Multi-chart Geometry Images Pedro Sander Harvard Harvard Hugues Hoppe Microsoft Research Hugues Hoppe Microsoft Research Steven Gortler Harvard Harvard John Snyder Microsoft Research John Snyder Microsoft Research Zoë Wood Caltech Caltech

Geometry representation semi-regularirregular completely regular

Basic idea cut parametrize

cut sample

cut [r,g,b] = [x,y,z] simple traversal to render store

Benefits of regularity l Simplicity in rendering n No vertex indirection n No texture coordinate indirection l Hardware potential l Leverage image processing tools for geometric manipulation l Simplicity in rendering n No vertex indirection n No texture coordinate indirection l Hardware potential l Leverage image processing tools for geometric manipulation

Limitations of single-chart Unavoidable distortion and undersampling Unavoidable distortion and undersampling long extremities high genus

Limitations of semi-regular Base charts effectively constrained to be equal size equilateral triangles

piecewise regular 400x160irregular Multi-chart Geometry Images

defineddefinedundefinedundefined l Simple reconstruction rules; for each 2-by-2 quad of MCGIM samples: n 3 defined samples render 1 triangle n 4 defined samples render 2 triangles (using shortest diagonal) l Simple reconstruction rules; for each 2-by-2 quad of MCGIM samples: n 3 defined samples render 1 triangle n 4 defined samples render 2 triangles (using shortest diagonal)

Multi-chart Geometry Images l Simple reconstruction rules; for each 2-by-2 quad of MCGIM samples: n 3 defined samples render 1 triangle n 4 defined samples render 2 triangles (using shortest diagonal) l Simple reconstruction rules; for each 2-by-2 quad of MCGIM samples: n 3 defined samples render 1 triangle n 4 defined samples render 2 triangles (using shortest diagonal)

Cracks in reconstruction l Challenge: the discrete sampling will cause cracks in the reconstruction between charts zippered

MCGIM Basic pipeline l Break mesh into charts l Parameterize charts l Pack the charts l Sample the charts l Zipper chart seams l Optimize the MCGIM l Break mesh into charts l Parameterize charts l Pack the charts l Sample the charts l Zipper chart seams l Optimize the MCGIM

Mesh chartification Goal: planar charts with compact boundaries Clustering optimization - Lloyd-Max (Shlafman 2002) : n Iteratively grow chart from given seed face. (metric is a product of distance and normal) n Compute new seed face for each chart. (face that is farthest from chart boundary) n Repeat above steps until convergence. Goal: planar charts with compact boundaries Clustering optimization - Lloyd-Max (Shlafman 2002) : n Iteratively grow chart from given seed face. (metric is a product of distance and normal) n Compute new seed face for each chart. (face that is farthest from chart boundary) n Repeat above steps until convergence.

Mesh chartification Bootstrapping n Start with single seed n Run chartification using increasing number of seeds each phase n Until desired number reached Bootstrapping n Start with single seed n Run chartification using increasing number of seeds each phase n Until desired number reached demo

Chartification Results l Produces planar charts with compact boundaries Sander et. al % stretch efficiency Our method 99% stretch efficiency

ParameterizationParameterization l Goal: Penalizes undersampling n L 2 geometric stretch of Sander et. al n Hierarchical algorithm for solving minimization l Goal: Penalizes undersampling n L 2 geometric stretch of Sander et. al n Hierarchical algorithm for solving minimization

ParameterizationParameterization l Goal: Penalizes undersampling n L 2 geometric stretch of Sander et. al n Hierarchical algorithm for solving minimization l Goal: Penalizes undersampling n L 2 geometric stretch of Sander et. al n Hierarchical algorithm for solving minimization Angle-preserving metric (Floater) (Floater)

Chart packing Goal: minimize wasted space l Based on Levy et al l Place a chart at a time (from largest to smallest) l Pick best position and rotation (minimize wasted space) l Repeat above for multiple MCGIM rectangle shapes n pick best Goal: minimize wasted space l Based on Levy et al l Place a chart at a time (from largest to smallest) l Pick best position and rotation (minimize wasted space) l Repeat above for multiple MCGIM rectangle shapes n pick best

Packing Results Levy packing efficiency 58.0% Our packing efficiency 75.6%

Sampling into a MCGIM l Goal: discrete sampling of parameterized charts into topological discs n Rasterize triangles with scan conversion n Store geometry l Goal: discrete sampling of parameterized charts into topological discs n Rasterize triangles with scan conversion n Store geometry

Sampling into a MCGIM Boundary rasterization Non-manifold dilation

Zippering the MCGIM l Goal: to form a watertight reconstruction

Zippering the MCGIM Algorithm: Greedy (but robust) approach n Identify cut-nodes and cut-path samples. n Unify cut-nodes. n Snap cut-path samples to geometric cut-path. n Unify cut-path samples. Algorithm: Greedy (but robust) approach n Identify cut-nodes and cut-path samples. n Unify cut-nodes. n Snap cut-path samples to geometric cut-path. n Unify cut-path samples.

Zippering: Snap l Snap n Snap discrete cut-path samples to geometrically closest point on cut-path l Snap n Snap discrete cut-path samples to geometrically closest point on cut-path

Zippering: Unify l Unify n Greedily unify neighboring samples l Unify n Greedily unify neighboring samples

How unification works l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance

How unification works l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance

How unification works l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance l Unify n Test the distance of the next 3 moves n Pick smallest to unify then advance

Geometry image optimization l Goal: align discrete samples with mesh features l Hoppe et. al l Reposition vertices to minimize distance to the original surface l Constrain connectivity l Goal: align discrete samples with mesh features l Hoppe et. al l Reposition vertices to minimize distance to the original surface l Constrain connectivity

Multi-chart results genus 2; 50 charts 478x133 Rendering PSNR 79.5

Multi-chart results genus 1; 40 charts 174x369 Rendering PSNR 75.6

Multi-chart results genus 0; 25 charts 281X228 Rendering PSNR 84.6

Multi-chart results genus 0; 15 charts 466x138 Rendering PSNR 83.8

478x133 irregular original single chart PSNR 68.0 multi- chart PSNR 79.5 demo Multi-chart results

Comparison to semi-regular Original irregular Semi-regularMCGIM

Comparison to semi-regular Original irregular mesh Semi-regular mesh PSNR 87.8 MCGIM mesh PSNR 90.2

SummarySummary l Contributions: n Overall: MCGIM representation –Rendering simplicity n Major: zippering and optimization n Minor: packing and chartification l Contributions: n Overall: MCGIM representation –Rendering simplicity n Major: zippering and optimization n Minor: packing and chartification

Future work l Provide: n Compression n Level-of-detail rendering control l Exploit rendering simplicity in hardware l Improve zippering l Provide: n Compression n Level-of-detail rendering control l Exploit rendering simplicity in hardware l Improve zippering