Particle-based T-Spline Level Set Evolution for 3D Object Reconstruction with Range and Volume Constraints

Transkript

1 Particle-based T-Spline Level Set for 3D Object Reconstruction with Range and Volume Constraints Robert Feichtinger (joint work with Huaiping Yang, Bert Jüttler) Institute of Applied Geometry, JKU Linz Plzen, 4-7th February / 27

2 Introduction 3D object reconstruction with range and volume constraints Implicitly defined surface Point set surface Range and volume constraints 2 / 27

3 Introduction - Properties Implicitly defined surfaces Adapt of topology Compute normals s Linearization of volume constraints Point set surfaces Sample points for discretization 2 / 27

4 Introduction - Related work (non-exhausting) s (Funck et al. 06, 07) s (Flöry et al. 07) Point set surfaces (Alexa et al. 07) Implicitly defined surfaces (Osher et al. 02) V = V Target A 2 / 27

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6 Given: An unorganized point cloud 4 / 27

7 Given: An unorganized point cloud 4 / 27

8 Given: An unorganized point cloud 4 / 27

9 Given: An unorganized point cloud 4 / 27

10 Given: An unorganized point cloud 4 / 27

11 Given: An unorganized point cloud 4 / 27

12 Speed function v(x) = v(κ, n, d) x point on the surface. κ and n are geometric properties of the surface at x. d is the unsigned distance function. 5 / 27

13 T-spline T-spline function (introduced by Sederberg et al. 03) f(x, τ) = n T i (x)c i (τ) x Ω R 3 i=1 where τ is a time-variable. Zero level-set Γ(f, τ) = {x Ω f(x, τ) = 0} 6 / 27

14 Grids of a tensor product B-spline and a T-spline: 7 / 27

15 Grids of a tensor product B-spline and a T-spline: 7 / 27

16 Grids of a tensor product B-spline and a T-spline: 7 / 27

17 Grids of a tensor product B-spline and a T-spline: 7 / 27

18 Grids of a tensor product B-spline and a T-spline: 7 / 27

19 Grids of a tensor product B-spline and a T-spline: 5148 B-spline control points 1484 T-spline control points 7 / 27

20 Least-squares problem ( actual normal x Γ velocity at x }{{} value of speed function at x ) 2 da min Discretization f = n f = (...)ċ i i=1 N ( actual normal velocity at x j j=1 value of speed function at x j ) 2 min Particle x j Γ. 8 / 27

21 Algorithm Initialization (initial T-spline zero-level set and particles, pre-computation of the unsigned distance field function). of the implicitly defined surface (one time step). Projection of the particles. Local resampling of the particles if necessary. Final refinement. 9 / 27

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23 Particles: Requirements and goals We need sample points along the zero level-set. Sample should be dense enough (close to uniform). Sample points should be fast to compute. Local resampling. Correct topology. 11 / 27

24 Marching triangulation (Hartmann 98). Needs information on normals (provided by the implicit surface) Can be done locally or globally Distance between neighbouring sample points in the initial particle set ρ ρ is a user defined constant called feature-size. 12 / 27

25 Criteria for resampling Distance between neighbouring sample points x i x j > 2 ρ 13 / 27

26 Criteria for resampling Distance between neighbouring sample points x i x j > 2 ρ 13 / 27

27 Criteria for resampling Normals of neighbouring sample points n i is the normal of f at x i n i n j < ǫ 13 / 27

28 Criteria for resampling Normals of neighbouring sample points n i is the normal of f at x i n i n j < ǫ 13 / 27

29 Example target points 14 / 27

30 Example 1 14 / 27

31 Example particles 14 / 27

32 Example 1 14 / 27

33 Example 1 14 / 27

34 Example 1 14 / 27

35 Example 1 need local resampling for topological change 14 / 27

36 Example 1 14 / 27

37 Example particles 14 / 27

38 Example 1 14 / 27

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40 T-spline ensures that f(x, τ) 0 for x inside Γ(f, τ). Define a constraint that forces a region A to lie inside or outside of the zero level-set. f(x i ) 0 x i A 16 / 27

41 f < 0 f > 0 A 16 / 27

42 Least-squares problem N 0 j=1 ( actual normal velocity at x j actual value of the T-spline function at x j ) 2 α ǫ (f(x j, τ)) min Sample points x j A α ǫ (f(x j, τ)) activator function. 17 / 27

43 Activator function 1 0,8 0,6 0,4 α ǫ = 1 A α ǫ = 0 α ǫ 0,2-1,4-1,2-1 -0,8-0,6-0,4-0,2 0 0 t 18 / 27

44 Example 2 19 / 27

45 Example 2 19 / 27

46 Example 2 19 / 27

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48 Specify a volume function V (τ) Control the volume change V (τ) during the evolution Volume preservation V (τ) = 0 21 / 27

49 Formulation as linear constraint ( actual normal where x Γ velocity at x }{{} ) da = V f = n f = (...)ċ i Use Lagrangian multipliers to add the constraint to the evolution. i=1 22 / 27

50 Example 3 input data V = 0.08 V = V = / 27

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52 Example 4 input data (48485 points) and grid 25 / 27

53 Example 4 without constraint 25 / 27

54 Example 4 using the data points for the range constraint time: 36 sec 25 / 27

55 Example 4 after displacement mapping time: 3 sec 25 / 27

56 Summary and conclusion Combination of the evolution of implicitly defined surfaces and point set surfaces. Strategies for resampling the point set surface. Range and volume constraints to represent a-priori knowledge 26 / 27

57 Thanks for your attention! 27 / 27