CENG 789 – Digital Geometry Processing
10- Least-Squares Solutions
Assoc. Prof. Yusuf Sahillioğlu
Computer Eng. Dept,
, Turkey
Outline
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 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
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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.
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Other Fitting Approaches
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 RANSAC: Randomized Sample Consensus
 Robust to outliers 
 Not good for models represented w/ many parameters 
 Here least-squares approach fails due to noise.
Other Fitting Approaches
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 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)
Other Fitting Approaches
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 RANSAC: Randomized Sample Consensus
 Robust to outliers 
 Not good for models represented w/ many parameters 
 Threshold for inliers 
 Sufficient inliers at right so stop.
Other Fitting Approaches
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 RANSAC: Randomized Sample Consensus
 Robust to outliers 
 Not good for models represented w/ many parameters 
 Popular in dominant plane detection in 3D scenes.
Other Fitting Approaches
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 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.
Other Fitting Approaches
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 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):
Other Fitting Approaches
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 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:
Other Fitting Approaches
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 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:
Potential Project Topics
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 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).