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Digital Geometry Processing: Least-Squares Solutions Overview

This class covers geometric problems, least-squares solutions, RANSAC algorithm, Hough Transform, and practical computational tools for image processing. Includes a new scale-adaptive ICP version and fitting approaches for modeling skull deformities.

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Digital Geometry Processing: Least-Squares Solutions Overview

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  1. CENG 789 – Digital Geometry Processing11- 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

  3. 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.

  4. 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.

  5. 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)

  6. 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.

  7. 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.

  8. 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.

  9. 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):

  10. 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:

  11. 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:

  12. 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).

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