Intensity

Intensity features

Intensity features measure pixel or voxel intensity on field, volume and object.

Measures the channel intensity value of all $n$ pixels and outputs specified aggregation (Mean by default) for the relevant pixels.

$X_{i}^{c h}$ or simply $X_{i}$ denotes $i$ -th pixel of channel $c h$ .

μ = \frac{\sum X _{i}}{n}

H (X) = - i = 1 \sum n P (X_{i}) l o g_{b} P (X_{i})

where: $P (X)$ is a distribution of intensity values.

K = \frac{n ( n + 1 ) ( n - 1 ) \sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{4}}{( n - 2 ) ( n - 3 ) ( \sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{2} ) ^{2}}

Mode: is the intensity value that is the most frequent.
Quantile: N-th quantile is the intensity that divides an ordered range of values into two parts such that the first part contains 100 x N % of values and the second part contains 100 x (1 - N) % of values. If quantile is set to 0.5, this function computes the median of values.
Skewness: describes symmetry of a distribution (see explanation):

S = \frac{n n - 1 \sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{3}}{n - 2 ( \sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{2} ) ^{\frac{3}{2}}}

Standard deviation: is a measure of the amount of variation of the values of a variable about its mean (see Wikipedia):

s = \frac{\sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{2}}{n - 1}

S = i = 1 \sum n X_{i}

Uniformity: is a measure of the uniformity (every value occurs roughly the same time) of the distribution of values:

U = i = 1 \sum n P (x_{i})^{2}

where: $P (X)$ is a distribution of intensity values

Variance: is the expected value of the squared deviation from the mean of a random variable (see Wikipedia):

s^{2} = \frac{\sum _{i = 1}^{n} ( X _{i} - X _{a vg} ) ^{2}}{n - 1}

Background estimate: estimates the background intensity.
Focus criterion: is a measure of contrast of neighboring pixels.
Otsu threshold: calculates the threshold intensity that separates the background from the foreground (see Wikipedia).
Gaussian fit [ $μ$ ]: $μ$ parameter of the gaussian fit of the pixel intensity distribution (see Wikipedia).
Gaussian fit [ $σ$ ]: $σ$ parameter of the gaussian fit of the pixel intensity distribution (see Wikipedia).
Gaussian fit [ $R^{2}$ ]: squared residuals ( $R^{2}$ ) of the gaussian fit of the pixel intensity distribution (see Wikipedia).

Measures the per pixel ratio of twos specified channel intensities ( $R$ , $G$ and outputs the average for all $n$ relevant pixels.

R = \frac{1}{n} i = 1 \sum n \frac{R _{i}}{G _{i}}

Pearson correlation coefficient measures linear correlation between two specified channel intensities $R$ and $G$ (see Wikipedia).

The value ranges from -1 to 1.

PC C_{RG} = \frac{\sum _{i = 1}^{n} ( R _{i} - R ˉ ) ( G _{i} - G ˉ )}{\sum _{i = 1}^{n} ( R _{i} - R ˉ ) ^{2} \sum _{i = 1}^{n} ( G _{i} - G ˉ ) ^{2}}

Measures following overlap features on two channel intensities $R$ and $G$ .

Manders overlap: describes overlap between two channels. It is not dependent on the relative strengths of the channels, but depends on the background.

MOC = \frac{\sum R _{i} G _{i}}{\sum R _{i}^{2} \sum G _{i}^{2}}

Manders overlap coefficients $K_{1}$ , $K_{2}$: The coefficients k1 and k2 describe intensity variations between channel 1 and channel 2. Values depend on the intensities of two channels. They are sensitive to the difference in the channel intensities.

K_{1} = \frac{\sum R _{i} G _{i}}{\sum R _{i}^{2}}, K_{2} = \frac{\sum R _{i} G _{i}}{\sum G _{i}^{2}}

Colocalization coefficients $M_{1}$ , $M_{2}$: describe contribution of each channel to the overall colocalization. Values are not sensitive to channel intensities. They can be used when the numbers of objects are not equal.

M_{1} = \frac{\sum R _{i, co l oc}}{\sum R _{i}}

where: R_{i, co l oc} = R_{i} if G_{i} > 0 else 0

M_{2} = \frac{\sum G _{i, co l oc}}{\sum G _{i}}

where: G_{i, co l oc} = G_{i} if R_{i} > 0 else 0