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Point Cloud Processing Has anyone seen the toothpaste? Given a - - PowerPoint PPT Presentation

Nov 15, 2022 282 likes •1.17k views

Point Cloud Processing Has anyone seen the toothpaste? Given a point cloud: how do you detect and localize objects? how do you map terrain? What is a point cloud? Point cloud: a set of points in 3-D space just a set of 3-d points

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Given a point cloud: – how do you detect and localize objects? – how do you map terrain?

Point Cloud Processing

Has anyone seen the toothpaste?

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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

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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 But, a mesh contains a lot more information Many depth sensors produce point clouds natively

Images from Course INF 555 slides, Ecole Polytechnique, Paris

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Time of flight sensors

Hokuyo UTM-30LX-EW Scanning Laser Range Finder

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Time of flight sensors

Slide from Course INF 555 slides, Ecole Polytechnique, Paris

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Time of flight sensors

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Structured light sensors

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Slide: John MacCormick, Dickinson University

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Slide: John MacCormick, Dickinson University

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Slide: John MacCormick, Dickinson University

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Slide: John MacCormick, Dickinson University

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Slide: John MacCormick, Dickinson University

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Slide: John MacCormick, Dickinson University

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Calculating surface normals

Point cloud Point cloud w/ surface normals (normals are subsampled)

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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
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Calculating surface normals

Let denote the r-ball about x Suppose we want to calculate the surface normal at is the set of points in the cloud within r of x Let C denote the set of points in the point cloud

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Calculating surface normals

Calculate the sample covariance matrix of the points in

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Calculating surface normals

Eigenvalues of Length = Principle axes of ellipse point in directions of corresponding eigenvectors Length =

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Calculating surface normals

So: surface normal is in the direction of the Eigenvector corresponding to the smallest Eigenvalue of

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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
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Calculating surface normals

Important note: the points alone do not tell us the sign

  • f the surface normal
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Calculating surface normals

Important note: the points alone do not tell us the sign

  • f the surface normal
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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?

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Calculating surface normals

Images from Course INF 555 slides, Ecole Polytechnique, Paris

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Calculating surface normals

Images from Course INF 555 slides, Ecole Polytechnique, Paris

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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

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Outlier removal

If points lie on a line, then is small If points are uniformly random, then is close to 1 Outlier removal: delete all points for which is above a threshold

Images from Course INF 555 slides, Ecole Polytechnique, Paris

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Point cloud registration: ICP

Images from Course INF 555 slides, Ecole Polytechnique, Paris

Find an affine transformation that aligns two partially overlapping point clouds

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ICP Problem Statement

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP: key idea

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Step 1: center the two point clouds

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Step 2: use SVD to get min t and R

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Step 2: use SVD to get min t and R

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP data association problem

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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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

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ICP example

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP Variants

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Selecting points to align

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Normal-space sampling

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Comparison: normal space sampling vs random

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Feature based sampling

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP: data association

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP: data association

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Closest point matching

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Normal shooting

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Point-to-plane distances

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Closest compatible point

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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ICP: summary

This slide from: Burgard, Stachniss, Bennewitz, Arras, U. Freiburg

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Another approach to alignment: RANSAC

This slide from: Kavita Bala, Cornell U.

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RANSAC

This slide from: Kavita Bala, Cornell U.

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How does regression work here?

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Image alignment problem

This slide from: Kavita Bala, Cornell U.

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Outliers

This slide from: Kavita Bala, Cornell U.

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RANSAC

This slide from: Kavita Bala, Cornell U.

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RANSAC key idea

This slide from: Kavita Bala, Cornell U.

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Counting inliers

This slide from: Kavita Bala, Cornell U.

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Counting inliers

This slide from: Kavita Bala, Cornell U.

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How do we find the best line?

This slide from: Kavita Bala, Cornell U.

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RANSAC

This slide from: Kavita Bala, Cornell U.

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This slide from: Kavita Bala, Cornell U.

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This slide from: Kavita Bala, Cornell U.

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This slide from: Kavita Bala, Cornell U.

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Using RANSAC to Fit a Sphere

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Using RANSAC to Fit a Sphere

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Using RANSAC to Fit a Sphere

Center? Radius?

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Using RANSAC to Fit a Sphere

How generate candidate spheres? How score spheres?

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Using RANSAC to Fit a Sphere

How generate candidate spheres?

  • 1. sample a point

How score spheres?

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Using RANSAC to Fit a Sphere

How generate candidate spheres?

  • 1. sample a point
  • 2. estimate surface normal

How score spheres?

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Using RANSAC to Fit a Sphere

How generate candidate spheres?

  • 1. sample a point
  • 2. estimate surface normal
  • 3. sample radius

radius How score spheres?

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Using RANSAC to Fit a Sphere

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 radius How score spheres?

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Using RANSAC to Fit a Sphere

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 radius How score spheres?

  • 1. count num pts within epsilon of

candidate sphere surface

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Using RANSAC to Fit a Cylinder

How generate candidate cylinders?

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Using RANSAC to Fit a Cylinder

How generate candidate cylinders?

  • 1. sample two pts
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Using RANSAC to Fit a Cylinder

How generate candidate cylinders?

  • 1. sample two pts
  • 2. estimate surface normal at both pts
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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

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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.

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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. 3x1 unit vector in direction of axis 3xn matrix of pts in 3d space 3xn matrix of pts projected onto plane

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RANSAC: the parameters

This slide from: Kavita Bala, Cornell U.

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RANSAC: how many rounds?

This slide from: Kavita Bala, Cornell U.

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RANSAC: how many parameters to sample?

This slide from: Kavita Bala, Cornell U.

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RANSAC Summary

This slide from: Kavita Bala, Cornell U.

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Hough transform

This slide from: Kavita Bala, Cornell U.

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Hough transform

This slide from: Kavita Bala, Cornell U.

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Hough transform

This slide from: Kavita Bala, Cornell U.

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Hough transform

This slide from: Kavita Bala, Cornell U.

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Hough transorm

This slide from: Kavita Bala, Cornell U.