Screened PoissonSurface Reconstruction

Misha KazhdanJohns Hopkins University

Hugues HoppeMicrosoft Research

Motivation3D scanners are everywhere:• Time of flight• Structured light• Stereo images• Shape from shading• Etc.

http://graphics.stanford.edu/projects/mich/

Geometryprocessing

Motivation

Parameterization

Decimation

Filtering

etc.

Surfacereconstruction

Implicit Function FittingGiven point samples:– Define a function with value zero at the points.– Extract the zero isosurface. >0

<0

0

F(q)Sample points

F(q)<0

F(q)>0

F(q) =0

Related work

[Kazhdan et al. 2006]

[Hoppe et al. 1992] [Curless and Levoy 1996]

[Calakli and Taubin 2011][Alliez et al. 2007]

[Carr et al. 2001]

… and many more …

Poisson Surface Reconstruction [2006]– Oriented points samples of indicator gradient.– Fit a scalar field to the gradients.

∇ 𝜒=𝑉

𝜒 (𝑞) 𝑉 (𝑞 )

𝜒=min𝐹

‖∇𝐹−𝑉‖2

(q)=-0.5

(q)=0.5

Poisson Surface Reconstruction [2006]1. Compute the divergence2. Solve the Poisson equation

𝑉 (𝑞 )

∇⋅ Δ−1

𝜒 (𝑞)

Poisson Surface Reconstruction [2006]1. Compute the divergence2. Solve the Poisson equation

Discretize over an octree Update coarse fine

coarse

fine

+

+

+

+

Solution Correction

𝑉 (𝑞 )

Δ−1

𝜒 (𝑞)

∇⋅

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

Screened PoissonPoisson

Outline• Introduction

• Better / faster reconstruction

• Evaluation

• Conclusion

Better ReconstructionAdd discrete interpolation to the energy:

– encouraged to be zero at samples

Adds a bilinear SPD term to the energy Introduces inhomogeneity into the system

𝐸 ( 𝜒 )=∫‖∇ 𝜒 (𝑞)−𝑉 (𝑞)‖2ⅆ𝑞  Gradient fitting Sample interpolation

[Carr et al.,…,Calakli and Taubin]

+𝜆∑𝑝∈𝑃

‖𝜒 (𝑝 )−0‖2

Better ReconstructionDiscretization:Choose basis to represent :

𝜒 (𝑞)=∑𝑖=1

𝑛

𝑥𝑖𝐵𝑖 (𝑞 )

𝐵𝑖− 1 (𝑞 ) 𝐵𝑖 (𝑞) 𝐵𝑖+1 (𝑞 ) 𝐵𝑖+2 (𝑞 )

𝑥𝑖 −1

𝑥𝑖

𝑥𝑖+1

𝑥𝑖+2

Better ReconstructionDiscretization:For an octree, use B-splines:– centered on each node– scaled to the node size

Better ReconstructionPoisson reconstruction:To compute , solve:

with coefficients given by:

Screened

^

𝐿𝑥=𝑏

𝑏𝑖=∫ ⟨∇𝐵𝑖 (𝑞 ) ,𝑉 (𝑞 ) ⟩ ⅆ𝑞𝐿𝑖𝑗=∫ ⟨∇𝐵 𝑖 (𝑞 ) ,∇𝐵 𝑗 (𝑞 ) ⟩ⅆ𝑞+𝜆∑

𝑝∈𝑃

𝐵𝑖 (𝑝 )𝐵 𝑗 (𝑝)Bi

Bj

Poisson reconstruction:Sparsity is unchangedEntries are data-dependent

Better Reconstruction

𝑏𝑖=∫ ⟨∇𝐵𝑖 (𝑞 ) ,𝑉 (𝑞 ) ⟩ ⅆ𝑞

Screened

^

𝐿𝑖𝑗=∫ ⟨∇𝐵 𝑖 (𝑞 ) ,∇𝐵 𝑗 (𝑞 ) ⟩ⅆ𝑞+𝜆∑𝑝∈𝑃

𝐵𝑖 (𝑝 )𝐵 𝑗 (𝑝)Bi

Bj

Bj

Bi

Faster Screened ReconstructionObservation:At coarse resolutions, no need to screen as precisely. Use average position,

weighted by point count.Bj

Bi BjBi

BjBi

Faster ReconstructionSolver inefficiency:Before updating, subtract constraints met at all coarser levels of the octree. complexity

coarse

fine

+

+

+

Solution Correction

𝜒 (𝑞)

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 ofthe solution visible to the finer basis.

Faster Reconstruction

Enrichment:Iterate fine coarse

Identify support of next-finer levelAdd 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

𝜒 (𝑞)

Outline• Introduction

• Better / faster reconstruction

• Evaluation

• Conclusion

AccuracyPoisson Screened Poisson

𝑥

𝑧 z

𝑥

SSD [Calakli & Taubin]

z

𝑥

𝑧 𝑧

AccuracyPoisson Screened Poisson SSD [Calakli & Taubin]

𝑥

𝑧

𝑥

𝑧

𝑥

𝑧

Performance

Input: 2x106 points

Solver Time Space

Poisson 89 sec 422 MB

Poisson (optimized) 36 sec604 MB

Screened Poisson 44 sec

SSD [Calakli & Taubin] 3302 sec 1247 MB

Input: 5x106 points

Performance

Solver Time Space

Poisson 412 sec 1498 MB

Poisson (optimized) 149 sec2194 MB

Screened Poisson 172 sec

SSD [Calakli & Taubin] 19,158 sec 4895 MB

Limitations Assumes clean data

Poisson Screened Poisson

𝑥

𝑧

Summary

Screened Poisson reconstruction:

Sharper reconstructions

Optimal-complexity solver

Future Work• Robust handling of noise• (Non-watertight reconstruction)• Extension to full multigrid

Data:Aim@Shape, Digne et al., EPFL,Stanford Shape Repository

Code:Berger et al., Calakli et al.,Manson et al.

Funding:NSF Career Grant (#6801727)

Thank You!

http://www.cs.jhu.edu/~misha/Code/PoissonRecon