Anisotropic Noise Alex Goldberg Matthias ZwickerFrédo Durand University of California, San DiegoMIT CSAIL PixelActive Inc

AnisotropicNoiseAnisotropicNoise

Alex GoldbergMatthias Zwicker Frédo Durand

University of California, San Diego MIT CSAIL

PixelActive Inc.

Procedural NoiseProcedural Noise

• Pioneered by Ken Perlin more than 20 years ago

• Powerful primitive for texture synthesis

• Valuable for small details

[Perlin, ‘99] [Perlin, ‘99]

Noise PropertiesNoise Properties

• Simple, irregular appearance

2D Noise

Noise PropertiesNoise Properties

• Simple, irregular appearance

– Octaves combine for complex textures

– Use directly or as input to another function

Noise Octaves Summed (Fractal) Noise

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+

+ + =

Noise TodayNoise Today

• Important for modern games– Increasing content demand

– Hand-created content time-consuming

• GPUs allow real-time noise

• Olano Noise (2005): Fast GPU results– 3D noise in two texture lookups

2D Noise Distortion2D Noise Distortion

• 2D noise textures still prevalent

• Unsightly parameterization artifacts

3D Noise

[Olano,’05]

2D Noise

[Olano,’05]

Stretching Artifact

Noise FilteringNoise Filtering

• 2D noise textures can use hardware anisotropic texture filtering

• 3D noise hard to filter without aliasing

3D Noise Filtering3D Noise Filtering

• Common approach: octave truncation

• Exclude octaves that would lead to aliasing

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Foreground




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Foreground Near-Background




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Foreground Near-Background Background

Anisotropic FilteringAnisotropic Filtering

• Circular pixel projects to an elliptical footprint

Surface Space

Screen Spacex

y


• Circular pixel projects to an elliptical footprint

Surface Frequency Space

Fourier Transform

Surface Space

Screen Spacex

y

Noise In The Frequency DomainNoise In The Frequency Domain

Spatial Domain

+ =+

Noise In The Frequency DomainNoise In The Frequency Domain

Spatial Domain

+ =+

Frequency Domain

Anisotropic Noise FilteringAnisotropic Noise Filtering

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Pixel Footprint Noise Spectrum Anisotropic-Filtered Noise Spectrum

Frequency domain multiplication = Anisotropic Filtering

Frequency Space Footprint

Octave Truncation Frequency AnalysisOctave Truncation Frequency Analysis


1st Octave Spectrum

1st Octave Spectrum 2nd Octave Spectrum


1st Octave Spectrum 2nd Octave Spectrum

Aliasing!


1st Octave Spectrum 2nd Octave Spectrum 3rd Octave Spectrum

Aliasing!


1st Octave Spectrum 2nd Octave Spectrum 3rd Octave Spectrum

Aliasing! Blurriness!


• Octave truncation:

• aliasing and blurriness

• isotropic filtering

Blurriness

Aliasing

Ideal Anisotropic Spectrum Truncated Spectrum

Spectrum ShowdownSpectrum Showdown

Perlin Noise SpectrumPerlin Noise Spectrum

• Perlin Noise not tightly band-limited

• Wide overlap makes octave truncation harder

Perlin Frequency Spectra

Overlaid Octaves

Perlin Noise SpectrumPerlin Noise Spectrum

• Perlin Noise not tightly band-limited

• Wide overlap makes octave truncation harder

Perlin Frequency Spectra

Overlaid Octaves

Wavelet Noise (Cook, DeRose 2005)Wavelet Noise (Cook, DeRose 2005)

• Tighter frequency extent than Perlin Noise

– But filtering still isotropic

• Aliasing / blurriness tradeoff high even for tightly band-limited functions

[Cook, DeRose,’05] [Cook, DeRose,’05]

Perlin Spectrum Wavelet Noise Spectrum


• 3D Noise: Uniform features, no anisotropic filtering

3D NoiseBlurriness


• 3D Noise: Uniform features, no anisotropic filtering

• 2D Noise: Anisotropic filtering, stretching artifacts

3D Noise 2D Noise

Stretching Artifact

Blurriness

Anisotropic NoiseAnisotropic Noise

• Band-limited, anisotropic filtering

• Uniform features on parameterized meshes

• Efficient implementation

Anisotropic Noise OutlineAnisotropic Noise Outline

• Anisotropic Noise Tiles

• Noise tile synthesis

• Parametric distortion compensation

• Anisotropic filtering

• GPU implementation

The Basic IdeaThe Basic Idea

• Partition the frequency domain into orientations

Partitioned Frequency Domain Noise Orientation Spectra

Frequency PaletteFrequency Palette

Spatial DomainTiles

FrequencyDomain

ORIENTATIONS

Frequency PaletteFrequency PaletteS

CA

LES

Spatial Domain

ORIENTATIONS

• Additional frequencies by scaling base tiles

Single Noise OctaveSingle Noise Octave

Noise Tiles Single Noise Octave

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Spatial D

omain

Frequency D

omain

Steerable NoiseSteerable Noise

• Approximate arbitrary frequency spectra

• Example: Elliptical spectrum

Target Spectrum Approximated Spectrum

Steerable NoiseSteerable Noise19

• No Fourier / Inverse Transform at runtime

• Tile frequency ranges known in advance

• Output is a linear blend of spatial noise tiles

Steering Texture



• Example: Elliptical spectra with steering texture

Approximated Spectra


Steering Texture

Approximated Spectra Output Noise


• Example: Elliptical spectra with steering texture







Noise Synthesis StepsNoise Synthesis Steps

1. Generate frequency-domain white noise

Frequency Domain


2. Multiply with oriented, band-limited filter masks

Frequency Domainwhite noise

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Filter Masks Noise Tile Spectra


3. Inverse Fourier transform yields spatial noise tiles

F-1 F-1 F-1 F-1

Noise Tile Spectra

Spatial Domain Noise Tiles

Extension to 3D noiseExtension to 3D noise

• Could be extended to 3D noise

• Output would be a set of volume textures

– Significant memory cost

• So let’s stick with 2D noise…







Parameterization DistortionsParameterization Distortions

• 2D noise suffers from parameterization artifacts

Distorted Mesh

Mesh Parameterization

Stretching Artifacts

Parameterization DistortionsParameterization Distortions

• 2D noise suffers from parameterization artifacts

Distorted Mesh


Stretching Artifacts

Frequency Space AnalysisFrequency Space Analysis

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Texture Space

Texture Frequency Space


u

v

Texture Space

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Local Object Space

Texture Frequency Space

u

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Texture Space

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Local Object Space


Texture Frequency Space Object Frequency Space


t

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Local Object Space

Object Frequency Space

t

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Local Object Space


t

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Local Object Space

Object Frequency Space Texture Frequency Space


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Local Object Space

u

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Texture Space


t

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Local Object Space

u

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Texture Space



Distortion Compensation GoalDistortion Compensation Goal

• Approximate target spectrum at this triangle

• Approach: use anisotropic spectrum control

Target Frequency Spectrum

Spectrum ApproximationSpectrum Approximation

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......

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* *

* *

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≈

• Compute tile weights and store per-vertex

• Multiple scales required

Target Spectrum

Noise Tile Spectra

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12 11

01 02

10

Spectrum ApproximationSpectrum Approximation

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......

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*

* *

* *

+ +

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+ +

≈

• Compute tile weights and store per-vertex

• Multiple scales required

Target Spectrum

Noise Tile Spectra

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12 11

01 02

10

Tile Weight ComputationTile Weight Computation

• Fast heuristic approach

• Approximate each subband as center point

• Evaluate target spectrum at each point

Target SpectrumSubband Centers

Spectral ResultsSpectral Results

Target Spectrum Heuristic Fit

• More orientations would produce a tighter fit

• Four works well in practice

Distortion Compensation ResultsDistortion Compensation Results

• No Distortion Compensation

Distorted Mesh



• With Distortion Compensation

Distorted Mesh




Distorted Mesh




Distorted Mesh




Distorted Mesh




Distorted Mesh




Distorted Mesh




Distorted Mesh


Distortion Compensation AnimationDistortion Compensation Animation

Distortion Compensation (uniform appearance)

No Distortion Compensation (noticeable stretching)








• Any 2D filtering approach can be used

• 2D hardware anisotropic filtering

– Memory bandwidth-intensive

• Our approach: view-dependent tile weights

– Relies only on bilinear filtering

– Minimal computational cost

Anisotropic Filtering RecapAnisotropic Filtering Recap

• Pixel projects to elliptical footprint

• Frequency footprint shows representable frequencies

Frequency Space Footprint

Anisotropic Filtering RecapAnisotropic Filtering Recap

Frequency Space Footprint Gaussian Filter Footprint

• Pixel projects to elliptical footprint

• Frequency footprint shows representable frequencies

Anisotropic Noise FilteringAnisotropic Noise Filtering

• Evaluate at subband centers for tile weights

– As with distortion compensation

Subband Centers Gaussian Filter Footprint

Anisotropic Noise ResultsAnisotropic Noise Results

Isotropic Filter (octave truncation)


Anisotropic Noise


Isotropic Filter (octave truncation)


Anisotropic Noise







GPU ImplementationGPU Implementation

• Distortion weights: CPU or vertex shader

• Antialiasing weights: pixel shader or vertex shader

• Noise tiles packed into a single RGBA texture

– 256x256x4 = 256KB

Separate Tile Textures Packed RGBA Texture

• Bottleneck: texture lookups

• One lookup per scale

– 3 scales for first octave usually sufficient

• Only one more lookup for each higher octave

– Olano Noise: 6 lookups for 3 octaves

– Anisotropic Noise: 5 lookups for 3 octaves

• Matches or outperforms Olano Noise

Performance CostPerformance Cost

Octaves AnisotropicNoise FPS

1 377

2 307

3 263

PerformancePerformance

• Timings (1680 x 1050 noise samples)

• GeForce 6800

• 4 orientations, 3 scales

3D Effects With 2D Noise3D Effects With 2D Noise

Cork-like WoodPine-like Wood Marble

• Most solid functions require uniform surface noise

• Satisfied by Anisotropic Noise

Filtering Nonlinear FunctionsFiltering Nonlinear Functions• Not technically correct for nonlinear functions

• Good results in practice

Animated NoiseAnimated Noise

• Texture coordinate shifting

• Pre-pass blend between 3 noise tiles

LimitationsLimitations

• Mesh parameterization required

– But can compensate for imperfect parameterizations

• Filtering imperfect for nonlinear functions

– But often produces good results in practice

• Additional per-vertex data for pre-computed weights

– But can compute in vertex shader

ConclusionConclusion

Anisotropic noise

• Fast, band-limited noise with anisotropic filtering

• Uniform features on paramerized meshes

• Steerable spectrum for anisotropic control

AcknowledgmentsAcknowledgments

Documents

Anisotropic Noise Alex Goldberg Matthias ZwickerFrédo Durand University of California, San DiegoMIT CSAIL PixelActive Inc