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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