3. Basic Methods II: Image operations

3.9. Linear Transfer Function

The simple but very useful linear transfer function assigns every original pixel value a new value on a linear scale. The endpoints of the function which determine what value is white ( \( p_{white} \) ), and what black ( \( p_{black} \) ) (initially 255 and 0) can, however, be altered. The transfer is described by:

\( f(p) = 0 ~ , ~ p \leq p_{black}\)

\( f(p) = (p-p_{black}) / (\frac{p_{white} - p_{black}}{p_{max}}) ~ , ~ p_{black} < p < p_{white}\)

\( f(p) = p_{white} ~ , ~ p > p_{white}\)

where \( p \) denotes the individual pixel value and \( p_{max} \) the maximum pixel value on the used intensity scale (always 255 for our brightness scale). Pixel values lower than \( p_{black} \) are drawn as black and values higher than \( p_{white} \) white.

Narrowing the band further and further lets certain areas of the image become entirely black or white. Information is lost either way. Completely white areas look over-exposed, because the maximum viable value has been reached. Information in the completely black areas vanishes, because we set a certain threshold value, or sensitivity in a practical sense, below which no values register. 

So why even use a linear transfer if information might only get lost? The entire output intensity range (all the different shades of gray) is applied to a more narrow band. This way the contrast in a certain intensity band increases and subtle details and soft gradients become visible.

Move the sliders on the right hand side of the picture below. Observe how narrow windows of allowed pixel values create high contrasts in different bands. The plot shows the transfer function, i.e. the input and output pixel values, respectively. Note that the pixel values are displayed on a scale of 0...1 instead of 0...255.