Point Cloud Processing
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- Lenard Owens
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1 Point Cloud Processing Has anyone seen the toothpaste? Given a point cloud: how do you detect and localize objects? how do you map terrain?
2 What is a point cloud? Point cloud: a set of points in 3-D space just a set of 3-d points Mesh: each point is a vertex of a triangulated face a set of vertices AND connectivity information Point cloud Mesh Images from Course INF 555 slides, Ecole Polytechnique, Paris
3 What is a point cloud? Point cloud: a set of points in 3-D space just a set of 3-d points Mesh: each point is a vertex of a triangulated face a set of vertices AND connectivity information Many depth sensors produce point clouds natively Point cloud But, a mesh contains a lot more information Mesh Images from Course INF 555 slides, Ecole Polytechnique, Paris
4 Time of flight sensors Hokuyo UTM-30LX-EW Scanning Laser Range Finder
5 Time of flight sensors Slide from Course INF 555 slides, Ecole Polytechnique, Paris
6 Time of flight sensors
7 Structured light sensors
8 Slide: John MacCormick, Dickinson University
9 Slide: John MacCormick, Dickinson University
10 Slide: John MacCormick, Dickinson University
11 Slide: John MacCormick, Dickinson University
12 Slide: John MacCormick, Dickinson University
13 Slide: John MacCormick, Dickinson University
14 Calculating surface normals Point cloud Point cloud w/ surface normals (normals are subsampled)
15 Calculating surface normals Question: How do we calculate the surface normal given only points? Answer: 1. Calculate the sample covariance matrix of the points within a local neighborhood of the surface normal 2. Take Eigenvalues/Eigenvectors of that covariance matrix
16 Calculating surface normals Let C denote the set of points in the point cloud Suppose we want to calculate the surface normal at Let denote the r-ball about x is the set of points in the cloud within r of x
17 Calculating surface normals Calculate the sample covariance matrix of the points in
18 Calculating surface normals Length = Eigenvalues of Length = Principle axes of ellipse point in directions of corresponding eigenvectors
19 Calculating surface normals So: surface normal is in the direction of the Eigenvector corresponding to the smallest Eigenvalue of
20 Calculating surface normals: Summary 1. calculate points within r-ball about x: 2. calculate covariance matrix: 3. calculate Eigenvectors: and Eigenvalues: (\lambda_3 is smallest) 4. v_3 is parallel or antiparallel to surface normal
21 Calculating surface normals Important note: the points alone do not tell us the sign of the surface normal
22 Calculating surface normals Important note: the points alone do not tell us the sign of the surface normal
23 Calculating surface normals How large a point neighborhood to use when calculating? Because points can be uneven, don't use k-nearest neighbor. it's important to select a radius r and stick w/ it. which what value of r to use?
24 Calculating surface normals Images from Course INF 555 slides, Ecole Polytechnique, Paris
25 Calculating surface normals Images from Course INF 555 slides, Ecole Polytechnique, Paris
26 Outlier removal Similar approach as in normal estimation: 1. calculate local covariance matrix 2. estimate Eigenvectors/Eigenvalues 3. use that information somehow... Images from Course INF 555 slides, Ecole Polytechnique, Paris
27 Outlier removal If points lie on a line, then is small If points are uniformly random, then Outlier removal: delete all points for which is close to 1 is above a threshold Images from Course INF 555 slides, Ecole Polytechnique, Paris
28 Point cloud registration: ICP Find an affine transformation that aligns two partially overlapping point clouds Images from Course INF 555 slides, Ecole Polytechnique, Paris
29 ICP Problem Statement This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
30 ICP: key idea This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
31 Step 1: center the two point clouds This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
32 Step 2: use SVD to get min t and R This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
33 Step 2: use SVD to get min t and R This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
34 ICP data association problem This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
35 ICP Algorithm Input: two point sets, X and P Output: translation t and rotation R that best aligns pt sets 1. Start with a good alignment 2. Repeat until t and R are small: 3. for every point in X, find its closest neighbor in P 4. find min t and R for that correspondence assignment 5. translate and rotate P by t and R 6. Figure out net translation and rotation, t and R Converges if the point sets are initially well aligned Besl and McKay, 1992
36 ICP example This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
37 ICP Variants This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
38 Selecting points to align This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
39 Normal-space sampling This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
40 Comparison: normal space sampling vs random This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
41 Feature based sampling This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
42 ICP: data association This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
43 ICP: data association This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
44 Closest point matching This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
45 Normal shooting This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
46 Point-to-plane distances This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
47 Closest compatible point This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
48 ICP: summary This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg
49 Another approach to alignment: RANSAC This slide from: Kavita Bala, Cornell U.
50 RANSAC This slide from: Kavita Bala, Cornell U.
51 How does regression work here?
52 Image alignment problem This slide from: Kavita Bala, Cornell U.
53 Outliers This slide from: Kavita Bala, Cornell U.
54 RANSAC This slide from: Kavita Bala, Cornell U.
55 RANSAC key idea This slide from: Kavita Bala, Cornell U.
56 Counting inliers This slide from: Kavita Bala, Cornell U.
57 Counting inliers This slide from: Kavita Bala, Cornell U.
58 How do we find the best line? This slide from: Kavita Bala, Cornell U.
59 RANSAC This slide from: Kavita Bala, Cornell U.
60 This slide from: Kavita Bala, Cornell U.
61 This slide from: Kavita Bala, Cornell U.
62 This slide from: Kavita Bala, Cornell U.
63 Using RANSAC to Fit a Sphere
64 Using RANSAC to Fit a Sphere
65 Using RANSAC to Fit a Sphere Radius? Center?
66 Using RANSAC to Fit a Sphere How generate candidate spheres? How score spheres?
67 Using RANSAC to Fit a Sphere How generate candidate spheres? 1. sample a point How score spheres?
68 Using RANSAC to Fit a Sphere How generate candidate spheres? 1. sample a point 2. estimate surface normal How score spheres?
69 Using RANSAC to Fit a Sphere radius How generate candidate spheres? 1. sample a point 2. estimate surface normal 3. sample radius How score spheres?
70 Using RANSAC to Fit a Sphere radius How generate candidate spheres? 1. sample a point 2. estimate surface normal 3. sample radius 4. estimate center to be radius distance from sampled point along surface normal How score spheres?
71 Using RANSAC to Fit a Sphere radius How generate candidate spheres? 1. sample a point 2. estimate surface normal 3. sample radius 4. estimate center to be radius distance from sampled point along surface normal How score spheres? 1. count num pts within epsilon of candidate sphere surface
72 Using RANSAC to Fit a Cylinder How generate candidate cylinders?
73 Using RANSAC to Fit a Cylinder How generate candidate cylinders? 1. sample two pts
74 Using RANSAC to Fit a Cylinder How generate candidate cylinders? 1. sample two pts 2. estimate surface normal at both pts
75 Using RANSAC to Fit a Cylinder How generate candidate cylinders? 1. sample two pts 2. estimate surface normal at both pts 3. get sample axis by taking cross product between two normals
76 Using RANSAC to Fit a Cylinder How generate candidate cylinders? 1. sample two pts 2. estimate surface normal at both pts 3. get sample axis by taking cross product between two normals 4. project points onto plane orthogonal to axis 5. fit a circle using a method similar to what we did for the sphere.
77 Using RANSAC to Fit a Cylinder 3xn matrix of pts projected onto plane How generate candidate cylinders? 1. sample two pts 2. estimate surface normal at both pts 3. get sample axis by taking cross product between two normals 4. project points onto plane orthogonal to axis 5. fit a circle using a method similar to what we did for the sphere. 3x1 unit vector in direction of axis 3xn matrix of pts in 3d space
78 RANSAC: the parameters This slide from: Kavita Bala, Cornell U.
79 RANSAC: how many rounds? This slide from: Kavita Bala, Cornell U.
80 RANSAC: how many parameters to sample? This slide from: Kavita Bala, Cornell U.
81 RANSAC Summary This slide from: Kavita Bala, Cornell U.
82 Hough transform This slide from: Kavita Bala, Cornell U.
83 Hough transform This slide from: Kavita Bala, Cornell U.
84 Hough transform This slide from: Kavita Bala, Cornell U.
85 Hough transform This slide from: Kavita Bala, Cornell U.
86 Hough transorm This slide from: Kavita Bala, Cornell U.
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