Sparse representation of terrains for procedural modeling

Transcription

1 Sparse representation of terrains for procedural modeling Eric Guérin, Julie Digne, Eric Galin and Adrien Peytavie LIRIS CNRS Geomod team Lyon, France

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3 Motivations Sparse modeling applied to terrain Many applications Inverse procedural modeling Compact representation Terrain synthesis Terrain amplification Perlin Noise Ridged

4 Related work Simulation Scalable Fast Control Geology [Benes 2006] Erosion Uplift [Hnaidi 2010] Sketching and editing Interactive Example editing based + - [Zhou 2007] + - Procedural Function Sparse [Génevaux 2013] 4

5 Overview Dictionary D Sparse Primitives Sparse construction tree S Procedural terrain Procedurallandform landformfeatures featuresdatabase database Procedural Analysis Terrain data T Reconstruction Blend Inverse procedural modeling Scene amplification and synthesis High level editing and sketching 5

6 Replace Blend Blend Final terrain River Mountain Bumpy hills Smooth hills 6

7 Sparse construction tree Extends [Génevaux15] «ling from feature Primitives» Hierarchical and functional model Combination of primitives and sub-trees Replace Blend Blend Final terrain River Mountain Bumpy hills Smooth hills

8 Sparse construction tree (2) Atoms of a dictionary are linearly combined to form sparse primitives Hierarchical terrain model Blend Sparse primitives Primitive f (p) Primitive Primitive j x i 0 i k x 0 i Dictionary of atoms D d (p) j d (p) k

9 Example T Ω Ω i j f (p) i f (p) j Sparse Primitives

10 About the sparsity represents the linear coefficients needed to reconstruct a primitive represents the number of nonzero coefficients, denoted s s has to be much smaller than N

11 Sparse decomposition (OMP) [Mallat93] e= T i D Γ β 2 2 Target patch Γ = 2 Γ = 4 Γ = 8 Γ = Γ

12 Dictionary learning Minimization problem Given a set of primitives F to represent and a sparsity s, find the dictionary D that minimizes the reconstruction error

13 Dictionary learning (2) K-SVD resolution [Aharon2006] In practice, we sometimes want to keep the original patches of the terrain intact

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15 Compact terrain representation Input : a terrain T Extract patches Optimize dictionary Store only D and X (sparse matrix) + mean altitudes of the patches

16 Compact terrain representation (2) Results (dictionary size 128 atoms)

17 Inverse procedural modeling Elevation map ~ d (p) k ~ d (p) j Analytic ~ dictionary D 1 Discretization d (p) k d j (p) Discrete dictionary D 2 Orthogonal Matching Pursuit on D Sparse matrix of coefficients X 3 ~ Reconstruction process using D Sparse Construction Tree T

18 Inverse procedural modeling (2) Perlin Noise Ridged

19 Terrain synthesis and amplification Low resolution Low resolution L OMP Correspondence Multi resolution dictionary Sparse matrix of coefficients X High resolution H Reconstruction High resolution

20 Terrain amplification

21 Terrain synthesis - Canyon Exemplar Sketch

22 Control Complete the tree with additional nodes Carve Blend Sparse Tree River banks Riverbed

23 Canyon with river

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25 Versatile model for terrain modeling Many perspectives Terrain stylization Include other data (color, vegetation, etc.) Volumetric data

26 Papaya Project Thank you!