3. Basic Methods II: Image operations

3.10. Log Transfer Function

A property of astronomical measurements in image form is often a high range of information values, i.e. the brightness of every pixel. The logarithmic scaling allows us to compress the information of an image with pixel values that run from a few to several orders of magnitude higher into a more readable and intuitive form.  

Did you know...?
The human eye and brain percieve light intensity strongly non-linear and almost logarithmic. This enables us to see in a wide range of lighting conditions. Mechanical light detectors on the other hand count photons in a linear fashion. In order to replicate an astronomical image "how we would see it" a log transfer function needs to be applied.

The corresponding transformation is given as follows. In order to match \( p_{max} \) it is necessary to scale the range of values in the logarithm:

\( f(p) = p_{max} (\frac{Log(p - p_{black})}{Log(p_{white} - p_{black})}) \)

Like before, pixel values smaller than \( p_{black} \) will be depicted as 0, and value greater than  \( p_{white} \) as \( p_{max} \). The image in the example below seems overall brighter since the pixel values in the middle of the range are relatively enhanced compared to very low and very high values of \( p \). Increasing the value for \(  p_{black} \) only a bit quickly separates the fainter regions in darkening them substantially. The real strength of the logarithmic transfer function is the reduction of a large value range (such as a nebula in a field of bright stars) of more than 256 steps in the applications here. Real images, as measured in astronomical observations, often feature millions of steps, depending on how many photons have been measured in a pixel.