Noise
Noise in general is an undesirable [1]signal that's mixed in with useful
signal and which we usually try to filter out, even though it can also be
useful, especially e.g. in [2]procedural generation. Typical example of
noise is flickering or static in video and audio signals, but it can also
take a form of e.g. random errors in transferred texts, irrelevant links
in web search results or imperfections in measuring of economic data. In
measurements we often talk about [3]signal to noise ratio (SNR) -- the
ratio that tells us how much noise there is in our data. While in
engineering and scientific measurements noise is almost always something
that's burdening us (e.g. [4]cosmic microwave background, the noise left
over from the [5]Big Bang, may interfere with our radio communication), in
some fields such as [6]computer graphics (and other computer [7]art) or
[8]cryptography we sometimes try to purposefully generate artificial noise
-- e.g. in [9]procedural generation noise is used to generate naturally
looking terrain, clouds, textures etc.
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2D binary white noise
Artificial Noise
In [10]computer science, especially e.g. in [11]computer graphics or audio
generation, we often want to generate artificial noise that looks like
real noise, in order to be able to generate data that look like real world
data. Noise may serve as a basis (first step, random initial state) for
generating something or it may be used to distort or modulate other data,
for example when drawing geometrical shapes we may want them to look as if
written by hand in which case we want to make them imperfect, add a bit of
noise to an otherwise perfect shape a computer would create. Adding noise
can help make rendered pictures look more natural, it can help us model
human speech that arises from noise created by our vocal cords, it can
help AI train itself to filter out real life noise etc.
Normally we don't want our noise to be completely [12]random but rather
[13]pseudorandom so that our programs stay [14]deterministic. Imagine for
example we are generating terrain [15]heightmap based on a 2D noise -- if
such function was truly random, the terrain in a certain place would be
different every time we returned to that place again, but if the noise is
pseudorandom ([16]seeded by the position coordinates), the terrain
generated at any given coordinate will be always the same.
There are different types of noise characterized by their properties such
as number of dimensions, [17]frequencies they contain, [18]probability
distribution of the values they contain etc. Basic division of noises may
be this:
* by type/[19]algorithm:
* value noises: The simplest type of noise, it arises by generating
a (pseudo)random value at each node of the spatial grid. Values
in between grid nodes are obtained by [20]interpolation.
* gradient noises: More sophisticated than value noise, rather than
a simple value it generates a [21]gradient at each grid point.
Specific algorithms/types of gradient noise are:
* [22]Perlin noise: One of the most common and basic gradient
noises, which however suffers from some [23]artifacts.
* [24]simplex noise: A more sophisticated noise than Perlin
noise, improves some of its properties, works with a
[25]simplex lattice of points rather than orthogonal grid.
* ...
* spectral noises: Noises that are generated in the [26]spectral
domain (see [27]Fourier transform):
* [28]anisotropic noise: Noise that has a specific
orientation, can be used e.g. for water waves that go in one
direction.
* ...
* [29]convolution noises: Noises that use [30]convolution of given
kernel with some simpler noise to create a larger texture.
* [31]Gabor noise
* ...
* other:
* [32]Voronoi noise: Noise that looks like a "honeycomb" or
structures created by living cells, it is created by
generating random point coordinates and then assigning each
space point (e.g. each pixel) a [33]distance to its closest
point. Different random [34]point sampling methods or
distance functions can help change the look of the noise.
Worley Noise is one type of Voronoi noise.
* Combinations and modifications of standard noises: we may
e.g. blend or [35]modulate two noise functions to create a
new complex type of noise.
* [36]midpoint displacement/[37]diamond square: Relatively
simple [38]algorithms for generating [39]fractal heightmaps,
working on the principle of repeatedly subdividing a line
(or in higher dimensions a square, cube etc.) and
(pseudo)randomly displacing the generated point.
* [40]wavelet noise: Noise further improving some
characteristics of Perlin noise.
* unorthodox noises created e.g. by [41]cellular automata, AI
etc.
* ...
* by frequencies:
* [42]fractal noise: Very important type of noise that similarly to
[43]fractals is composed of differently scaled versions of itself
-- this noise looks like (and can be used to simulate) clouds,
mountains and other structures found in nature. It is created by
taking some basic noise function (e.g. Perlin noise or simplex
noise) and overlaying (adding) multiple versions of it that
differ by frequency and amplitude (just as e.g. mountains are
composed of big and tall hills that have on them progressively
smaller and less tall hills up to the microscopic level). These
different individual layers are called octaves: ith octave has
the amplitude p^i (where p is a constant from 0 to 1 called a
persistence) and frequency of 2^i.
* [44]white noise: Noise containing "same amount of all
frequencies" -- this is basically the simplest kind of noise we
get when we generate a sequence of independent ([45]uncorrelated)
(pseudo)random numbers with uniform probability distribution.
* [46]pink noise: Energy density of frequencies decreases
proportionally with 1/frequency, i.e. it basically has strong
high frequencies and weak low frequencies.
* [47]blue noise
* ...
* by other properties:
* symmetry/invariance and other function properties: A noise may or
may not be invariant against various operations such as shifting,
rotation or scaling, i.e. when we apply this operation it will
look basically the same. For example Perlin noise outputs value 0
at each grid point and is non-zero only between the grid point,
so it is NOT invariant towards shifting. Noise functions are just
mathematical [48]functions, so we may examine noises just as we
examine functions (they may be continuous, bounded etc.).
* explicit vs implicit: Values of an implicit noise can relatively
simply and quickly be computed at any given point in space
whereas explicit noises require processing as a whole and
therefore storage of the whole generated noise in memory.
* tiling: Similarly to [49]procedural [50]textures, a noise
generated by an algorithm may be tiling, i.e. not having visible
seams when repeated in certain dimensions.
* ...
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fractal noise
1D fractal noise composed of 3 octaves
Code
A super simple "poor man's noise" that can be of use sometimes is coin
flip noise which works simply like this: start with value 0, in each step
output current value and randomly change it by +1 or -1. It's basically a
[51]random walk, a next best thing to the simplest [52]white noise, so
watch out, it will only work for most basic things, generalizing to a 2D
noise will be awkward, you won't know how high or low the values will
actually go, also the frequency properties probably won't be ideal. { Not
sure actually what spectrum to expect, have to check that out, TODO.
~drummyfish } You may at least try to play around with changing the value
even by more than one using e.g. [53]normal probability distribution.
TODO: code for the above, maybe even a one liner for white noise
TODO: actual Perlin noise etc., also some nice noise that's just adding
some random sine waves in a fractal fashion, like a one line formula
Links:
1. signal.md
2. procgen.md
3. snr.md
4. cmb.md
5. big_bang.md
6. graphics.md
7. art.md
8. cryptography.md
9. procgen.md
10. compsci.md
11. graphics.md
12. randomness.md
13. pseudorandomness.md
14. determinism.md
15. heightmap.md
16. seed.md
17. frequency.md
18. probability.md
19. algorithm.md
20. interpolation.md
21. gradient.md
22. perlin_noise.md
23. artifact.md
24. simplex_noise.md
25. simplex.md
26. spectrum.md
27. fourier_transform.md
28. anisotropic_noise.md
29. convolution.md
30. convolution.md
31. gabor_noise.md
32. voronoi.md
33. distance.md
34. point_sampling.md
35. modulation.md
36. midpoint_displacement.md
37. diamond_square.md
38. algorithm.md
39. fractal.md
40. wavelet_noise.md
41. cellular_automaton.md
42. fractal_noise.md
43. fractal.md
44. white_noise
45. correlation.md
46. pink_noise.md
47. blue_noise.md
48. function.md
49. procgen.md
50. texture.md
51. random_walk.md
52. white_noise.md
53. normal_distribution.md