Arithmetic

Arithmetic Mean

Computes the standard average of pixel values, ideal for general blending or smoothing between multiple images.
Defined as the sum of all inputs divided by the number of inputs:

$\frac{1}{\mathrm{n}}\sum _{i=0}^n{{\rm I}}_i\left(\iota \right)=\frac{{\iota }_1+{\iota }_2+\mathrm{...}+{\iota }_n}{\mathrm{n}}$

Geometric Mean

Calculates the $n\mathrm{th}$ root of the product of all input values, preserving proportional balance between pixels. This method is especially useful for averaging multiplicative data or images with wide dynamic ranges.
Defined as:

${\left(\prod _{i=0}^n{{\rm I}}_i\left(\iota \right)\right)}^{\frac{1}{n}}={({\iota }_1\times {\iota }_2\times \mathrm{...}\times {\iota }_n)}^{\frac{1}{n}}=\sqrt[n]{({\iota }_1\times {\iota }_2\times \mathrm{...}\times {\iota }_n)}$

Harmonic Mean

Emphasizes smaller values, producing results that are less influenced by bright outliers. Commonly used when averaging rates or ratios, it is particularly stable for combining images with inverse intensity relationships.
Defined as:

$\frac{n}{\sum _{i=0}^n\frac{1}{\max ({{\rm I}}_i\left(\iota \right),\epsilon )}}$

Note: The Harmonic Mean operates only on positive, non‐zero input values. The Epsilon parameter ( $\epsilon$ ) ensures numerical stability by clamping near‐zero or negative pixel values before division.

Median

Identifies the middle pixel value among all inputs when they are ordered by intensity. Unlike the arithmetic mean, the median is unaffected by extreme values, making it ideal for denoising, temporal filtering, and exposure balancing.
For example, when the data set has an odd number of entries:

$\mathrm{median}[1,3,3,6,7,8,9]=6$

Or, when the data set has an even number of entries:

$\mathrm{median}[1,2,3,4,5,6,8,9]=(4+5)÷2=4.5$

Mode

Selects the most frequently occurring pixel value among inputs. This emphasizes recurring features or dominant tones, often used when consolidating multiple nearly identical images or pattern repetitions.
A tolerance value ( $\epsilon$ ) defines the range within which pixel values are considered equivalent.
For example:

$\mathrm{mode}[1,2,2,2,3,3,4,7,9]=2$

Mid‐Range

Calculates the midpoint between the minimum and maximum pixel values, providing a simple yet fast approximation of the central intensity. While less robust to outliers than the median, it can be useful for quick tone balancing or establishing exposure baselines.
Defined as:

$\frac{\min \left({{\rm I}}_1,{{\rm I}}_2,\mathrm{...},{{\rm I}}_n\right)+\max \left({{\rm I}}_1,{{\rm I}}_2,\mathrm{...},{{\rm I}}_n\right)}{2}$