Intensity

Intensity features

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

Intensity

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 $c h$ .

Aggregations

Mean: is expected value (see Wikipedia):

μ = \frac{\sum X _{i}}{n}

Maximum: highest intensity
Minimum: lowest intensity
Entropy: describes uncertainty of information (see Wikipedia):

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.

Kurtosis: describes relative frequency of extreme values (see explanation):

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}

Sum: is the addition of all values:

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).

Ratio

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}}

MSE

Measures the mean squared difference between two specified channel intensities $R$ and $G$ over all $n$ relevant pixels or voxels.

MS E_{RG} = \frac{1}{n} i = 1 \sum n (R_{i} - G_{i})^{2}

The value is non-negative. A value of 0 means that the two channels have identical intensities at every relevant pixel.

Pearson

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}}

Manders

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

See the paper:
E. M. M. MANDERS, F. J. VERBEEK, J. A. ATEN: “Measurement of co-localization of objects in dual-colour confocal images” (1993)

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

Costes Background

Automatically estimates a pair of intensity thresholds for two fluorescence channels. Intensities below these thresholds are treated as uncorrelated background for colocalization analysis.

Use the two resulting values as the channel thresholds for a subsequent Costes-style colocalization measurement. For example, thresholded Manders coefficients can be calculated from voxels that are above the thresholds in both channels. This avoids choosing the two thresholds manually and makes the analysis more reproducible.

The method searches for the thresholds at which the lower-intensity voxels no longer have a positive Pearson correlation. It returns only the two thresholds; it does not calculate a colocalization coefficient or perform the Costes randomization significance test.

In this context, background means the low-intensity, uncorrelated part of the two-channel data. It is not a general-purpose estimate of detector offset or uniform image background. Such background should be corrected before the measurement. If a binary mask is provided, only voxels inside the mask are analyzed.

See the paper:
S. V. Costes et al.: “Automatic and Quantitative Measurement of Protein-Protein Colocalization in Live Cells” (2004)

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