OCTAVE SUMMING
A commonly used image creation tool is fractal noise made with octave summing.
Octave summed noise is used to create semi-random surface perturbation that have details at a variety of scales.  Many real-world surfaces have details on a range of scales, and octave summed fractal noise is often used to give computer generated objects a more photoreal appearance.  The basic idea is to combine random noise fields at a variety of frequencies and amplitudes to get a wide range of detail.  The term "octave summing" refers to the audio analogy of adding up many octaves of sound, with each octave being twice the frequency and half the volume of the previous octave.  Instead of sound, the following examples are built with perlin noise functions.


simple noise
a simple 1d noise function
Frequency = N
Amplitude = N

take a simple noise function, double the frequency, and halve the amplitude


double F noise
a simple 1d noise function
Frequency = N*2
Amplitude = N*1/2

combine these two noise functions to get detail on two size scales


2 octave noise
a two octave compund noise function

this octave summing can be repeated numerous times to get a function that has details at a wide variety of scales


8 octave noise
an eight octave compound noise
amplitude = 1/frequency

these functions have fine detail for high resolution image creation or for getting closeup images of detailed areas


animated zoom
eight octave compound noise detail zoom
All of the above visual examples were built with perlin noise functions, but the same technique can be used with sine waves or other functions.  For ocean surfaces, sine waves make more realistic motion.

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