Anisotropic Noise Alex Goldberg Matthias ZwickerFrédo Durand University of California, San DiegoMIT CSAIL PixelActive Inc.
Procedural Noise Pioneered by Ken Perlin more than 20 years ago Powerful primitive for texture synthesis Valuable for small details [Perlin, 99]
Noise Properties Simple, irregular appearance 2D Noise
Noise Properties Simple, irregular appearance – Octaves combine for complex textures – Use directly or as input to another function Noise Octaves Summed (Fractal) Noise =
Noise 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 Distortion 2D noise textures still prevalent Unsightly parameterization artifacts 3D Noise [Olano,05] 2D Noise [Olano,05] Stretching Artifact
Noise Filtering 2D noise textures can use hardware anisotropic texture filtering 3D noise hard to filter without aliasing
3D Noise Filtering Common approach: octave truncation Exclude octaves that would lead to aliasing + + Foreground
3D Noise Filtering Common approach: octave truncation Exclude octaves that would lead to aliasing + + ForegroundNear-Background
3D Noise Filtering Common approach: octave truncation Exclude octaves that would lead to aliasing + + ForegroundNear-Background Background
Anisotropic Filtering Circular pixel projects to an elliptical footprint Surface Space Screen Space x y
Anisotropic Filtering Circular pixel projects to an elliptical footprint Surface Frequency Space Fourier Transform Surface Space Screen Space x y
Noise In The Frequency Domain Spatial Domain + = +
Noise In The Frequency Domain Spatial Domain + = + Frequency Domain
Anisotropic Noise Filtering = * Pixel Footprint Noise Spectrum Anisotropic-Filtered Noise Spectrum Frequency domain multiplication = Anisotropic Filtering
Frequency Space Footprint Octave Truncation Frequency Analysis
1 st Octave Spectrum
2 nd Octave Spectrum Octave Truncation Frequency Analysis
1 st Octave Spectrum 2 nd Octave Spectrum Aliasing! Octave Truncation Frequency Analysis
1 st Octave Spectrum 2 nd Octave Spectrum 3 rd Octave Spectrum Aliasing! Octave Truncation Frequency Analysis
1 st Octave Spectrum 2 nd Octave Spectrum 3 rd Octave Spectrum Aliasing! Blurriness! Octave Truncation Frequency Analysis
Octave truncation: aliasing and blurriness isotropic filtering Blurriness Aliasing Ideal Anisotropic Spectrum Truncated Spectrum Spectrum Showdown
Perlin Noise Spectrum Perlin Noise not tightly band-limited Wide overlap makes octave truncation harder Perlin Frequency Spectra Overlaid Octaves
Perlin Noise Spectrum Perlin Noise not tightly band-limited Wide overlap makes octave truncation harder Perlin Frequency Spectra Overlaid Octaves
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] Perlin Spectrum Wavelet Noise Spectrum
Noise Today 3D Noise: Uniform features, no anisotropic filtering 3D Noise Blurriness
Noise Today 3D Noise: Uniform features, no anisotropic filtering 2D Noise: Anisotropic filtering, stretching artifacts 3D Noise 2D Noise Stretching Artifact Blurriness
Anisotropic Noise Band-limited, anisotropic filtering Uniform features on parameterized meshes Efficient implementation
Anisotropic Noise Outline Anisotropic Noise Tiles Noise tile synthesis Parametric distortion compensation Anisotropic filtering GPU implementation
The Basic Idea Partition the frequency domain into orientations Partitioned Frequency Domain Noise Orientation Spectra
Frequency Palette Spatial Domain Tiles Frequency Domain ORIENTATIONS
Frequency Palette SCALES Spatial Domain ORIENTATIONS Additional frequencies by scaling base tiles
Single Noise Octave Noise Tiles Single Noise Octave = Spatial Domain Frequency Domain
Steerable Noise Approximate arbitrary frequency spectra Example: Elliptical spectrum Target Spectrum Approximated Spectrum
Steerable Noise 19 No Fourier / Inverse Transform at runtime Tile frequency ranges known in advance Output is a linear blend of spatial noise tiles
Steering Texture Steerable Noise Approximate arbitrary frequency spectra Example: Elliptical spectra with steering texture Approximated Spectra
Steerable Noise Steering Texture Approximated Spectra Output Noise Approximate arbitrary frequency spectra Example: Elliptical spectra with steering texture
Anisotropic Noise Outline Anisotropic Noise Tiles Noise tile synthesis Parametric distortion compensation Anisotropic filtering GPU implementation
Noise Synthesis Steps 1. Generate frequency-domain white noise Frequency Domain
Noise Synthesis Steps 2. Multiply with oriented, band-limited filter masks Frequency Domain white noise * * * * = = = = Filter Masks Noise Tile Spectra
Noise Synthesis Steps 3. Inverse Fourier transform yields spatial noise tiles F -1 Noise Tile Spectra Spatial Domain Noise Tiles
Extension to 3D noise Could be extended to 3D noise Output would be a set of volume textures – Significant memory cost So lets stick with 2D noise…
Anisotropic Noise Outline Anisotropic Noise Tiles Noise tile synthesis Parametric distortion compensation Anisotropic filtering GPU implementation
Parameterization Distortions 2D noise suffers from parameterization artifacts Distorted Mesh Mesh Parameterization Stretching Artifacts
Parameterization Distortions 2D noise suffers from parameterization artifacts Distorted Mesh Mesh Parameterization Stretching Artifacts
Frequency Space Analysis u v Texture Space Texture Frequency Space
Frequency Space Analysis u v Texture Space t s Local Object Space Texture Frequency Space
u v Texture Space t s Local Object Space Frequency Space Analysis Texture Frequency SpaceObject Frequency Space
Frequency Space Analysis t s Local Object Space Object Frequency Space t s Local Object Space
Frequency Space Analysis t s Local Object Space Object Frequency Space Texture Frequency Space
Frequency Space Analysis t s Local Object Space u v Texture Space Object Frequency Space Texture Frequency Space
t s Local Object Space u v Texture Space Frequency Space Analysis Object Frequency Space Texture Frequency Space
Distortion Compensation Goal Approximate target spectrum at this triangle Approach: use anisotropic spectrum control Target Frequency Spectrum
Spectrum Approximation * * ** ** Compute tile weights and store per-vertex Multiple scales required Target Spectrum Noise Tile Spectra
Spectrum Approximation * * ** ** Compute tile weights and store per-vertex Multiple scales required Target Spectrum Noise Tile Spectra
Tile Weight Computation Fast heuristic approach Approximate each subband as center point Evaluate target spectrum at each point Target SpectrumSubband Centers
Spectral Results Target SpectrumHeuristic Fit More orientations would produce a tighter fit Four works well in practice
Distortion Compensation Results No Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results With Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results No Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results With Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results No Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results With Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results No Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Results With Distortion Compensation Distorted Mesh Mesh Parameterization
Distortion Compensation Animation Distortion Compensation (uniform appearance) No Distortion Compensation (noticeable stretching)
Anisotropic Noise Outline Anisotropic Noise Tiles Noise tile synthesis Parametric distortion compensation Anisotropic filtering GPU implementation
Anisotropic Filtering 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 Recap Pixel projects to elliptical footprint Frequency footprint shows representable frequencies Frequency Space Footprint
Anisotropic Filtering Recap Frequency Space FootprintGaussian Filter Footprint Pixel projects to elliptical footprint Frequency footprint shows representable frequencies
Anisotropic Noise Filtering Evaluate at subband centers for tile weights – As with distortion compensation Subband Centers Gaussian Filter Footprint
Anisotropic Noise Results Isotropic Filter (octave truncation)
Anisotropic Noise Results Anisotropic Noise
Anisotropic Noise Results Isotropic Filter (octave truncation)
Anisotropic Noise Results Anisotropic Noise
Anisotropic Noise Outline Anisotropic Noise Tiles Noise tile synthesis Parametric distortion compensation Anisotropic filtering GPU implementation
GPU 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 Cost
OctavesAnisotropic Noise FPS Performance Timings (1680 x 1050 noise samples) GeForce orientations, 3 scales
3D Effects With 2D Noise Cork-like WoodPine-like WoodMarble Most solid functions require uniform surface noise Satisfied by Anisotropic Noise
Filtering Nonlinear Functions Not technically correct for nonlinear functions Good results in practice
Animated Noise Texture coordinate shifting Pre-pass blend between 3 noise tiles
Limitations 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
Conclusion Anisotropic noise Fast, band-limited noise with anisotropic filtering Uniform features on paramerized meshes Steerable spectrum for anisotropic control
Acknowledgments Paul Green MIT pre-reviewers Arash Keshmirian Hugues Hoppe Microsoft New Faculty Fellowship Sloan Fellowship