \sum_{j=1}^{N_s} \frac{\text{GLSZM}(i, j)}{i^2 j^2} $$

9. Small Area High Gray Level Emphasis (SAHGLE)

Highlights small regions with high intensity, potentially indicating bright, small lesions:

$$ \text{SAHGLE} = \frac{1}{N_z} \sum_{i=1}^{N_g} \sum_{j=1}^{N_s} \frac{\text{GLSZM}(i, j) \cdot i^2}{j^2} $$

10. Large Area Low Gray Level Emphasis (LALGLE)

Focuses on large, uniform dark zones within the image:

$$ \text{LALGLE} = \frac{1}{N_z} \sum_{i=1}^{N_g} \sum_{j=1}^{N_s} \frac{\text{GLSZM}(i, j) \cdot j^2}{i^2} $$

11. Large Area High Gray Level Emphasis (LAHGLE)

Detects large, bright homogeneous zones—often related to aggressive pathology:

$$ \text{LAHGLE} = \frac{1}{N_z} \sum_{i=1}^{N_g} \sum_{j=1}^{N_s} \text{GLSZM}(i, j) \cdot i^2 \cdot j^2 $$

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Gray Level Dependence Matrix (GLDM) Features

Notation

$ \text{GLDM}(i, j) $: Number of voxels with gray level $ i $ and dependence $ j $
$ N_g $: Number of gray levels
$ N_d $: Number of dependence levels
$ N_s $: Total number of voxel entries in the GLDM matrix
$ i $: Gray level index
$ j $: Dependence level index

1. Small Dependence Emphasis (SDE)

Emphasizes small dependence counts. High SDE indicates many small neighborhoods of similar intensity:

$$ \text{SDE} = \frac{1}{N_s} \sum_{j=1}^{N_d} \frac{p_j}{j^2} $$

2. Large Dependence Emphasis (LDE)

Highlights large dependence regions. High LDE implies coarse textures with extended similar intensity areas:

$$ \text{LDE} = \frac{1}{N_s} \sum_{j=1}^{N_d} p_j \cdot j^2 $$

3. Gray Level Non-Uniformity (GLN)

Measures non-uniformity of gray levels. Low GLN means uniform gray-level distribution:

$$ \text{GLN} = \frac{1}{N_s} \sum_{i=1}^{N_g} p_i^2 $$

4. Dependence Non-Uniformity (DN)

Measures non-uniformity of dependence sizes. Higher DN reflects dominance of certain dependence levels:

$$ \text{DN} = \frac{1}{N_s} \sum_{j=1}^{N_d} p_j^2 $$

5. Dependence Entropy (DE)

Captures the randomness in dependence distribution. Higher DE implies more texture complexity:

$$ \text{DE} = -\sum_{i=1}^{N_g} \sum_{j=1}^{N_d} \text{GLDM}(i,j) \log_2(\text{GLDM}(i,j) + \epsilon) $$

6. Dependence Variance (DV)

Variance of dependence size around the mean. Indicates spread of dependence levels:

$$ \text{DV} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} (j - \mu_j)^2 \cdot \text{GLDM}(i,j) $$

7. Gray Level Variance (GLV)

Variance of gray levels in the matrix. Measures intensity dispersion:

$$ \text{GLV} = \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} (i - \mu_i)^2 \cdot \text{GLDM}(i,j) $$

8. Large Dependence High Gray Level Emphasis (LDHGLE)

Highlights large dependent regions with high intensity values:

$$ \text{LDHGLE} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} i^2 \cdot j^2 \cdot \text{GLDM}(i,j) $$

9. Large Dependence Low Gray Level Emphasis (LDLGLE)

Emphasizes large areas of low gray levels, useful for highlighting dark homogeneous textures:

$$ \text{LDLGLE} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} \frac{j^2}{i^2} \cdot \text{GLDM}(i,j) $$

10. Small Dependence High Gray Level Emphasis (SDHGLE)

Captures bright regions with small dependence, potentially identifying small, intense lesions:

$$ \text{SDHGLE} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} \frac{i^2}{j^2} \cdot \text{GLDM}(i,j) $$

11. Small Dependence Low Gray Level Emphasis (SDLGLE)

Focuses on small, low-intensity zones often associated with fine dark textures:

$$ \text{SDLGLE} = \frac{1}{N_s} \sum_{i=1}^{N_g} \sum_{j=1}^{N_d} \frac{1}{i^2 j^2} \cdot \text{GLDM}(i,j) $$

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Neighborhood Gray Tone Difference Matrix (NGTDM) Features

Notation

$ s(i) $: Average absolute difference between gray level $ i $ and the mean of its neighbors
$ p(i) $: Probability of gray level $ i $ in the region of interest
$ N_g $: Number of gray levels with non-zero probability
$ \epsilon $: Small constant to prevent division by zero

1. Coarseness

Measures the level of texture smoothness. A higher value implies a coarser and less detailed texture:

$$ \text{Coarseness} = \frac{1}{\epsilon + \sum_{i} p(i) \cdot s(i)} $$

2. Contrast

Quantifies the intensity variation between different gray levels. Higher values indicate greater variation or edges:

$$ \text{Contrast} = \left( \frac{1}{N_g (N_g - 1)} \right) \cdot \left( \sum_{i=1}^{N} s(i) \right) \cdot \left( \sum_{i} \sum_{j} p(i) \cdot p(j) \cdot (i - j)^2 \right) $$

3. Busyness

Measures the rate of gray level change across the image. High values suggest rapid changes and busy textures:

$$ \text{Busyness} = \frac{ \sum_{i} p(i) \cdot s(i) }{ \sum_{i} \sum_{j} | i \cdot p(i) - j \cdot p(j) | } $$

4. Complexity

Indicates how varied and unpredictable the texture is. High complexity means diverse and intricate structures:

$$ \text{Complexity} = \sum_{i} \sum_{j \ne i} \left( \frac{ |i - j| }{ N_g (p(i) + p(j)) } \right) \cdot (p(i) \cdot s(i) + p(j) \cdot s(j)) $$

5. Strength

Measures how prominent or strong the texture features are in terms of intensity and spatial arrangement:

$$ \text{Strength} = \frac{ \sum_{i} \sum_{j} (p(i) + p(j)) \cdot (i - j)^2 }{ \sum_{i} s(i) } $$

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Wavelet-Based Radiomic Features

Wavelet-based radiomic features involve performing a 3D discrete wavelet transform (DWT) on the masked MRI tumor region. The transformed subbands isolate texture patterns at various spatial resolutions and orientations. First-order statistics are then calculated on each subband to quantify intensity distribution and variability.

Wavelet Subbands

Each wavelet decomposition generates 8 subbands, representing combinations of high (H) and low (L) pass filtering across 3 spatial dimensions (x, y, z):

LLL: Low-pass filtered in all directions (approximation)
LLH, LHL, LHH: One or two high-pass directions
HLL, HLH, HHL: High-pass in x and/or y
HHH: High-pass in all directions (fine details)

Notation

$ x_i $: Intensity of voxel $ i $ in the wavelet subband
$ N $: Total number of non-zero (tumor) voxels in the subband

1. Mean

Average voxel intensity:

$$ \text{Mean} = \frac{1}{N} \sum_{i=1}^{N} x_i $$

2. Variance

Intensity dispersion around the mean:

$$ \text{Variance} = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 $$

where $ \mu $ is the mean intensity.

3. Skewness

Asymmetry of the intensity distribution:

$$ \text{Skewness} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i - \mu}{\sigma} \right)^3 $$

4. Kurtosis

Peakedness or flatness of the distribution:

$$ \text{Kurtosis} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i - \mu}{\sigma} \right)^4 $$

5. Energy

Sum of squared intensities (signal strength):

$$ \text{Energy} = \sum_{i=1}^{N} x_i^2 $$

6. Root Mean Square (RMS)

Magnitude of voxel intensities:

$$ \text{RMS} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} x_i^2 } $$

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Gabor Filter Features

Gabor filters are used to extract texture features from an image by convolving the image with sinusoidal waves modulated by a Gaussian envelope. They are parameterized by frequency, orientation, and standard deviation.

Notation

$ x, y $: Coordinates in the filter window
$ \theta $: Orientation of the filter (in radians)
$ f $: Spatial frequency of the sinusoidal wave
$ \sigma $: Standard deviation of the Gaussian envelope
$ G(x,y) $: Value of the Gabor filter at position $(x,y)$

Mathematical Equation

Gabor filter definition:

$$ G(x,y) = \exp\left(-\frac{X_r^2 + Y_r^2}{2\sigma^2}\right) \cdot \cos\left(2 \pi f X_r\right) $$

where

$$ \begin{cases} X_r = x \cos \theta + y \sin \theta \\ Y_r = -x \sin \theta + y \cos \theta \end{cases} $$

Feature Calculations

Mean Amplitude: Average absolute response of the Gabor filter over the tumor region

Energy: Sum of squared filter responses:

$$ \text{Energy} = \sum_i |R_i|^2 $$

Variance: Variance of the filter responses

Standard Deviation: Standard deviation of the filter responses

Orientation Entropy: Entropy of responses across orientations:

$$ H = -\sum_{k} p_k \log_2(p_k) $$

where $ p_k $ is the normalized mean response for orientation $ k $

Dominant Orientation: Orientation $ \theta_k $ corresponding to the maximum mean response

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Fourier-Based Features

Notation

$ P(u,v) $: Power spectrum at frequency coordinates $(u,v)$
$ R $: Radial distance from the center frequency
$ \hat{P}(u,v) $: Normalized power spectrum
$ \epsilon $: Small positive constant to avoid $\log(0)$
$ \theta $: Angle in frequency domain

1. Spectral Energy

Total energy of the power spectrum:

$$ E = \sum_{u,v} P(u,v) $$

2. Spectral Entropy

Measures the randomness in frequency distribution:

$$ H = - \sum_{u,v} \hat{P}(u,v) \log_2 \hat{P}(u,v) \quad \text{where} \quad \hat{P}(u,v) = \frac{P(u,v)}{\sum_{u,v} P(u,v)} $$

3. Radial Power Spectrum

Average power at radius $r$:

$$ R = \sqrt{(u - u_c)^2 + (v - v_c)^2} $$

$$ \text{Radial Power at } r = \text{mean}\left\{ P(u,v) \mid r - 0.5 \leq R < r + 0.5 \right\} $$

4. Low and High Frequency Power

Sum of power inside inner 25% radius (low frequencies):

$$ P_{\text{low}} = \sum_{R \leq 0.25 R_{\max}} P(u,v) $$

Sum of power outside outer 25% radius (high frequencies):

$$ P_{\text{high}} = \sum_{R \geq 0.75 R_{\max}} P(u,v) $$

5. Frequency Centroid

Weighted average radius representing center of frequency mass:

$$ C = \frac{\sum_{u,v} R \cdot P(u,v)}{\sum_{u,v} P(u,v)} $$

6. Dominant Frequency

Radius at which the power spectrum attains maximum value:

$$ D = \arg\max_{u,v} P(u,v) \quad \rightarrow \quad R_D = \sqrt{(u - u_c)^2 + (v - v_c)^2} $$

7. Texture Periodicity

Measure of regularity by autocorrelation of power spectrum:

$$ \text{Periodicity} = \frac{\sigma(\text{Autocorr}_{\text{center}})}{\mu(\text{Autocorr}_{\text{center}})} $$

where $\sigma$ and $\mu$ are standard deviation and mean of the autocorrelation values in the central region.

8. Directional Frequency Components

Average power in four principal angular directions $\theta_k \in \{0^\circ, 45^\circ, 90^\circ, 135^\circ\}$:

$$ P_{\theta_k} = \text{mean}\left\{ P(u,v) \mid |\angle(u,v) - \theta_k| \leq 22.5^\circ \right\} $$

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Tamura Texture Features

Notation

$ I(x,y) $: Intensity at pixel $(x,y)$
$ M_k(x,y) $: Average intensity in a window of size $ (2k+1) \times (2k+1) $ centered at $(x,y)$
$ \mu $, $ \sigma^2 $: Mean and variance of intensities
$ m_4 $: Fourth central moment of intensity
$ G_x, G_y $: Gradient components in x and y directions
$ \theta $: Gradient orientation
$ h(\theta) $: Histogram of gradient orientations
$ N $: Number of bins in histogram

1. Coarseness

Measures the size of texture primitives by evaluating average differences at multiple scales:

$$ M_k(x,y) = \frac{1}{(2k+1)^2} \sum_{i=-k}^k \sum_{j=-k}^k I(x+i, y+j) $$

Horizontal and vertical differences at scale $k$:

$$ E_h^k(x,y) = | M_k(x+ k + 1, y) - M_k(x - k, y) |, \quad E_v^k(x,y) = | M_k(x, y + k + 1) - M_k(x, y - k) | $$

At each pixel, select scale $S_{best}(x,y)$ maximizing these differences:

$$ S_{best}(x,y) = \arg\max_k \max \left( E_h^k(x,y), E_v^k(x,y) \right) $$

Coarseness is average of $ S_{best} $ over the region:

$$ \text{Coarseness} = \frac{1}{N} \sum_{x,y} S_{best}(x,y) $$

2. Contrast

Measures intensity variation, considering variance and kurtosis:

$$ \mu = \frac{1}{N} \sum I(x,y), \quad \sigma^2 = \frac{1}{N} \sum (I(x,y) - \mu)^2, \quad m_4 = \frac{1}{N} \sum (I(x,y) - \mu)^4 $$

Contrast defined as:

$$ \text{Contrast} = \frac{\sigma^2}{m_4^{1/4}} $$

3. Directionality

Quantifies the strength of texture direction based on gradient histogram:

$$ G_x = \frac{\partial I}{\partial x}, \quad G_y = \frac{\partial I}{\partial y} $$

$$ \theta = \arctan\left(\frac{G_y}{G_x}\right), \quad \theta \in [0^\circ, 180^\circ) $$

$$ h(\theta_i) = \text{normalized histogram of } \theta $$

Directionality is the sum of squared histogram peaks $P$:

$$ \text{Directionality} = \sum_{i \in P} h(\theta_i)^2 $$

4. Line-Likeness

Measures similarity of gradient directions in neighboring pixels using co-occurrence matrix $C$:

$$ C(a,b) = \frac{\# \text{neighbor pairs with directions } a \text{ and } b}{\text{total pairs}} $$

Line-likeness computed as diagonal dominance:

$$ \text{LineLikeness} = \frac{\sum_a C(a,a)}{\sum_{a,b} C(a,b)} $$

5. Regularity

Based on variance of local coarseness and contrast over subregions:

$$ \text{Regularity} = \frac{1}{1 + \mathrm{Var}(\text{Coarseness}) + \mathrm{Var}(\text{Contrast})} $$

6. Roughness

Combined measure of coarseness and contrast:

$$ \text{Roughness} = \text{Coarseness} + \text{Contrast} $$