Lapped Textures

Lapped Textures Princeton University Princeton University Microsoft Research Emil Praun Adam Finkelstein Hugues Hoppe

Goal mesh geometry ? textured surface “example” image

Goal • Little user effort • No apparent seams • No obvious periodicity • Low distortion • Local texture control • Anisotropy

Previous 2D Texture Synthesis Histogram equalization [Heeger ’96] Laplacian block shuffling [de Bonet ’97] Pixel template matching [Efros ’99] [Wei ’00]

Previous 2D Texture Synthesis Histogram equalization [Heeger ’96] Laplacian block shuffling [de Bonet ’97] Pixel template matching [Efros ’99] [Wei ’00] Random pasting of image blocks [Xu ’00]

Previous 3D Texturing • Volumetric textures: • Noise functions [Perlin ’85, Worley ’96] • Solid textures by example [Ghazanfarpour ’96] • Synthesizing texture on a surface: • Reaction-diffusion [Turk ’91, Witkin ’91] • Cellular textures [Fleischer ’95] • Covering surface with triangular tiles [Neyret ’99]

Approach texture patch surface

Key Idea: Patch Pasting texture patch “lapped textures” surface

Anisotropic vs. Isotropic images

PROCESS

Algorithm • Cut texture patches from input texture • [Specify texture direction & scale on mesh] • REPEAT • Pick texture patch T, mesh point P • Grow surface patch S around P to size(T) • Flatten S over T; paste T on mesh • Update mesh coverage info • UNTIL mesh is covered

Algorithm texture patch surface

Issues • Texture patch creation • Specifying direction field • Surface patch growth • Patch parametrization • Face coverage estimation • Texture storage and rendering

Texture Patch Creation

Less Structure  Splotch

Direction Field: User-specified

Direction Field: User-specified • Angle - Only for anisotropic patches. • Can be set for only small number of faces – Gaussian radial basis function is used for interpolation. • Since the normal lies within the plane of the face we get an orthogonal basis : • S = T x N. (N – normal to the plane).

Direction Field:Local to Patch

Patch Growth • 1. Choose a random point on a non fully covered triangle. • 2. Add triangles in a BFS order to cover the texture patch. • 3. Stop when the patch is fully coverd. • Condintions: • The surface patch homeomorphic to a disk. • The distortion should not exceed a given limit. • We add edge only if it is within the texture patch.

Patch Growth

Polygonal Hulls • Start w/ boundary pixels, edges between neighbors • Conservatively simplify polygons

Align Patch to Direction Field texture patch surface

Tangential Vector Field

Optimizing the Parametrization Least squares best match to unit axes Sparse linear system. No explicit fairness functional

^ (T) t ^ s (S) Optimizing the Parametrization  (A) A (B) T S B C (C) face in 3D 2D texture space

Optimizing the Parametrization • For each mesh face f ={A,B,C} the vector T lies within the face plain. • Barycentric coordinates: • T = A + B + C and  +  +  = 0.

^ (T) t ^ s (S) Optimizing the Parametrization  (A) A (B) T S B C (C) face in 3D 2D texture space ^ dT = (A) + (B) + (C) - t We wish to minimize :

Result of Optimization

Coverage estimation Render patch triangles Flag covered triangles Remember 1 pixel per uncovered triangle • off-screen buffer

Texture Storage and Rendering • Method 1: Texture Atlas • Pre-composite into a global texture map. • -- OR -- • Method 2: Runtime pasting • Composite at run-time using hardware

Texture Atlas • Texture Atlas is a set of charts {1,…, k}. • Each chart j is a mapping from a subset U j of the mesh to the Euclidean plane (the texture plane). • {U1,…, Uk} is a cover of the surface. • If i  j then j and i don’t overlap except on the edges.

Texture Atlas • Patches of triangles with similar normals • 2D packing problem for arbitrary polygons

Lapped Textures

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