Lecture 6 Introduction to Numerical Geometry. Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2018

Transcription

1 Lecture 6 Introduction to Numerical Geometry Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2018

2 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

3 Introduction Landscape HORSE Computer vision Pattern recognition 2D world Computer graphics 3D world Image processing Geometry processing

4 Shapes VS Images Geometry Parametrization Sampling Euclidean (flat) Global Uniform Cartesian Non-Euclidean (curved) Local Uniform is not well-defined

5 Shapes VS Images Representation Deformations Array of pixels Cloud of points, mesh, etc, etc. Rotation, affine, projective, etc. Wealth of non rigid deformations

6 Non rigid world from macro to nano Organs Nano machines Proteins Microorganisms Animals

7 Invariant similarity SIMILARITY TRANSFORMATION

8 Topics Metric spaces Canonical forms Local features Shape Geometric Representation words & expressions

9 Tools Metric geometry Fast marching Iterative closest point algorithms Multidimensional scaling Convex optimization

10 Materials A. M. Bronstein et al., Numerical geometry of non rigid shapes, Springer 2008

11 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

12 Distances Euclidean Manhattan Geodesic

13 Metric A function satisfying for all Non-negativity: Indiscernability: Symmetry: Triangle inequality: if and only if is called a metric space B A C AB BC + AC

14 Metric balls Open ball: Closed ball: Euclidean ball L 1 ball L ball

15 Connectivity The space is connected if it cannot be divided into two disjoint nonempty open sets, and disconnected otherwise Connected Stronger property: path connectedness Disconnected

16 Examples of metrics Euclidean Path length

17 Homeomorphisms A bijective (one-to-one and onto) continuous function with a continuous inverse is called a homeomorphism Homeomorphisms copy topology homeomorphic spaces are topologically equivalent Torus and cup are homeomorphic

18 Homeomorphisms Topology of Latin alphabet a b d e o p q c f h k l m n r s t u v w x y z homeomorphic to homeomorphic to i j homeomorphic to

19 Isometries Two metric spaces and are equivalent if there exists a distance-preserving map (isometry) satisfying Such and are called isometric, denoted Isometries copy metric geometries isometric spaces are equivalent from the point of view of metric geometry

20 Isometries Euclidean isometries

21 Isometries Euclidean isometries Rotation Translation Reflection

22 Isometries Geodesic isometries

23 Similarity as metric Human and monkey are similar Two deformations of a human are equivalent ~ Human is twice more similar to monkey than to dog Shape space

24 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

25 Metric for discrete geometry Discretization Continuous world Surface Discrete world Sampling Metric Topology Discrete metric (matrix of distances) Discrete topology (connectivity)

26 Metric for discrete geometry How to compute the intrinsic metric? So far, we represented itself. Our model of non-rigid shapes as metric spaces involves the intrinsic metric Sampling procedure requires as well. We need a tool to compute geodesic distances on.

27 Metric for discrete geometry Shortest path problem 183 Paris 346 Brussels Bern Munich Prague 194 Vienna

28 Metric for discrete geometry Shapes as graphs Sample the shape at vertices. Represent shape as an undirected graph set of edges representing adjacent vertices. Define length function measuring local distances as Euclidean ones,

29 Metric for discrete geometry Shapes as graphs Path between is an ordered set of connected edges where and. Path length = sum of edge lengths

30 Metric for discrete geometry Geodesic distance Shortest path between Length metric in graph Approximates the geodesic distance on the shape. Shortest path problem: compute and between any. Alternatively: given a source point, compute the distance map.

31 Metric for discrete geometry Bellman s principle of optimality Let be shortest path between and a point on the path. Then, and are shortest sub-paths between, and. Richard Bellman ( ) Suppose there exists a shorter path. Contradiction to being shortest path.

32 Metric for discrete geometry Edsger Wybe Dijkstra ( )

33 Metric for discrete geometry Dijkstra s algorithm Initialize and for the rest of the graph; Initialize queue of unprocessed vertices. While Find vertex x with smallest value of, For each unprocessed adjacent vertex, Remove from. Return distance map.

34 Metric for discrete geometry Troubles with the metric Grid with 4-neighbor connectivity. True Euclidean distance Shortest path in graph (not unique) Increasing sampling density does not help.

35 Metric for discrete geometry Metrication error 4-neighbor topology Manhattan distance 8-neighbor topology Continuous Euclidean distance Graph representation induces an inconsistent metric. Increasing sampling size does not make it consistent. Neither does increasing connectivity.

36 Metric for discrete geometry Discrete solution Continuous solution Stick to graph representation Change connectivity Consistency guaranteed under certain conditions Stick to given sampling Compute distance map on the surface New algorithm!

37 Metric for discrete geometry To solve the above issue, we can use fast marching methods A continuous variant of Dijkstra s algorithm Consistently approximate the intrinsic metric on the surface Source point

38 Metric for discrete geometry Usages of fast marching Geodesic distances Minimal geodesics Voronoi tessellation & sampling Offset curves

39 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

40 How good is a sampling?

41 Sampling density How to quantify density of sampling? is an -covering of if Alternatively: for all, where is the point-to-set distance.

42 Sampling efficiency Are all points necessary? An -covering may be unnecessarily dense (may even not be a discrete set). Quantify how well the samples are separated. is -separated if d ( x, x ) r X ' x, x for all. i i j X For, an -separated set is finite if is compact. j ' ' Also an r covering!

43 Farthest point sampling Good sampling has to be dense and efficient at the same time. Find a r-separated and r-covering of X. Achieved using farthest point sampling.

44 Farthest point sampling Farthest point

45 Farthest point sampling Start with some. Determine sampling radius If stop. Find the farthest point from X Add to

46 Farthest point sampling Outcome: -separated -covering of. Produces sampling with progressively increasing density. A greedy algorithm: previously added points remain in. There might be another -separated -covering containing less points. In practice used to sub-sample a densely sampled shape. Straightforward time complexity: number of points in dense sampling, number of points in. Using efficient data structures can be reduced to.

47 Sampling as representation Sampling represents a region on as a single point. Region of points on closer to than to any other Voronoi region (Dirichlet or Voronoi-Dirichlet region, Thiessen polytope or polygon, Wigner-Seitz zone, domain of action).

48 Voronoi decomposition Voronoi region Voronoi edge Voronoi vertex A point can belong to one of the following Voronoi region ( is closer to than to any other ). Voronoi edge ( is equidistant from and ). Voronoi vertex ( is equidistant from three points ).

49 Voronoi decomposition

50 Voronoi decomposition Voronoi regions are disjoint. Their closure covers the entire. Cutting along Voronoi edges produces a collection of tiles. The tiles are topological disks (are homeomorphic to a disk).

51 Voronoi tessellations in Nature

52 Delaunay tessellation Define connectivity as follows: a pair of points whose Voronoi cells are adjacent are connected The obtained connectivity graph is dual to the Voronoi diagram and is called Delaunay tesselation Boris Delaunay ( ) Voronoi regions Connectivity Delaunay tesselation

53 Delaunay tessellation For a set P of points in the (d-dimensional) Euclidean space, a Delaunay triangulation is a triangulation DT(P) such that no point in P is inside the circumhypersphere of any simplex in DT(P) It is known that there exists a unique Delaunay triangulation for P if P is a set of points in general position In the plane, the Delaunay triangulation maximizes the minimum angle

54 Delaunay tessellation This triangulation does not meet the Delaunay condition (the circumcircles contain more than three points) Flipping the common edge produces a Delaunay triangulation for the four points

55 Shape representation Cloud of points Graph Triangular mesh

56 Triangular meshes A structure of the form consisting of Vertices Edges Faces is called a triangular mesh The mesh is a purely topological object and does not contain any geometric properties The faces can be represented as an matrix of indices, where each row is a vector of the form, and

57 Example of triangular mesh Vertices Edges Coordinates Faces Topological Geometric

58 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

59 A fairy tale shape similarity problem

60 Extrinsic shape similarity

61 Extrinsic shape similarity Given two shapes and, find the degree of their incongruence. Compare and as subsets of the Euclidean space. Invariance to rigid motion: rotation, translation, (reflection): is a rotation matrix, is a translation vector

62 How to get rid of Euclidean isometries? How to remove translation and rotation ambiguity? Find some canonical placement of the shape in. Extrinsic centroid (center of mass, or center of gravity): x Set to resolve translation ambiguity. Three degrees of freedom remaining 0 X X xdx dx

63 How to get rid of Euclidean isometries? Find the direction in which the surface has maximum extent. Maximize variance of projection of onto is the covariance matrix is the first principal direction

64 How to get rid of Euclidean isometries? Project on the plane orthogonal to. Repeat the process to find second and third principal directions.

65 How to get rid of Euclidean isometries? Canonical basis span a canonical orthogonal basis for in.

66 How to get rid of Euclidean isometries? Direction maximizing = largest eigenvector of. and correspond to the second and third eigenvectors of. admits unitary diagonalization. d d d T 1 where T. U 2 T 3 Principal component analysis (PCA), or Karhunen-Loéve transform (KLT), or Hotelling transform.

67 Second order geometric moments Eigenvalues of are second-order moments of. Second-order geometric moments of : In the canonical basis, mixed moments vanish. Ratio describe eccentricity of. Magnitudes of express shape scale.

68 How to get rid of Euclidean isometries? Examples Without self-alignment With self-alignment by using PCA

69 Outline Introduction Basic concepts in geometry Discrete geometry Metric for discrete geometry Sampling Rigid shape analysis Euclidean isometries removal ICP based shape matching

70 Iterative closest point (ICP) algorithms Given two point sets and, find the best motion { } N i i 1 sr( n) t m { n } M j j 1 j { m } N i i 1, min ( ), bringing as close as possible to : d m n d sr n t m ICP i j j i srt,, (, s Rt,) ( ), j i d sr n t m is some shape-to-shape distance. { m } N i i 1 { m } N i i 1 Minimum = extrinsic dissimilarity of and. Minimizer = best alignment between and. ICP is a family of algorithms differing in The choice of the shape-to-shape distance. { n } M j j1 { n } M j j1 The choice of the numerical minimization algorithm.

71 Iterative closest point (ICP) algorithms [s,r,t] = ICP (,{ n } M ) (suppose N<M) calculate the point correspondences calculate the error: 2 While not convergent N i1 m n 2 i i N mi, nii 1 Evaluate s, R and T according to the pairs Apply s, R and T to to get Let n ' j nj End Return s, R, T { } N i i 1 m j j 1 n j Re-calculate the point correspondences 1 2 re-calculate the error: 2 N i1 m i n ' j n i m, n (closest point) N mi, nii 1 i i i N

72 Iterative closest point (ICP) algorithms [s,r,t] = ICP (,{ n } M ) (suppose N<M) calculate the point correspondences calculate the error: 2 While not convergent End N i1 m n 2 i i N mi, nii 1 Evaluate s, R and T according to the pairs Apply s, R and T to to get Let n ' j nj n j Re-calculate the point correspondences 1 2 re-calculate the error: 2 Return s, R, T { } N i i 1 m j j 1 N i1 m i n ' j n i m, n (closest point) Can be efficiently computed by using Delaunay triangulation N mi, nii 1 i i i N

73 Iterative closest point (ICP) algorithms [s,r,t] = ICP (,{ n } M ) (suppose N<M) { } N i i 1 calculate the point correspondences calculate the error: m j j 1 2 While not convergent N i1 m n 2 i i N mi, nii 1 (closest point) End Evaluate s, R and T according to the pairs Apply s, R and T to to get Let n ' j nj n j n ' j N mi, nii 1 m, n Re-calculate the point correspondences 1 2 re-calculate the error: 2 N i1 m i n i i i i N How? Return s, R, T

74 Iterative closest point (ICP) algorithms Problem definition: Given a set of point correspondence pairs N m, n, how to evaluate s, R and T to minimize i ii 1 N 2 i i i1 ( ) 2 m sr n T

75 We assume that there is a similarity transform between point N N sets and n mii 1 ii 1 Find s, R and T to minimize N N 2 2 i i i i1 i1 Let Iterative closest point (ICP) algorithms e m sr n T ( ) 2 Note: R is an orthogonal matrix. 1 1 m m n n m m m n n n N (1) N N ' ' i, i, i i, i i i1 N i1 Note that: N N ' mi, i1 i1 0 n 0 ' i

76 Then: ' ' ' ' e m sr( n ) T m m sr( n n) T m m sr( n ) sr( n) T i i i i i i i m sr( n ) T m sr( n) m sr( n ) e ' ' ' ' i i i i e0 T msr( n) (1) can be rewritten as: N N N N 2 2 N N N 2 i ( i) 2 i 2 ( i) e m sr( n ) e m sr( n ) 2 e m sr( n ) Ne 2 2 ' ' ' ' ' ' 2 i i i 0 i i 0 i i 0 i1 i1 i1 i1 msrn e m e srn Ne ' ' ' ' i1 i1 i1 N i1 Iterative closest point (ICP) algorithms ' ' 2 2 i ( i) 0 m sr n Ne Variables are separated and can be minimized separately. 2 e T m sr n is independent from 0 ' ' { mi, ni} 0 0 ( ) If we have s and R, T can be determined.

77 Then the problem simplifies to: how to minimize N 2 ' ' mi sr ni i1 Iterative closest point (ICP) algorithms ( ) 2 Consider its geometric meaning here. We revise the error item as a symmetrical one: N 2 N N N 2 1 ' ' 1 ' 2 ' ' ' 2 mi srn ( ) i m 2 i mi Rn ( i) s Rn ( i) i1 s s i1 i1 i1 1 s N N N ' 2 ' ' ' 2 mi 2 mi R( ni) s ni i1 i1 i1 P D Q Variables are separated P2DsQ s Q P s s 2( PQ D) Thus, 2

78 Iterative closest point (ICP) algorithms 2 1 P s Q P 0 s s Q N i1 N Then the problem simplifies to: how to maximize i1 N ' ' i ( i) i1 N N ' ' ' T N ' ' ' i i i i i i i1 i1 i1 D m R n m Note that: D is a real number. n ' i ' i 2 2 Determined!. D m Rn m Rn trace Rn m trace RH T N i1 ' H ni m ' i T Now we are looking for an orthogonal matrix R to maximize the trace of RH.

79 Iterative closest point (ICP) algorithms Lemma For any positive semi definite matrix C and any orthogonal matrix B: tracec tracebc Proof: T From the positive definite property of C, A, C AA where A is a non singular matrix. Let be the ith column of A. Then a i T T T i i trace BAA trace A BA a Ba According to Schwarz inequality:, i xy x y a Ba a Ba a a a B Ba a a Hence, T trace BAA T T a a trace AA that is, trace BC T T T T T T i i i i i i i i i i i i i tracec

80 Iterative closest point (ICP) algorithms Consider the SVD of H UV According to the property of SVD, Uand V are orthogonal matrices, and is a diagonal matrix with nonnegative elements. Now let X VU T N ' H ni m i1 Note that: X is orthogonal. ' i T T We have T T T XH VU UV V V which is positive semi definite. Thus, from the lemma, we know: for any orthogonal matrix B trace( XH ) trace( BXH ) trace( XH ) trace( H ) for any orthogonal matrix It s time to go back to our objective now R should be X

81 Now, s, R and T are all determined. ' ' 1 N T T i i i H n m U V T R VU 2 ' 1 2 ' 1 N i i N i i m s n ( ) T m sr n Iterative closest point (ICP) algorithms

82 ICP Matching An Example bottle1~bottle2: ac1~ac2: bottle1 bottle2 bottle1~ac1: bottle1~ac2: bottle2~ac1: bottle2~ac2: ac1 ac2

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