C08 - 3D-3D Geometry

• mroe robust to noise

• -> as we use more points and work with an overdetermined case…

1:

• N points in both first and second coordinate system

• -> compute R,T between both…

2:

• N+N points in a single coordinate system

• -> i.e. point set moves in single coord

• -> compute R,T to “compensate” movement of points

1. 3D-3D correspondance establishment

2. transformation establishment

• SE(3) -> only rotatoin and translation; same scale

• Sim(3) -> Rotation, translation and scale

• given two point clouds

• and 3D-3D correspondances

• => estimate the motion

• two point sets in 3D (different frames)

  • can be obtained by e.g. triangulation or stereo vision…

  • can also be virtual (i.e. EPnP)

• three 3D-3D correspondances

• =>DLT

• provided two sets of points (must not necesarrily be the same amount)

• goal: find the optimal translation and rotation

  • minimizing the sum of squared error

• where xi and pi are initially unknown but sought point correspondances

1. non-iterative method

   • correct correspondences are known

   • R, t can be directly calculated closed form (DLT)

2. iterative method

   • correct correspondences not known

   • generally impossible to determine optimal R,t in one step!

2D-2D:

• motion estimation from feature correspondences

  • -> i.e. simlar but we have 2D feature corresp. and not 3D point correspondences

• minimal solution: 5 feature correspondences

• popular algos:

  • 5-point

  • 8-point

3D-2D:

• motion estimation from 3D-2D feature correspondances

  • -> i.e. one set in 2D features, one set in 3D points (features)

• popular algos:

  • EPNP (at least 6 point corresp.)

  • DLT (at least 6 point corresp.)

  • P3P (minimal case with 3 point corresp.)

3D-3D:

• both correspondences in 3D space

• minimal solution: 3 point correspondences

SE(3):

• non-iterative

• iterative closest point (ICT)

Sim(3):

• horns method

• umeyamas method

• compute center of mass

• point set normalization

• SVD

• decompose into rotation and translation

-> i.e. DLT…

assume cardinality is the same!

compute matrix W by element wise multiplication of the corresponding points (non-iterative -> we assume we know the correspondances…)

-> assumption same cardinality…

decompose it using SVD

-> where the sigmas are the singluar values

REMEMBER!

• iteratively align two point sets

• -> converges if cprresponding points are “close enough”

• treat pair of points with smallest distance as temporary 3D-3D correspondence

• -> given this, we can compute trhansforamtion efficiently (non-iterative method…) with SVD

=> e.g. ICP is an iteration of non-iteration method where the correspondances are the at the current iteration closest points…

1. determine correspondences based on currently smallest distance

2. compute R,t via SVD (non-iterative method)

3. apply R and t to the points of the set to be registered

4. compute error E(R,t)

-> t is distance of projected center of mass…

1. if error decreased and error > threshold

   1. repeat these steps

2. otherwise: stop and output final alignment

• weighting the correspondences (mainly for high accuracy)

• rejectin outlier pairs (mainly for higher robustness)