CSE 681 Perlin Noise Turbulence Functions

1
CSE 681Perlin Noise (Turbulence) Functions
2
Noise

• Nature has a random or fractal look
• Level of detail property
• Mountain range
• Coastline

Ken Musgrave
Gunther Berkus via Mojoworld
3
Level Of Detail

• Level of detail defines fractional dimensions
• Detail at all levels of magnification

4
Perlin Noise

• Ken Perlin (1985) defined Perlin noise that could
  be used to define natural looking textures

Ken Musgrave
Ken Perlin, 1985
5
Perlin Noise

• Goal Textures that look natural
• Tools (fractal properties)
• Randomness
• Level of detail (scale)

6
Noise Function

• Random values 0 1 defined on an integer
  (discrete) lattice
• Use a random number generator

7
Resolution

• Take examples on a 2D integer lattice
• What resolution (or scale) do we use?
• Remember, we want to evaluate at any (s, t, r)

10 x 10
64 x 64
256 x 256
8
Noise Function Interpolation
9

10
11
12
13
14

9
Level of Detail Magnification

• What if we zoom into the noise texture?

10
Level of Detail Minification

• What if we zoom out of the noise texture?

Sample too sparsely - no continuity
11
Turbulence

• Idea Construct a noise function by adding
  together noise functions sampled at different
  scales
• Scale sample resolution frequency
• Function values (random) amplitude

12
Turbulence
13
1D Turbulence Sin Wave Examplesin(x)
14
1D Turbulence Example sin(x), 1/2 sin(2x)
Half amplitude Double Frequency
15
1D Turbulence Example sin(x), 1/2 sin(2x),
1/4 sin(4x)
16
1D Turbulence Example Add sin(x) 1/2 sin(2x)
17
1D Turbulence Example Add sin(x) 1/2
sin(2x) 1/4 sin(4x)
18
1D Turbulence Example Add sin(x) 1/2
sin(2x) 1/4 sin(4x) 1/8 sin(8x)
19
1D Turbulence Noise Example
20
2D Turbulence Example
21
Building a 1D Turbulence Function

• Define noise functions on the interval 0 1
  with increasing resolutions
• Lets assume power of two resolutions

0
1
20
21

• Evaluate at a location x Î 0 1
• Interpolate on each scale
• Evaluate at any location

22
23
22
Building a 1D Turbulence Function

• Pre-compute noise functions into textures
• Sample and interpolate on the textures
• Large storage cost
• One 3D texture per resolution
• Compute on-the-fly

23
Building a 1D Turbulence Function

• double turbulence(double x)
• double value 0
• double f
• for (f MINFREQ f lt MAXFREQ f 2)
• value noise(x f) / f
• return(value)

Frequency a power of two

• Random number generator
• Can add smoothing here

Amplitude
24
The Noise Function

• double noise(double x)
• int int_x (int)x
• double fract_x x int_x
• double r1 random(int_x)
• double r2 random(int_x 1)
• return interpolate(r1, r2, fract_x)
• Noise depends on the location x
• Each integer lattice location must be assigned
  only one random value
• An integer lattice value may be used many times
  for texture sampling
• How do we define random to ensure this?

25
Random Number Invariance

• We want to ensure that the random numbers
  generated at each scale and integer lattice
  location does not change at each function
  evaluation
• Use a seeded random number generator
• Use a prime number generator (one per scale)
• Pre-compute random numbers

26
Random Number Invariance

• Assume our highest texture resolution is 256
• Construct a table of random numbers
• NoiseTable256 random values 0, 1
• Construct a table of index values
• Index256 Random permutations of values 0,
  255
• random(i) NoiseTableIndexi, where i is an
  integer lattice location

27
Building a 3D Turbulence Function
Use a 256 x 256 x 256 Array
Deposit random values at integer grid points
28
Random Number Generation

• Construct 1D tables
• NoiseTable256 random values 0, 1
• Index256 Random permutations of values 0,
  255
• Random number depends upon integer lattice
  location
• define PERM(v) Indexv 255
• define INDEX(ix, iy, iz) PERM(ix PERM(iy
  PERM(iz)))
• double random(i, j, k)
• return NoiseTableINDEX(i, j, k)

29
3D Volume Interpolation
d110
30
3D Turbulence Function

• double turbulence(double x, double y, double z)
• double value 0
• double f
• for (f MINFREQ f lt MAXFREQ f 2)
• value noise(x f, y f, z f) / f
• return(value)
• noise(x, y, z)
• Get integer (ix, iy, iz) and fractional parts
  (fx, fy, fz)
• Get cell lattice noise values
• d000,d001,d010,d011, d100,d101,d110,d111
• Do the trilinear interpolation by fx, fy, fz

31
Smoothing

• Smooth out the sharp function slopes between the
  lattice values
• Why ? - Theres a discontinuity in the
  derivatives between neighboring lattice element
  values
• Alternative is to use higher order interpolation
• Cubics, cosine interpolation, hermite splines
  can be expensive!

before
after
32
Wood and Marble Textures

• Marble
• texture cos( x turbulence(x, y, z))
• Weight this on two colors (black and white here)
• Wood
• g turbulence (x, y, z) 20, grain g -
  int(g)
• Use this with our concentric rings

33
More Marble

• double marble(x, y, z)
• return undulate(sin(2pz Aturbulence(x, y,
  z)))
• undulate(double x)
• if (x lt -0.4)
• return 0.15 2.857 SQR(x 0.75)
• else if ((x lt 0.4)
• return 0.95 2.8125 SQR(x)
• else
• return 0.26 2.66 SQR(x - 0.7)
• define SQR(x) x x

34
More Marble

• double marble(x, y, z)
• return (1 sin(pz turbulence( 3p/ 2 x, 3p/2
  y, 3p/2 z))) / 1.25 / 2
• Given a base color compute a texture color as
• white (1, 1, 1)
• t marble(x, y, z)
• baseColor (1 - t) white t