Screened Poisson Surface Reconstruction Misha Kazhdan Hugues Hoppe

Screened Poisson Surface Reconstruction Misha Kazhdan Hugues Hoppe

Screened Poisson Surface Reconstruction Misha Kazhdan Hugues Hoppe Johns Hopkins University Microsoft Research Motivation 3D scanners are everywhere: Time of flight Structured light Stereo images

Shape from shading Etc. http://graphics.stanford.edu/projects/mich/ Motivation Surface reconstruction Geometry processing tio a z ri

n ete m ra Pa ation Decim Filte rin etc.

g Implicit Function Fitting Given point samples: Define a function with value zero at the points. Extract the zero isosurface. >0 F(q) =0 F(q)<0 0 F(q)>0

Sample points F(q) <0 Related work [Hoppe et al. 1992] [Curless and Levoy 1996] [Carr et al. 2001] [Kazhdan et al. 2006]

[Alliez et al. 2007] [Calakli and Taubin 2011] and many more Poisson Surface Reconstruction [2006] Oriented points samples of indicator gradient. Fit a scalar field to the gradients. 2 =min =

(q)=0.5 (q)=-0.5 ( ) ( ) Poisson Surface Reconstruction [2006] 1. Compute the divergence 2. Solve the Poisson equation

( ) ( ) 1 Poisson Surface Reconstruction [2006] 1. Compute the divergence 2. Solve the Poisson equation fine Discretize over an octree

Update coarse fine + + ( ) ( ) + 1 +

coarse Solution Correction Poisson Surface Reconstruction [2006] Properties: Supports noisy, non-uniform data Over-smoothes Solver time is super-linear Screened Poisson Reconstruction Higher fidelity at same triangle count Faster solver time is linear

Poisson Screened Poisson Outline Introduction Better / faster reconstruction Evaluation Conclusion Better Reconstruction Add discrete interpolation to the energy: 2

+ ( ) 0 2 ( )= ( ) ( ) Gradient fitting Sample interpolation [Carr et al.,,Calakli and Taubin] encouraged to be zero at samples Adds a bilinear SPD term to the energy Introduces inhomogeneity into the system Better Reconstruction Discretization:

Choose basis to represent : 1 ( ) ( ) +1 ( ) +2 ( ) 1 +2

+1 ( ) = ( ) =1 Better Reconstruction Discretization: For an octree, use B-splines: centered on each node scaled to the node size Better Reconstruction

Screened Poisson reconstruction: ^ To compute , solve: = with coefficients given by: = ( ) , ( ) ( ) = , ( ) + ( ) ( )

Bi Bj Better Reconstruction Screened Poisson reconstruction: ^ Sparsity is unchanged Entries are data-dependent

Bj Bi Bi = ( ) , ( ) ( ) = , ( ) + ( ) ( ) Bj

Faster Screened Reconstruction Observation: At coarse resolutions, no need to screen as precisely. Use average position, weighted by point count. Bj Bi Bi Bj B Bi j

Faster Reconstruction Solver inefficiency: fine Before updating, subtract constraints met at all coarser levels of the octree. complexity + ( ) +

+ Solution coarse Correction Faster Reconstruction Regular multigrid: Function spaces nest can upsample coarser solutions to finer levels Faster Reconstruction Adaptive multigrid:

Function spaces do not nest coarser solutions need to be stored explicitly Faster Reconstruction Naive enrichment: Complete octree Faster Reconstruction Observation: Only upsample the part of the solution visible to the finer basis. Faster Reconstruction

Enrichment: Iterate fine coarse Identify support of next-finer level Add visible functions Faster Reconstruction Original Enriched Faster Reconstruction Adaptive Poisson solver:

+ Update coarse fine + Get supported solution Adjust constraints + + + ( )

Solve residual + + + + + Solution + Correction

Visible Solution Outline Introduction Better / faster reconstruction Evaluation Conclusion Accuracy Poisson Screened Poisson

SSD [Calakli & Taubin] z z Accuracy Poisson

SSD [Calakli & Taubin] Screened Poisson

Performance Solver Time Poisson 89 sec Poisson (optimized) 36 sec Space

422 MB 604 MB Screened Poisson SSD [Calakli & Taubin] Input: 2x106 points 44 sec 3302 sec 1247 MB Performance Solver Time

Space Poisson 412 sec 1498 MB Poisson (optimized) 149 sec 2194 MB Screened Poisson 172 sec

Input: 5x106 points SSD [Calakli & Taubin] 19,158 sec 4895 MB Limitations Assumes clean data Poisson Screened Poisson Summary Screened Poisson reconstruction: Sharper reconstructions

Optimal-complexity solver Future Work Robust handling of noise (Non-watertight reconstruction) Extension to full multigrid Data: Thank You! [email protected], Digne et al., EPFL, Stanford Shape Repository Code:

Berger et al., Calakli et al., Manson et al. Funding: NSF Career Grant (#6801727) http://www.cs.jhu.edu/~misha/Code/PoissonRecon

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