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Table 1 Definitions of QUS measurements and their mathematical formulas

From: Quantitative ultrasound imaging of Achilles tendon integrity in symptomatic and asymptomatic individuals: reliability and minimal detectable change

QUS measurements computed from a grayscale histogram Mathematical formulas
First order statistics
 Echogenicity
 Mean (\( \overline{\mathrm{x}} \))
Mean of grayscale values of micro pixels encompassed within the ROI (from 0 (black) to 255 (white) inclusively). \( \overline{\mathrm{x}} = {\displaystyle {\sum}_{\mathrm{x}=1}^{\mathrm{M}}}{\displaystyle {\sum}_{\mathrm{y}=1}^{\mathrm{N}}}\frac{\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{M}\mathrm{N}} \)
 Variance (σ2) Dispersion around the mean of the grayscale values of micro pixels encompassed within the ROI. \( {\upsigma}^2=\frac{{\displaystyle {\sum}_{\mathrm{X}=1}^{\mathrm{M}}}{\displaystyle {\sum}_{\mathrm{y}-1}^{\mathrm{N}}}{\left\{\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)-\overline{\mathrm{x}}\right\}}^2}{\mathrm{M}\mathrm{N}} \)
 Skewness (Sk) Reflects the asymmetry of the grey level frequency distribution curve around its mean. A high coefficient (in absolute value) translates in a shifted distribution relative to the mean, while a zero coefficient indicates a symmetric distribution. In a positively skewed distribution, pixels intensities are biased toward lower values (shifted distribution to the left). In a negatively skewed distribution, pixels intensities are biased toward higher values (shifted distribution to the right). \( {\mathrm{S}}_{\mathrm{k}}=\frac{1}{\mathrm{M}\mathrm{N}}\frac{{\displaystyle {\sum}_{\mathrm{x}=1}^{\mathrm{M}}}{\displaystyle {\sum}_{\mathrm{y}=1}^{\mathrm{N}}}{\left\{\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)-\overline{\mathrm{x}}\right\}}^3}{\upsigma^3} \)
 Kurtosis (Kt) Reflects the flatness of the grey level frequency distribution curve around its mean. A diffuse distribution will translate in a lower kurtosis value. Distribution concentrated around its mean will translate in a higher kurtosis value. \( {\mathrm{K}}_{\mathrm{t}}=\frac{1}{\mathrm{M}\mathrm{N}}\frac{{\displaystyle {\sum}_{\mathrm{x}=1}^{\mathrm{M}}}{\displaystyle {\sum}_{\mathrm{y}=1}^{\mathrm{N}}}{\left\{\mathrm{I}\left(\mathrm{x},\mathrm{y}\right)-\overline{\mathrm{x}}\right\}}^4}{\upsigma^4} \)
 Entropy (E) Reflects disorder in a ROI. It considers the number of grey levels in a ROI, and the proportions of each grey level. There is an increased entropy when multiple grey level values are present in the ROI. Vice-versa, entropy equals zero if an image has a single grey level value for all its micro pixels. \( \mathrm{E}=-{\displaystyle {\sum}_{\mathrm{i}=0}^{255}}{\displaystyle {\sum}_{\mathrm{j}=0}^{255}}\mathrm{p}\left(\mathrm{i},\mathrm{j}\right){ \log}_2\left(\mathrm{p}\left(\mathrm{i},\mathrm{j}\right)\right) \)
QUS Measurements computed from a co-occurence matrix  
Texture parameters
 Contrast (I con ) Contrast measures the difference of intensity between the grey level values of neighboring micro pixels. There is a reduced contrast in a constant image with lesser local variations of the grey level intensities. On the contrary, contrast is higher in an image containing a large amount of local sudden variations in the values of grey level intensities. \( {\mathrm{I}}_{\mathrm{con}}={\displaystyle {\sum}_{\mathrm{i}=0}^{255}}{\displaystyle {\sum}_{\mathrm{j}=1}^{255}}{\left|\mathrm{i}-\mathrm{j}\right|}^2\ \mathrm{p}\left(\mathrm{i},\mathrm{j}\right) \)
 Energy (I eng ) Energy is linked to the regularity and consistency of the patterns in an image. High energy is measured in a constant and steady picture. Vice-versa, low energy is found in an image in which the contacts of grey level values are diverse, uncoordinated and random. \( {\mathrm{I}}_{\mathrm{eng}}={\displaystyle {\sum}_{\mathrm{i}=0}^{255}}{\displaystyle {\sum}_{\mathrm{j}=0}^{255}}\mathrm{p}{\left(\mathrm{i},\mathrm{j}\right)}^2 \)
 Homogeneity (I hmg ) Homogeneity is increased in an image with a large number of pixels having the same grey level values, with little grayscale transition (i.e., increased when there is a large area of the same color). \( {\mathrm{I}}_{\mathrm{hmg}}={\displaystyle {\sum}_{\mathrm{i}=0}^{255}}{\displaystyle {\sum}_{\mathrm{j}=0}^{255}}\frac{1}{1+{\left(\mathrm{i}-\mathrm{j}\right)}^2}\mathrm{p}\left(\mathrm{i},\mathrm{j}\right) \)
  1. I(x,y) denotes the grayscale intensity at the x,y coordonates in a ROI comprising M rows and N columns. p(i,j) represent the element of a grey-level co-occurrence matrix and denotes the probability that grayscale intensities i and j are adjacent