1 numerical geometry of non-rigid shapes a journey to non-rigid world objects metric model of shapes...
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1Numerical geometry of non-rigid shapes A journey to non-rigid world
objects
Metric model of shapes
non-rigid
Alexander BronsteinMichael Bronstein
Numerical geometry of
2Numerical geometry of non-rigid shapes A journey to non-rigid world
Raffaello Santi, School of Athens, Vatican
3Numerical geometry of non-rigid shapes A journey to non-rigid world
Metric model
Shape
metric space
Similarity
Distance between metric
spaces and .
Invariance
isometry w.r.t.
4Numerical geometry of non-rigid shapes A journey to non-rigid world
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
-isometric
‘‘
-similar =‘‘ In which metric?
5Numerical geometry of non-rigid shapes A journey to non-rigid world
Examples of metrics
GeodesicEuclidean Diffusion
6Numerical geometry of non-rigid shapes A journey to non-rigid world
Isometry w.r.t. Euclidean metric = rigid motion
Two shapes differing by a Euclidean isometry are congruent
ROTATION TRANSLATION REFLECTION
Rigid isometry: congruence
7Numerical geometry of non-rigid shapes A journey to non-rigid world
Hausdorff distance
Distance
from to .
Distance
from to .
Hausdorff distance between subsets of a metric space
8Numerical geometry of non-rigid shapes A journey to non-rigid world
Best rigid alignment: find minimum Hausdorff distance between
and over all Euclidean transformations
Iterative closest point
9Numerical geometry of non-rigid shapes A journey to non-rigid world
Iterative closest point
Find closest point correspondence
Optimal alignment between corresponding points
Update
10Numerical geometry of non-rigid shapes A journey to non-rigid world
A fairy tale shape similarity problemA fairy tale shape similarity problem
11Numerical geometry of non-rigid shapes A journey to non-rigid world
And now, non-rigid similarity…
Non-rigid similarityRigid similarity
Part of the same metric space Two different metric spaces
SOLUTION: Find a representation of and
in a common metric space
12Numerical geometry of non-rigid shapes A journey to non-rigid world
Canonical forms
Elad & Kimmel, 2003
Non-rigid shape similarity
= Rigid similarity of canonical forms
Compute canonical formsCompare canonical forms as rigid shapes
13Numerical geometry of non-rigid shapes A journey to non-rigid world
Isometric embedding
Ideal isometric embedding
Elad & Kimmel, 2003
Embed metric space into Euclidean metric space
14Numerical geometry of non-rigid shapes A journey to non-rigid world
Mapmaker’s problem
15Numerical geometry of non-rigid shapes A journey to non-rigid world
Mapmaker’s problem
A sphere has non-zero curvature, therefore, it is
not isometric to the plane (a consequence of
Theorema egregium)
Karl Friedrich Gauss (1777-1825)
Bad news: exact canonical forms usually do
not exist (embedding error)
16Numerical geometry of non-rigid shapes A journey to non-rigid world
Minimum-distortion embedding
Elad & Kimmel, 2003
Best possible embedding with minimum distortion
17Numerical geometry of non-rigid shapes A journey to non-rigid world
Multidimensional scaling
BBK, I. Yavneh, 2005G. Rosman, BBK, 2007
Different distortion criteria
Non-linear non-convex optimization problem
Efficient numerical methods (multiscale, multigrid, vector extrapolation)
Heuristics to prevent local convergence