04 - Normal Estimation, Curves

Transcription

1 04 - Normal Estimation, Curves Acknowledgements: Olga Sorkine-Hornung

2 Normal Estimation

3 Implicit Surface Reconstruction Implicit function from point clouds Need consistently oriented normals < 0 0 > 0

4 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

5 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

6 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

7 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

8 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

9 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

10 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

11 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

12 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

13 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

14 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

15 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

16 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

17 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

18 Normal Estimation Assign a normal vector n at each point cloud point x Estimate the direction by fitting a local plane Find consistent global orientation by propagation (spanning tree)

19 Local Plane Fitting For each point x in the cloud, pick k nearest neighbors or all points in r-ball: Find a plane Π that minimizes the sum of square distances:

20 Local Plane Fitting For each point x in the cloud, pick k nearest neighbors or all points in r-ball: Find a plane Π that minimizes the sum of square distances:

21 Linear Least Squares? y x

22 Linear Least Squares? Find a line y = ax+b s.t. y x But we would like true orthogonal distances!

23 Best Fit with SSD y yʹ xʹ x SSD = sum of squared distances (or differences)

24 Principle Component Analysis (PCA) PCA finds an orthogonal basis that best represents a given data set y yʹ xʹ z x x y PCA finds the best approximating line/plane/orientation (in terms of Σdistances 2 )

25 Notations Input points: Looking for a (hyper) plane passing through c with normal n s.t.

26 Notations Input points: Centroid: m Vectors from the centroid:

27 Centroid: 0-dim Approximation It can be shown that: m m minimizes SSD m will be the origin of the (hyper)-plane Our problem becomes:

28 Hyperplane Normal Minimize!

29 Hyperplane Normal Minimize!

30 Hyperplane Normal Constrained minimization Lagrange multipliers Matrix Cookbook!

31 Hyperplane Normal Constrained minimization Lagrange multipliers

32 Hyperplane Normal Constrained minimization Lagrange multipliers What can be said about n??

33 Hyperplane Normal Constrained minimization Lagrange multipliers n is the eigenvector of S with the smallest eigenvalue

34 Summary Best Fitting Plane Recipe Input: Compute centroid = plane origin Compute scatter matrix The plane normal n is the eigenvector of S with the smallest eigenvalue

35 What does the Scatter Matrix do? Let s look at a line l through the center of mass m with direction vector v, and project our points x i onto it. The variance of the projected points xʹi is: x i y i l m v xʹi l l l l Original set Small variance Large variance

36 What does the Scatter Matrix do? The scatter matrix measures the variance of our data points along the direction v x i y i l m v xʹi l l l l Original set Small variance Large variance

37 Principal Components Eigenvectors of S that correspond to big eigenvalues are the directions in which the data has strong components (= large variance). If the eigenvalues are more or less the same there is no preferable direction.

38 Principal Components There s no preferable direction S looks like this: There s a clear preferable direction S looks like this: Any vector is an eigenvector µ is close to zero, much smaller than λ

39 Normal Orientation PCA may return arbitrarily oriented eigenvectors Wish to orient consistently Neighboring points should have similar normals

40 Normal Orientation Build graph connecting neighboring points Edge (i,j) exists if xi knn(xj) or xj knn(xi) Propagate normal orientation through graph For neighbors xi, xj : Flip nj if ni T nj < 0 Fails at sharp edges/corners Propagate along safe paths (parallel tangent planes) Minimum spanning tree with angle-based edge weights Surface reconstruction from unorganized points, Hoppe et al., SIGGRAPH

41 Elementary Differential Geometry

42 Differential Geometry Motivation Describe and analyze geometric characteristics of shapes e.g. how smooth?

43 Differential Geometry Motivation Describe and analyze geometric characteristics of shapes e.g. how smooth? how shapes deform

44 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood

45 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood manifold point

46 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood manifold point

47 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood continuous 1-1 mapping manifold point

48 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood manifold point continuous 1-1 mapping non-manifold point

49 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood manifold point continuous 1-1 mapping non-manifold point x

50 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood

51 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood continuous 1-1 mapping

52 Differential Geometry Basics Geometry of manifolds Things that can be discovered by local observation: point + neighborhood v continuous 1-1 mapping u If a sufficiently smooth mapping can be constructed, we can look at its first and second derivatives Tangents, normals, curvatures, curve angles Distances, topology

53 Curves

54 How do we model shapes? Delcam Plc. [CC BY-SA 3.0 ( or GFDL ( via Wikimedia Commons

55 Building blocks: curves and surfaces By Wojciech mula at Polish Wikipedia, CC BY-SA 3.0,

56 A modeling session Demo with Keynote for 2D Demo with Blender for 3D

57 Modeling curves We need mathematical concepts to characterize the desired curve properties Notions from curve geometry help with designing user interfaces for curve creation and editing

58 2D parametric curve must be continuous

59 A curve can be parameterized in many different ways

60 Tangent vector Parametrization-independent!

61 Arc length How long is the curve between and? How far does the particle travel? Speed is, so: Speed is nonnegative, so is non-decreasing

62 Arc length parameterization Every curve has a natural parameterization:

63 Arc length parameterization Every curve has a natural parameterization: Isometry between parameter domain and curve Tangent vector is unit-length:

64 Curvature How much does the curve turn per unit?

65 Curvature How much does the curve turn per unit?

66 Curvature profile Given, we can get up to a constant by integration. Integrating reconstructs the curve up to rigid motion.

67 Curvature of a circle Curvature of a circle: = Unit Normal Osculating circle

68 Frenet Frame

69 Curvature normal Points inward useful for evaluating curve quality

70 Smoothness Two kinds, parametric and geometric: Parametrization-Independent

71 Smoothness example

72 Recap on parametric curves

73 Turning Angle from start tangent to end tangent: If curve is closed, the tangent at the beginning is the same as the tangent at the end

74 Turning numbers measures how many full turns the tangent makes.

75 Gauss map Point on curve maps to point on unit circle.

76 Space curves (3D) In 3D, many vectors are orthogonal to T B are the Frenet frame N is torsion: non-planarity T

77 Discrete Differential Geometry of Curves Some references: see

78 Discrete Planar Curves

79 Discrete Planar Curves Piecewise linear curves Not smooth at vertices Can t take derivatives Generalize notions from the smooth world for the discrete case! There is no one single way

80 Tangents, Normals For any point on the edge, the tangent is simply the unit vector along the edge and the normal is the perpendicular vector

81 Tangents, Normals For vertices, we have many options

82 Tangents, Normals Can choose to average the adjacent edge normals

83 Tangents, Normals Weight by edge lengths

84 Inscribed Polygon, p Connection between discrete and smooth Finite number of vertices each lying on the curve, connected by straight edges.

85 The Length of a Discrete Curve Sum of edge lengths p 2 p 3 p 4 p 1

86 The Length of a Continuous Curve Take limit over a refinement sequence h = max edge length

87 Curvature of a Discrete Curve Curvature is the change in normal direction as we travel along the curve no change along each edge curvature is zero along edges

88 Curvature of a Discrete Curve Curvature is the change in normal direction as we travel along the curve normal changes at vertices record the turning angle!

89 Curvature of a Discrete Curve Curvature is the change in normal direction as we travel along the curve normal changes at vertices record the turning angle!

90 Curvature of a Discrete Curve Curvature is the change in normal direction as we travel along the curve same as the turning angle between the edges

91 Curvature of a Discrete Curve Zero along the edges Turning angle at the vertices = the change in normal direction α 1 α 2 α 1, α 2 > 0, α 3 < 0 α 3

92 Total Signed Curvature α 1 α 2 Sum of turning angles α 3

93 Discrete Gauss Map Edges map to points, vertices map to arcs.

94 Discrete Gauss Map Turning number well-defined for discrete curves.

95 Discrete Turning Number Theorem For a closed curve, the total signed curvature is an integer multiple of 2π. proof: sum of exterior angles

96 Turning Number Theorem Continuous world Discrete world k:

97 Curvature is scale dependent is scale-independent

98 Discrete Curvature Integrated Quantity! Cannot view α i as pointwise curvature α 1 α 2 It is integrated curvature over a local area associated with vertex i

99 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1

100 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1 A 2

101 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1 A 2 The vertex areas A i form a covering of the curve. They are pairwise disjoint (except endpoints).

102 Structure Preservation Arbitrary discrete curve discrete analogue of continuous theorem total signed curvature obeys discrete turning number theorem even coarse mesh (curve) which continuous theorems to preserve? that depends on the application

103 Convergence Consider refinement sequence length of inscribed polygon approaches length of smooth curve in general, discrete measure approaches continuous analogue which refinement sequence? depends on discrete operator pathological sequences may exist in what sense does the operator converge? (pointwise, L 2 ; linear, quadratic)

104 Recap Structure- preservation Convergence For an arbitrary (even coarse) discrete curve, the discrete measure of curvature obeys the discrete turning number theorem. In the limit of a refinement sequence, discrete measures of length and curvature agree with continuous measures.

105 Thank you