Geometric Algebra Computing Analysis of point clouds Dr. Dietmar Hildenbrand

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1 Geometric Algebra Computing Analysis of point clouds Dr. Dietmar Hildenbrand Technische Universität Darmstadt

2 Literature Book Foundations of Geometric Algebra Computing, Dietmar Hildenbrand Computers & Graphics 2005 : volume 29, no. 5, october, 2005 : "Geometric Computing in Computer Graphics using Conformal Geometric Algebra" by Dietmar Hildenbrand GRAPP 2008, Madeira : "ANALYSIS OF POINT CLOUDS Using Conformal Geometric Algebra" by Dietmar Hildenbrand and Eckhard Hitzer. VISAPP 2010: ESTIMATION OF CURVATURES IN POINT SETS BASED ON GEOMETRIC ALGEBRA by H. Seibert, D. Hildenbrand et. al Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 2

3 Inner Product Calculations in 5D conformal GA Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 3

4 The geometric product of 2 basis vectors (revisited) geometric algebra G p, with n = p + q q define e i e j = 1 1 eij = e i e j = e j e i for for for i = i = i j {1,..., p} j { p + 1,..., j n} Note : Conformal Geometric Algebra = G 4,1 : Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 4

5 The two additional base vectors Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 5

6 The two additional base vectors are null vectors Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 6

7 The inner product between conformal vectors Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 7

8 The distance between points Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 8

9 Distance between point and plane Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 9

10 Point inside or outside of a sphere? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 10

11 Distance measure: Inner product of point and sphere Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 11

12 Analysis of point clouds normals curvatures Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 12

13 Analysis of point clouds normals curvatures What are the most interesting local fittings of geometric objects? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 13

14 Fitting of geometric objects into point clouds Plane fitting Normal vector Sphere fitting curvature Note.: osculating circle in tangent direction Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 14

15 Curvature =0 at point p i Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 15

16 Curvature >0 at point p i Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 16

17 Infinite curvature at point p i Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 17

18 Overview Conventional fitting of spheres Fitting of spheres in GA The role of infinity Planes as a limit of spheres Fitting of spheres or planes in GA Fitting of osculating circles in point clouds Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 18

19 Conventional fitting of spheres [Eberly, 2007] Note: sphere isn t one algebraic expression Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 19

20 Conventional fitting of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 20

21 Conventional fitting of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 21

22 Conventional fitting of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 22

23 Conventional fitting of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 23

24 Benefits of geometric algebra Easy computations with algebraic objects describing spheres, planes and circles Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 24

25 Fitting a sphere to 3D points Distance measure for the fitting? point P sphere S Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 25

26 Inner product of point and sphere revisited Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 26

27 Distance measure: Inner product of point and sphere Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 27

28 Fitting a sphere to 3D points [Dissertation Hildenbrand ] Correction: r^ Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 28

29 Fitting a sphere to 3D points Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 29

30 Fitting a sphere to 3D points Correction: Missing sum sign Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 30

31 Fitting a sphere to 3D points Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 31

32 Fitting of sphere or plane into a point cloud In CGA : 5D vector with shortest distance measure to the points? Planes and spheres are vectors Inner product as a distance measure Least squares approach Result: eigen vectors of 5x5 matrix Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 32

33 Plane as a specific sphere sphere plane Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 33

34 Plane as a limit of spheres What happens with infinitely increasing radius? at first: What is the role of infinity? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 34

35 Origin sphere with infinite radius? Exercise? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 35

36 Point at infinity? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 36

37 Point at infinity? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 37

38 Plane at infinity? Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 38

39 Plane as a limit of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 39

40 Plane as a limit of spheres Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 40

41 Vectors in GA sphere plane Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 41

42 Fitting of sphere or plane into a point cloud Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 42

43 Fitting of sphere or plane into a point cloud Least squares approach with constraint s =1 (Lagrange) in bilinear form with Introduce L Necessary condition Eigen vector of B with smallest eigen value Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 43 [ Matrix Analysis, Horn/Johnson]

44 Results Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 44

45 Results Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 45

46 Curvature estimation for point clouds 2D: Osculating circle 3D -> curvature = inverse radius Same for the osculating circle in tangent direction Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 46

47 Curvature estimation for point clouds Estimate osculating circle locally in point cloud depending on different tangent directions In more detail: Estimate osculating circle or line locally in point cloud depending on different tangent directions Includes vanishing curvature Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 47

48 Example hyperbolic point Curvatures with different signs Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 48

49 Sphere example Curvatures with same signs Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 49

50 Algorithm for circle/line fit Estimate normal vector n locally at x Determine points P i in desired tangent direction Estimate sphere in the points P i with center point in normal direction -> the radius of the sphere describes the curvature in the desired direction Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 50

51 Curvature in one tangent direction In the estimation radius Result is plane/line (no curvature) Otherwise: osculating circle Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 51

52 Curvature = circle fit in all directions Determine curvature in eight directions Approximate curvatures Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 52

53 Future work Linear equation instead of Eigen vector determination Bachelor thesis Roman Getto: Verbesserte Hauptkrümmungsbestimmung in Punktwolken durch optimiertes Schmiegekreisfitting auf Grundlage der Geometrischen Algebra Least Squares completely in GA Recognition of geometric objects (cylinder, torus etc.) Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 53

54 Thanks for your attention Technische Universität Darmstadt Computer Science Department Dietmar Hildenbrand 54

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