Notes on Power Functions

Introduction

Bitmaps, be they single channel greyscale or RGB are essentially an array of values where each value is between 0 and 1.

Greyscale Roughness Map — *Greyscale Head Roughness Map*

In a shader, when we use a parameter such as ‘Roughness Power’, this parameter is used in an operation which takes a value $b$ from the bitmap and raises it to the power of the parameter $n$ i.e. $b^n$ (here $b$ is referred to as the base and $n$ as the exponent. This operation is usually referred to as Exponentiation or a Power Law.

Blender uses the Math->Power ‘Converter’ type shadernode (light blue) to do this (shown below for illustration).

This type of operation is known as Gamma Correction where ‘gamma’ ( $\gamma$ ) is the exponent, $V_{\mathrm{in}}$ is the base (i.e. the input bitmap) and $A$ is a constant of proportionality which gives Gamma Corrected $V_{\mathrm{out}}$ (i.e. the output bitmap) in the expression:

$V_{\mathrm{out}} = A V_{\mathrm{in}}^{\gamma}$

Exponent Effects

When the base value you are raising to a power is between 0 and 1 (as is the case with a bitmap), the result can never be lower then 0 or greater than 1. Thus all the values in a bitmap can be safely tuned in a non-linear manner by only varying the exponent.

Important

The value of the exponent can have a dramatic effect on the bitmap. Consider the examples below:

When the exponent is 0.2 ( $y = x^{0.2}$ i.e. the quintic root $x^{1/5}$ ) smaller values are amplified much more than the higher values (with a roughness map, this would mean that the smoother areas are made somewhat rougher; with a RGB image, the darker areas would be significantly lightened)

When the exponent is 1 ( $y = x^{1}$ $\equiv$ $y = x$ ) this is the neutral case where the operation makes no changes to the bitmap.

When the exponent is above 1 but still small, we can see a slight reduction in lower end values and a slight increase in higher end values e.g with $y = x^{2}$ .

As the exponent becomes higher (e.g with $y = x^{5}$ ) we observe a very significant reduction of all but the highest end values (with a roughness map this would smooth out all but the roughest areas; with a RGB image, the lighter areas would be significantly darkened)

The value of the exponent should therefore be adjusted carefully.