Improving Landmark Positions for Evolutionary Morphing Dan Alcantara Nina Amenta

Outline What is evolutionary morphing? Blending process Improving the results Problems encountered & future directions

What is evolutionary morphing? Method of visualizing an evolutionary tree. Relies on shape analysis theory from Geometric Morphometrics.

Theory basics Is there an optimal position for the points representing curves?

Overview of the morphing process 1) Important points on the models are hand-marked as landmarks. Curves are approximated by semi-landmarks. 2) Models are aligned so that corresponding landmarks are close to each other using a Generalized Procrustes Alignment. 3) A thin-plate spline warps the models so that corresponding landmarks lie on top of each other. Explain how the alignment doesn’t distort anything. Target landmarks come from the weights. 4) The models are blended together using weights calculated from the tree.

Associated metrics Generalized Procrustes Alignment minimizes squared distances between corresponding landmarks. Thin-plate spline minimizes distortion created when warping from one model to another.

Distortion created by the thin-plate spline Bending energy increases as the plane gets more distorted.

Bookstein’s minimization method Find all of the tangent lines at the semi-landmarks. 2) Slide semi-landmarks along their tangent lines to minimize the bending energy. 3) Reproject the landmarks back onto their respective curves. 4) Re-align using the new landmark points and repeat the method until convergence.

Observations about semi-landmark sliding Calculated minimums don’t lie on the skull. Bending energy may increase once reprojected. Semi-landmarks tend to spread out evenly. Actual minimum; not on skull Reprojection location on skull

Sliding results

Future plans Completely extend the method to 3D features. Utilize the metric from the Generalized Procrustes Alignment. May be “more correct” according to some morphologists.

References Fred L. Bookstein. Landmark Methods for Forms Without Landmarks: Localizing Group Differences in Outline Shape. Proceedings of the Workshop on Mathematical Methods in Biomedical Image Analysis, June 1996, pp 279-289. W.D.K. Green. The thin-plate spline and images with curving features. Proceedings in Image Fusion and Shape Variability Techniques, pp 79-87. David F. Wiley, et al. Evolutionary Morphing. To appear in IEEE Visualization 2005.

Acknowledgements Nina Amenta for letting me work with her the past year. Lab mates for helping me with various problems I’ve come across. Stephen Frost for providing more insight into the sliding process. AGEP program