1. Numerical geometry of objects
non-rigid
objects
Metric model of shapes
Alexander Bronstein
Michael Bronstein 1

2. Raffaello Santi, School of Athens, Vatican

3. Distance between metric spaces and .
Metric model 
Shape
Similarity
Invariance
metric space
Distance between metric spaces and
isometry w.r.t.

4. ‘ ‘ -similar = -isometric In which metric? Isometry
Two metric spaces and are isometric if there exists a bijective distance preserving map such that
Two metric spaces and are -isometric if there exists a map which is
distance preserving
surjective ‘ ‘
-similar =
-isometric
In which metric?

5. Examples of metrics
Euclidean
Geodesic
Diffusion

6. Rigid isometry: congruence
Isometry w.r.t. Euclidean metric = rigid motion
ROTATION
TRANSLATION
REFLECTION
Two shapes differing by a Euclidean isometry are congruent 6

7. Hausdorff distance between subsets of a metric space
from to .
Distance
from to .

8. Iterative closest point
Best rigid alignment: find minimum Hausdorff distance between and over all Euclidean transformations

9. Iterative closest point
Find closest point correspondence
Optimal alignment between corresponding points
Update

10. A fairy tale shape similarity problem

11. And now, non-rigid similarity…
Part of the same metric space
Two different metric spaces
SOLUTION: Find a representation of and
in a common metric space

12. ? Canonical forms Non-rigid shape similarity
Compare canonical forms as rigid shapes
Compute canonical forms
Non-rigid shape similarity
= Rigid similarity of canonical forms
Elad & Kimmel, 2003

13. Ideal isometric embedding
Embed metric space into Euclidean metric space
Ideal isometric embedding
Elad & Kimmel, 2003

14. Mapmaker’s problem ?

15. Mapmaker’s problem
A sphere has non-zero curvature, therefore, it is not isometric to the plane (a consequence of Theorema egregium)
Bad news: exact canonical forms usually do not exist (embedding error)
Karl Friedrich Gauss
( )

16. Best possible embedding with minimum distortion
Minimum-distortion embedding
Best possible embedding with minimum distortion
Elad & Kimmel, 2003

17. Multidimensional scaling
Different distortion criteria
Non-linear non-convex optimization problem
Efficient numerical methods (multiscale, multigrid, vector extrapolation)
Heuristics to prevent local convergence
BBK, I. Yavneh, 2005
G. Rosman, BBK, 2007