Noise

Noise

Wanho Choi (wanochoi.com)

Signal

• What conveys information (or message)

about the behavior of some phenomenon from a source to an observer

Frequency

• The number of occurrences of a repeating event ‣ _{Per unit time: temporal freq.}

‣ _{Per unit length: spatial freq.}

time (or length)

Frequency Domain

• Fourier Transform

‣ _{Spatial domain → Frequency domain}

https://ranabasheer.files.wordpress.com/2014/03/ft.gif http://hometheaterhifi.com/wp-content/uploads/2015/05/up-sampling-fig3-sm.jpg

Colored Noises

https://en.wikipedia.org/wiki/Colors_of_noise

white noise pink noise _{red noise}

brown(ian) noise

Random Number Generator

• Pseudo (=not real) random number generator • Unpredictable

• Deterministic • Reproducible

Random Number Generator

for( int i=0; i<1000; ++i ) {

cout << drand48() << endl; } 0.000985395 0.176643 0.0913306 0.487217 0.454433 0.831292 0.56806 0.0508319 0.0189148 0.298197 . . .

Good Noise

• 𝒇(𝒑) = 𝑣 (𝒑∈ℝn, 𝑣∈ℝ), (𝑣∈[-1.0,1.0], average=0.0)

• Random to the human eye (Not every noise is equal.) • Controllable frequency and amplitude

• Reproducible (always same results)

• Smooth (no sharp feature, C2-continuity) • Correlated (coherency: 𝒑₁≈𝒑₂ → 𝒇₁≈𝒇₂)

• Band-limited (concentrated region in freq. domain.) • Invariant (indiscernible translation and rotation)

Coherent Noises

• Value noise • Perlin noise • Simplex noise • etc. http://libnoise.sourceforge.net/noisegen/index.html

Lattice & Value Noise

Lattice & Value Noise

Lattice & Value Noise

Interpolation

Names of Polynomials

degree name formula

1 linear (or monic) ax+b

2 quadratic ax2+bx+c

3 cubic ax3+bx2+cx+d

4 quartic ax4+bx3+cx2+dx+e

5 quintic ax5+bx4+cx3+dx2+ex+f

6 sextic (or hexic) ax6+bx5+cx4+dx3+ex2+fx+g

7 septic ax7+bx6+cx5+dx4+ex3+fx2+gx+h

8 octic ax8+bx7+cx6+dx5+ex4+fx3+gx2+hx+i

9 nonic ax9+bx8+cx7+dx6+ex5+fx4+gx3+hx2+ix+j

Interpolation

closest neighborhood linear interpolation smooth interpolation http://libnoise.sourceforge.net/noisegen/

Interpolation

• The flaw is especially apparent when this function

is used to generate a bump-map texture.

http://libnoise.sourceforge.net/noisegen/

Lattice & Value Noise

• A lattice noise acts as a low pass filter. • Noise looks like blurry.

Infinite Domain

• Periodicity by (the first value) = (the last value)

Value Noise

• Divide the n-dimensional space into a lattice. • Assign a random value to each lattice point.

‣ _{Real value range: -1.0 ~ +1.0}

• Get the noise value of a point by interpolating ‣ _{Two lattice values when n=1.}

‣ _{Four lattice values when n=2.} ‣ _{Eight lattice values when n=3.}

Perlin(’s Gradient) Noise

• Divide the n-dimensional space into a lattice.

• Assign a random gradient to each lattice point. ‣ _{Gradient: unit vector}

‣ _{Lattice value: (gradient vector) • (distance vector)} ‣ _{Distance vector: (lattice point) - (query point)}

• Get the noise value of a point by interpolating ‣ _{Two lattice values when n=1.}

‣ _{Four lattice values when n=2.} ‣ _{Eight lattice values when n=3.}

Picking Gradients

• It needs to have enough variation to conceal the

fact that the function if not truly random.

• But, too much variation will cause unpredictable

behavior for the noise function.

• Predefined set of unit vectors (2D:8, 3D:12, 4D:32)

Why Gradient Noise?

• Non-aligned light and dark areas to a lattice

value noise gradient noise

http://libnoise.sourceforge.net/noisegen/ rotation invariant

Improved Perlin Noise

• Published at SIGGRAPH 2002 after almost 20 years

cubic interpolation quintic interpolation

(2t3_+3t2_{) (6t5-15t4+10t3 )}

http://http.developer.nvidia.com/GPUGems/elementLinks/fig05-07.jpg

Original Perlin Noise Improved Perlin Noise

discontinuous

of 2nd _{order derivative} (zero curvature)

Improved Perlin Noise

cubic interpolation quintic interpolation

Improved Perlin Noise

cubic interpolation quintic interpolation

Cons. of Perlin Noise

• Zero at any lattice point

⇒ Value-gradient noise (value+gradient)

• Slow for higher dimensions

Simplex Noise

• Lower computational complexity • Linear dimensional scalibility

Simplex

• A simplest polygon tessellating n-D space • A hyper-tetrahedron with n+1 corners

‣ _{1D: line segment} ‣ _{2D: triangle}

‣ _{3D: tetrahedron}

Simplex

simplex vertex count hyper-cube vertex count

1D line 2 line 2 2D triangle 3 square 4 3D tetrahedron 4 hexahedron 8 4D pentachoron 5 tesseract 16

O(n)

O(2

n

)

Simplex noise lattice points Perlin noise lattice points

Simplex Noise

• Step1) Find the simplex including the query point.

Simplex Noise

Simplex vs Perlin

Worley(=Cell) Noise

• Point based approach (But, It also use grid.) • Noise value = the distance to the closest point

Fractal

• If you look at things in nature, they look like fractals.

• Various level of details in a mountain

‣ _{Large variations (mountains)}

‣ _{Medium variations (hills)}

‣ _{Small variations (boulders)}

‣ _{Tiny variations (stones)}

‣ _{. . .}

Fractal

• Frequency: the number of cycles in unit length • Amplitude: the maximum absolute value

• Octave: successive coherent-noises • Lacunarity: scale factor of frequency

• Persistence(or gain): scale factor of amplitude

float value = 0.f;

for( int i=0;i<numOctaves;++i ) { value += noise( x*f, y*f, z*f ) * a; f *= lacunarity;

a *= persistence; }

fBm

• Fractal Brownian Motion

• The summation of successive octaves of noise,

each with higher frequency and lower amplitude.

http://libnoise.sourceforge.net/noisegen/

+ + +

fBm

• Fractal Brownian Motion

• The summation of successive octaves of noise,

each with higher frequency and lower amplitude.

+ + =

Procedural Textures

http://lodev.org/cgtutor/randomnoise.html http://devmag.org.za/2009/04/25/perlin-noise/

• Play with sin(), cos(), abs(), …

while combining octaves.