Noises & TurbulencesNoises & Turbulences

Why Noises?Why Noises?

• One of the most popular procedural technique• Very few things in nature are regular • Great variety of noises exists• The simplest way = white noise:

out = random();– A very bad way of generating noise

• No memory• No spatial relation to anything• Not visually interesting

What is Noise?What is Noise?

• Defined by Defined by noise functionnoise function

• RR →→ RR function functionstatistically invariant to translationstatistically invariant to translation

• RR22 → R→ R function function+ statistically invariant to rotationstatistically invariant to rotation

• Any Any RRnn → R→ R function function+ sharp frequency spectrumsharp frequency spectrum

Lattice NoisesLattice Noises

• Need to relate noise to space– Output = noise(point coord);

• Family of lattice noises:– Generate a 1D/2D/3D grid– Compute random values at grid points– Store values– During rendering: interpolate neighbors

• Questions to make a practical algorithm– What are the values at the nodes?– Interpolation strategy?

• First noise in CGFirst noise in CG• Ken Perlin: Ken Perlin: An Image SynthesizerAn Image Synthesizer, 1985, 1985• Originally defined as a lattice gradient noiseOriginally defined as a lattice gradient noise• Interpolation dependentInterpolation dependent

different interpolation exampledifferent interpolation example

Perlin NoisePerlin Noise

Perlin Noise HistoryPerlin Noise History

• Tron 1981Tron 1981

Lattice Noise TypesLattice Noise Types

• Value noise– n-linear interpolation in between random values

• Gradient noise– Store gradient vectors in the nodes– Values are zero at the nodes

• Value-gradient noisea) weighted sum of value & gradientb) Hermite interpolation

• Lattice convolution noise– convolution interpolation

Value-Gradient ComparisonValue-Gradient Comparison

Value / ValGrad 0.25 / ValGrad 0.5 / ValGrad 0.75 / GradientValue / ValGrad 0.25 / ValGrad 0.5 / ValGrad 0.75 / Gradient

Ridged NoiseRidged Noise

Value / ValGrad 0.25 / ValGrad 0.5 / ValGrad 0.75 / GradientValue / ValGrad 0.25 / ValGrad 0.5 / ValGrad 0.75 / Gradient

Palette modulation used for interpolationPalette modulation used for interpolation

• Most Simple = Most Simple = nn-linear-linear

1D linear1D linear 2D bilinear 2D bilinear

InterpolationsInterpolations

Interpolation Blending FunctionsInterpolation Blending Functions

• Linear interpolation on Linear interpolation on ff(0), (0), ff(1):(1):ff((xx) = (1-) = (1-xx) ) ff(0) + (0) + xx ff(1)(1)

uses blending functions (1-uses blending functions (1-xx), ), xx

• In general:In general:ff((xx) = u() = u(xx) ) ff(0) + v(x) (0) + v(x) ff(1)(1)

• Requirements on blending functions (Requirements on blending functions (xx 0,10,1))a)a) uu((xx) + ) + vv((xx) = 1) = 1

b)b) uu(0) = 1 (0) = 1 vv(0) = 0(0) = 0

c)c) uu(1) = 0(1) = 0 vv(1) = 1(1) = 1

Smooth InterpolationsSmooth Interpolations

• CC22 continuity = ? continuity = ?• Additional requirements on blending functionsAdditional requirements on blending functions• Null derivations:Null derivations:

d)d) uu’(0) = 0’(0) = 0 vv’(0) = 0’(0) = 0

e)e) uu’(1) = 0’(1) = 0 vv’(1) = 0’(1) = 0

• One Solution: Hermite cubics POne Solution: Hermite cubics P11, P, P44

• uu((xx) = 2) = 2xx33 + 3 + 3xx22 + 1 + 1• vv((xx) = -2) = -2xx33 + 3 + 3xx22

• ff((xx) = 2( ) = 2( ff(0) – (0) – ff(1) )x(1) )x33 + 3( + 3( ff(1) – (1) – ff(0) )(0) )22 + + ff(0)(0)

• Zero derivates Zero derivates flats in lattice points flats in lattice points • Use derivates aligned to neighborsUse derivates aligned to neighbors

zero derivateszero derivates aligned derivates aligned derivates

1D Cubic Interpolations1D Cubic Interpolations

2D Cubic Interpolations2D Cubic Interpolations

zero derivateszero derivates aligned derivates aligned derivates

Composed InterpolationsComposed Interpolations

• Blending in 1D: Blending in 1D: ff((xx) = (1-) = (1-xx) ) ff(0) + (0) + xx ff(1)(1)

• What about What about ff((xx) = (1-) = (1-xx) ) uu00((xx) + ) + xx uu11((xx) )

• Set requirementsSet requirements• Solution = smoothest cubic interpolationSolution = smoothest cubic interpolation

Cosine InterpolationCosine Interpolation

• Blending functionsBlending functions• uu((xx) = ( cos() = ( cos(xx) + 1) / 2) + 1) / 2• vv((xx) = ( 1 – cos() = ( 1 – cos(xx) ) / 2) ) / 2

• Slower computationSlower computation

Sharp Interpolations ExampleSharp Interpolations Example

Deformed Lattice InterpolationsDeformed Lattice Interpolations

• Radial interpolationRadial interpolation– ((rr, , ) coordinate system) coordinate system– radius, angleradius, angle– principle same as for ortho-latticesprinciple same as for ortho-lattices

Lattice Noises in 3DLattice Noises in 3D

• Stored lattice = memory consumptiveStored lattice = memory consumptive• Store onlyStore only

– 256 random values256 random values– Permutation tablePermutation table

• Referencing:Referencing:Value table index:

P[ ix+P[ iy+P[iz] ] ]

P - permutation table

• Suppressed periodic structureSuppressed periodic structure

Convolution NoisesConvolution Noises

1.1. Generate random latticeGenerate random lattice

2.2. Apply convolution filter until necessaryApply convolution filter until necessary• Gaussian convolutionGaussian convolution

• Values are averaged Values are averaged averaged noiseaveraged noise

K

1

9

1 1 1

1 1 1

1 1 1

Averaged NoiseAveraged Noise

• Control = number of iterationsControl = number of iterations

• Modifications: anisotropic filtersModifications: anisotropic filters

K

1

55

0 0 5 0 0

0 1 10 1 0

1 2 10 2 10

0

1

0

10

5

1

0

0

0

→→ →→

Other Noise TypesOther Noise Types

• Spot Noise (van Wijk, 1991)Spot Noise (van Wijk, 1991)• Sparse convolution noise (Lewis 1989)Sparse convolution noise (Lewis 1989)

– Interpolation of randomly located random valuesInterpolation of randomly located random values– Suppresses grid artifactsSuppresses grid artifacts

• fBm (Fractal Brownian Move)fBm (Fractal Brownian Move)– frequency basedfrequency based

• Integral NoiseIntegral Noise– noise(noise(xx) = ) = ff( noise(( noise(xx-1) )-1) )

• Voronoi NoiseVoronoi Noise

Spot NoiseSpot Noise

• 1D spot noise function:1D spot noise function:

• Composition of spot function Composition of spot function hh((xx): ):

→→

Integral Noise Integral Noise

• nn-th integration of white noise-th integration of white noise

Integral Noise AlgorithmIntegral Noise Algorithm

• Init: randomly generate Init: randomly generate ff(0), (0), ff11(0), … (0), … ffnn-1-1(0)(0)• In step In step xx::

0.0. ffnn(x) = random((x) = random(xx) ) ……i. i. f fii(x) = (x) = ffii((xx-1) + -1) + ffi+i+11((xx))……n.n. f f(x) = (x) = ff00(x)(x)n+1.n+1. remember remember ff((xx), ), ff11((xx), … ), … ffnn-1-1((xx))

• ffii(x) = (x) = ii-th derivate of -th derivate of ff((xx))

2D Integral Noise Example2D Integral Noise Example

• Vertical, horizontal edge = 1D noiseVertical, horizontal edge = 1D noise• Inner points are zig-zag traversedInner points are zig-zag traversed• point[x,y] = composition of point[x-1,y] & point[x,y-1] point[x,y] = composition of point[x-1,y] & point[x,y-1]

Voronoi NoiseVoronoi Noise

1.1. Randomly distribute Randomly distribute nn points points

2.2. Visualize distance to the nearest neighbourVisualize distance to the nearest neighbour

TurbulencesTurbulences

• Image modulation functionsImage modulation functions• Output = Output = ff( ( xx + turb( + turb(xx) )) )• Usual requirement = continuityUsual requirement = continuity• Modulation sourceModulation source

– Simple imageSimple image– Explicit functionExplicit function

• Original function = identityOriginal function = identity

• Lower turbulence is 3 times greater than upperLower turbulence is 3 times greater than upper

1D Turbulence Example1D Turbulence Example

Perlin TurbulencePerlin Turbulence

• kk = smallest number satisfying = smallest number satisfying1/21/2kk+1+1 < pixel size < pixel size

• Sum of noise functionsSum of noise functions• Fractal character: Fractal character:

– doubling frequencydoubling frequency– Halving amplitudeHalving amplitude

• Noise function = Perlin noiseNoise function = Perlin noise

turb xnoise xi

ii

k

( )( )

2

20

• ii = octave number (frequency = 2 = octave number (frequency = 2 ii))

Composing Perlin TurbulenceComposing Perlin Turbulence

1D Perlin Turbulence1D Perlin Turbulence

• Frequency self-similarityFrequency self-similarity• Not visually attractive in 1DNot visually attractive in 1D

2D Perlin Turbulence2D Perlin Turbulence

• Noise functions:Noise functions:

• Composition:Composition:

Perlin Turbulence ExamplePerlin Turbulence Example

• Direct visualizationDirect visualization– fractal terrainsfractal terrains– cloudsclouds

Turbulence Modulated ImagesTurbulence Modulated Images

• Image to modulate = clampImage to modulate = clamp• Turbulence used: 2D averaged noiseTurbulence used: 2D averaged noise

Perlin Turbulence CloudsPerlin Turbulence Clouds

→→

• Clamp = sin(y)Clamp = sin(y)• Perlin turbulencePerlin turbulence

smaller turbulence larger turbulencesmaller turbulence larger turbulence

MarbleMarble

WoodWood

• Clamp = rotation sineClamp = rotation sine• Brown paletteBrown palette

Other Turbulence ExamplesOther Turbulence Examples

FireFire

• 3D turbulence (2D + time) of a flame3D turbulence (2D + time) of a flame

Fire AnimationFire Animation

LiteratureLiterature

• David Ebert, Kent Musgrave, Darwyn Peachey, David Ebert, Kent Musgrave, Darwyn Peachey, Ken Perlin, and Worley. Ken Perlin, and Worley. Texturing and Modeling: A Procedural Texturing and Modeling: A Procedural ApproachApproach. . Academic Press, October 1994. Academic Press, October 1994. ISBN 0-12-228760-6.ISBN 0-12-228760-6.

• Ken Perlin: www.noisemachine.com• Tons of examples on www