1. CENG 789 – Digital Geometry Processing 11- Least-Squares Solutions
Assoc. Prof. Yusuf Sahillioğlu
Computer Eng. Dept, , Turkey

2. Outline In this class we will
First warm up with some geometric problems (distances, intersections, etc.)
Then formulate and solve some least-squares problems (fitting, transformations, etc.)
This book is used in the preparation of some of the following slides
A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing, Cohen-Or et al., 2015

7. This new scale-adaptive
version of ICP has been
introduced in this paper:
Skuller: A Volumetric Shape
Registration Algorithm for
Modeling Skull Deformities,
Sahillioglu and Kavan,
Medical Image Analysis,
2015.

13. Other Fitting Approaches
RANSAC: Randomized Sample Consensus
Robust to outliers 
Not good for models represented w/ many parameters 
Here least-squares approach fails due to noise.

14. Other Fitting Approaches
RANSAC: Randomized Sample Consensus
Robust to outliers 
Not good for models represented w/ many parameters 
Fit model to d (=2 for line-fitting) random pnts, check # inliers (low)

15. Other Fitting Approaches
RANSAC: Randomized Sample Consensus
Robust to outliers 
Not good for models represented w/ many parameters 
Threshold for inliers 
Sufficient inliers at right so stop.

16. Other Fitting Approaches
RANSAC: Randomized Sample Consensus
Robust to outliers 
Not good for models represented w/ many parameters 
Popular in dominant plane detection in 3D scenes.

17. Other Fitting Approaches
Hough Transform: each sample votes for all models (line, plane, ..) it supports. Vote is cast in the space of model parameters. Look for the model parameters w/ many votes.
Robust to outliers 
Not good for models represented w/ many parameters 
Possible optimization for point clouds: at each point, vote only for planes that are roughly aligned with the estimated local normal.

18. Other Fitting Approaches
Hough Transform: each sample votes for all models (line, plane, ..) it supports. Look for the model parameters w/ many votes.
Cool demo by Wikipedia (parameters: angle & distance from origin):

19. Other Fitting Approaches
Hough Transform: each sample votes for all models (line, plane, ..) it supports. Look for the model parameters w/ many votes.
Cool demonstration by S. Chaudhuri:

20. Other Fitting Approaches
Hough Transform: each sample votes for all models (line, plane, ..) it supports. Look for the model parameters w/ many votes.
Cool demonstration by S. Chaudhuri:

21. Potential Project Topics
Fitting a surface to a 3D point cloud in the Least-Squares sense (slide 12).
Implementing the Least-Squares Meshes paper or any other paper citing/cited-by by this paper (google scholar to see the citing papers).