Describe the bug

In the current implementation of the Geotrellis Sigmoidal contrast, the following formula is used to calculate the contrast for a given raster value (source) :

val numer = 1/(1+math.exp(beta*(alpha-u))) - 1/(1+math.exp(beta))
val denom = 1/(1+math.exp(beta*(alpha-1))) - 1/(1+math.exp(beta*alpha))
val gu = math.max(0.0, math.min(1.0, numer / denom))

When you look at the sigmoidal contrast formula that is defined by ImageMagick, we get the following equation:

( 1/(1+exp(β*(α-u))) - 1/(1+exp(β*(α)) ) / ( 1/(1+exp(β*(α-1))) - 1/(1+exp(β*α)) )

As you can see, the numer part of the original equation is missing a multiplication by alpha in its second diff term.The correct calculation of the numer part should be:

val numer = 1/(1+math.exp(beta*(alpha-u))) - 1/(1+math.exp(beta*alpha))
val denom = 1/(1+math.exp(beta*(alpha-1))) - 1/(1+math.exp(beta*alpha))
val gu = math.max(0.0, math.min(1.0, numer / denom))

Screenshots

The first screenshot shows the original equation, the second one the updated formula as defined by ImageMagick. As you can see, when applying a lower contrast value of beta, the higher values of the rasters will be cut off. This behaviour is less apparent when using higher beta values. However the consequence is that it makes the sigmoidal contrast function very unpredictable.