Digital Image Statistics

See how to compute Image Statistics

The Sample Statistics Analysis has the purpose to compute and present the following statistical parameters from images previously selected:

  • Moments
  • Median
  • Covariance and Correlation Matrices
  • Autocorrelation Matrix
  • Cross Correlation Matrix

The results will be presented in the window in a graphical and textual formats.

To access the Statistics... option go to the SPRING main menu, and select the Image in the bar menu. To operate this function, an image has to be previously selected.

Moments

If X1, X2, ..., XN are the N values that the X variable can take, than it is possible to define:

Order of the Moment "r" by:

,

and the Order of the Moment "r", centered in the average, by:

,

where (1st order moment), is the data average value.

The moments of the 3rd and 4th order are computed to r values equal to 3 and 4 respectively and, the centered moment in the average order 2 defines the variance of a numerical data set.

Average

The average of a set N with numerical data X1, X2, ..., XN is represented by and defined by:

,

which is the 1st order moment.

Median

The median of a set N of ordered numbers in increasing order is the central value (if N is odd) or the average of the two central values (N is even).

Examples:

3,4,4,5,6,8,8,8,10 -  has median 6

5,5,7,9,11,12,13,17 -  has median 10.

Mode

The mode is the value that appears most frequently among the numerical values in the set. The mode might not exist and, even if it exists, it might not be unique.

Examples:

1,1,3,3,5,7,7,7,11,13  - has mode 7

3,5,8,11,13,18 -  has no mode

3,5,5,5,6,6,7,7,7,11,12  - has two modes: 5,7 (bimodal).

Standard Deviation and Variance

They measure the dispersion degree of the numerical data around the average value.

The Standard Deviation of a data set X1, ..., Xn is defined by:

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The Variance is the square of the standard deviation:

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Covariance

The covariance value between two sets of numerical data a and b, with N points is defined by:

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This similarity degree between the two sets a and b, that is, how the data is correlated to themselves. The higher this value, the higher the correlation degree between the data.

Correlation Coefficient

The correlation coefficient measures the similarity between two numerical data sets over an absolute scale of [-1, 1]. It is computed dividing the covariance value by the square root of the product of the standard deviation of the data sets a and b:

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Variation Coefficient

The variation effect or dispersion related to the average can be measured by the relative dispersion, defined by:

Relative Dispersion = Absolute Dispersion/Average

If the absolute dispersion is the standard deviation, the relative dispersion is called variation coefficient v:

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The variation coefficient is not useful when the average is close to zero.

Asymmetric Moment Coefficient

This is the deviation degree or symmetry axis distance in a distribution. To asymmetric distributions the average will be placed in the longest tail of the distribution. This coefficient can be defined using the 3rd moment centered in the average and the standard deviation:

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Kurtosis Coefficient

It measures the flattening of a data distribution, and can be defined by dividing the 4th degree moment, centered in the average by the square of the variance. That is:

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