Robust Poisson Surface Reconstruction

Transcription

1 Robust Poisson Surface Reconstruction V. Estellers, M. Scott, K. Tew, and S. Soatto Univeristy of California, Los Angeles Brigham Young University June 2, /19

2 Goals: Surface reconstruction from noisy oriented point clouds scann registration reconstruct object pointcloud surface Motivation: New challenges as datasets grown in size, but lose in accuracy Techniques: formulation as convex minimization problem - model: global, robust to outliers - level-set representation: efficient and adaptive - algorithm: fast, scale well with size 2/19

3 Input Oriented point cloud: {x i, n i } Output Surface s indicator function χ, implicit representation S = {χ = 0} (a) pointcloud {x i, n i } (b) Indicator function χ (c) S = {χ = 0} KEY: pointcloud samples χ, i.e., χ(x i ) = n i 3/19

4 Variational Model State of the Art 1 min χ α 2 N χ(x k ) 2 + β 2 k=1 N k=1 χ(x k ) n k Hχ 2 F R 3 Our Model min χ R 3 f (n χ) + α 2 { N 1 χ(x k ) 2 f (v) = 2 v 2 2 v 2 < ɛ ɛ( v 2 ɛ 2 ) v 2 ɛ k=1 1 F. Calakli and G. Taubin. SSD: Smooth Signed Distance Surface Reconstruction. Computer Graphics Forum, /19

5 Variational Model State of the Art 1 min χ R 3 n χ α 2 N χ(x k ) 2 k=1 Our Model min χ R 3 f (n χ) + α 2 { N 1 χ(x k ) 2 f (v) = 2 v 2 2 v 2 < ɛ ɛ( v 2 ɛ 2 ) v 2 ɛ k=1 1 M. Kazhdan and H. Hoppe. Screened poisson surface reconstruction. ACM Transactions on Graphics, /19

6 Variational Model State of the Art 1 min χ R 3 n χ α 2 N χ(x k ) 2 k=1 Our Model min χ R 3 f (n χ) + α 2 { N 1 χ(x k ) 2 f (v) = 2 v 2 2 v 2 < ɛ ɛ( v 2 ɛ 2 ) v 2 ɛ k=1 1 M. Kazhdan and H. Hoppe. Screened poisson surface reconstruction. ACM Transactions on Graphics, /19

7 Huber vs least squares: rounding corners min χ R 3 n χ α 2 N χ(x k ) 2 k=1 (a) pointcloud (b) least-squares (c) our model 5/19

8 Huber vs least squares: Shrinking Bias min χ R 3 n χ α 2 N χ(x k ) 2 k=1 (a) pointcloud (b) least-squares (c) our model 6/19

9 Finite-Element Discretization 7/19

10 Efficient Representation FE discretization with non-uniform splines: χ(x) = n c A N A (x) A=1 why splines? Good analytical and numerical properties piecewise smooth, compactly supported, positive, bounded fast evaluation of basis and derivatives by Cox-De Bor why non-uniform? knot vector {ξ 1,..., ξ n+p+1 } tiles domain in polynomial pieces defines resolution and smoothness how non uniform? Local refinement add new knots to subdivide elements h-refine repeat existing knots to change the polynomial degree p-refine 8/19

11 Efficient Representation FE discretization with non-uniform splines: χ(x) = n c A N A (x) A=1 why splines? Good analytical and numerical properties piecewise smooth, compactly supported, positive, bounded fast evaluation of basis and derivatives by Cox-De Bor why non-uniform? knot vector {ξ 1,..., ξ n+p+1 } tiles domain in polynomial pieces defines resolution and smoothness how non uniform? Local refinement add new knots to subdivide elements h-refine repeat existing knots to change the polynomial degree p-refine 8/19

12 Efficient Representation FE discretization with non-uniform splines: χ(x) = n c A N A (x) A=1 why splines? Good analytical and numerical properties piecewise smooth, compactly supported, positive, bounded fast evaluation of basis and derivatives by Cox-De Bor why non-uniform? knot vector {ξ 1,..., ξ n+p+1 } tiles domain in polynomial pieces defines resolution and smoothness how non uniform? Local refinement add new knots to subdivide elements h-refine repeat existing knots to change the polynomial degree p-refine 8/19

13 Higher dimensional spaces Inefficient representation with tensor-product bases N p,q A (x, y) = N p i (x) M q j }{{} (y) {ξ i } n i=1 }{{} {(ξ i,υ j )} n,m i,j=1 }{{} {υ j } m j=1 A = i n+j Spline forest: only the 0-level set of χ describes S M. Scott, D. Thomas, and E. Evans. Isogeometric Spline Forests. Computer Methods in Applied Mechanics and Engineering, /19

14 Dynamic Spline Refinement Everything should be as simple as possible, but not simpler Start with first-order splines over coarse grid and refine dynamically 1 solve the minimization problem with coarse basis, 2 refine basis over elements violating the fidelity criterion 3 express χ with new basis and return to 1 10/19

15 Refinement criteria (a) pointcloud (b) point density (c) interpolation error (d) reconstruct. error 11/19

16 Minimization 12/19

17 Exploiting Convexity Discretize the integral with quadrature: given {p i, w i } Q i=1 R 3 f ( χ n) Q w i f ( χ(p i ) n(p i )) i=1 min c Q i=1 w i f ( A c A N A (p i ) n(p i ) ) + α 2 N k=1[ A c A N A (x k ) ] 2 Let P R 3Q n be sparse matrix with P ij = N j (p i ) min c,v min c,v Q w i f (V i n(p i )) + α 2 i=1 F (V ) + G(c) s.t. V = Pc N k=1[ A c A N A (x k ) ] 2 s.t. V = Pc 13/19

18 Experimental Results 14/19

19 Synthetic Data: Huber vs least squares Accuracy (c) least squares (d) Huber 15/19

20 Quantitative Comparison on Synthetic Data Table: Average reconstruction time and Haussdorf distance d D (10 2 distance units) between the point cloud and the reconstructed surface. d D subsampled PC perturbed PC bunny cow horse cube bunny cow horse cube Poisson SSD Poisson ours (Screened) Poisson surface reconstruction, Kazhdan, Hoppe. ACM TOG, (2006) 2013 SSD: Smooth Signed Distance Surface Reconstruction, Calakli, Taubin, Computer Graphics Forum, /19

21 Qualitative Comparison on Clean Data (a) comparison (b) SSD (c) ours 17/19

22 Qualitative Comparison on Real Data (a) object (b) pointcloud (c) pointcloud 18/19

23 Qualitative Comparison on Real Data (a) ours (b) Poisson 2013 (c) SSD 18/19

24 Conclusions Robust surface reconstruction with a spline forest robust model that avoids the shrinking bias of least-squares efficient discretization with hierarchical B-spline forest dynamic refinement guided by the reconstructed surface 19/19

25 SSD: mixed Finite Elements and Finite Differences Inside each cell, χ is linearly interpolated from the cell corners. χ(x) = 8 χ 1 w i (x)χ i = w T i=1 χ 8 χ constant within the cell χ(ω) = χ Ω Hessian is a distribution supported on the faces of the cell: χ 8 H lm = 1 lm ( χ(ω l ) χ(ω m )) Ω l, Ω m neighbor cells 1/2

26 Poisson Reconstruction Uniform splines, i.e., with a uniform knot vector Each B-spline is centered at a leaf of an octree B-splines loose their defining property: they are not longer a basis. 2/2

27 F. Calakli and G. Taubin. SSD: Smooth Signed Distance Surface Reconstruction. Computer Graphics Forum, A. Chambolle and T. Pock. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. Journal of Mathematical Imaging and Vision, M. Kazhdan and H. Hoppe. Screened poisson surface reconstruction. ACM Transactions on Graphics, M. Scott, D. Thomas, and E. Evans. Isogeometric Spline Forests. Computer Methods in Applied Mechanics and Engineering, /2