1. Isometry invariant similarity
Lecture 7
© Alexander & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. Invariant similarity
SIMILARITY
TRANSFORMATION 2

3. Equivalence
Equal
Congruent
Isometric

4. Equivalence
Equivalence is a binary relation on the space of shapes which
for all satisfies
Reflexivity:
Symmetry:
Transitivity:
Can be expressed as a binary function
if and only if
Quotient space is the space of equivalence classes

5. Equivalence

6. All deformations of the human shape are “the same”
Equivalence
All deformations of the human shape are “the same”

7. Similarity
Shapes are rarely truly equivalent (e.g., due to acquisition noise or since
most shapes are rigid)
We want to account for “almost equivalence” or similarity
-similar = -isometric (w.r.t. some metric)
Define a distance on the shape space quantifying the degree of
dissimilarity of shapes

8. A monkey shape is more similar to a deformation of a monkey shape…
Similarity
…than to a human shape
A monkey shape is more similar to a deformation of a monkey shape…

9. Isometry-invariant distance
Non-negative function satisfying for all
Similarity: and are isometric;
and are -isometric
(In particular, if and only if )
Symmetry:
Triangle inequality:
Corollary: is a metric on the quotient space
Given discretized shapes and
sampled with radius
Consistency to sampling:

10. Compute Hausdorff distance over all isometries in
Canonical forms distance
Minimum-distortion
embedding
Minimum-distortion
embedding
Compute Hausdorff distance over all isometries in
No fixed embedding space will give distortion-less canonical forms

11. Gromov-Hausdorff distance
Isometric
embedding
Isometric
embedding
Gromov-Hausdorff distance: include into minimization
Mikhail Gromov

12. Properties of Gromov-Hausdorff distance
Metric on the quotient space of isometries of shapes
Similarity: and are isometric;
and are -isometric
Consistent to sampling: given discretized shapes
and sampled with radius
Generalization of Hausdorff distance:
Hausdorff distance between subsets of a metric space
Gromov-Hausdorff distance between metric spaces
Gromov, 1981

13. Alternative definition I (metric coupling)
where
is the disjoint union of and
the (semi-) metric satisfies and
Mémoli, 2008

14. Alternative definition I (metric coupling)
Optimization over translates into finding the values of
A lot of constraints!
Mémoli, 2008

15. Correspondence A subset is called a correspondence between and
if for every there exists at least one such that
and similarly for every there exists such that
Particular case: given and

16. Correspondence distortion
The distortion of correspondence is defined as
In the particular case of , consider the following cases for
If the distortion is

17. Correspondence distortion (cont)
Case 1
Case 3
Case 2
Otherwise, the distortion is given by
Therefore,

18. Alternative definition II (correspondence distortion)
Proof sketch
1. Show that for any there exists with
Since , by definition of , and are subspaces of some such that
Let
By triangle inequality, for

19. Alternative definition II (correspondence distortion)
2. Show that for any
Let
It is sufficient to show that there is a (semi-)metric on the disjoint union
such that , , and
Construct the metric as follows
(in particular, for ).

20. Alternative definition II (correspondence distortion)
First,
For each
Since for ,
Second, we need to show that is a (semi-)metric on
On and , it is straightforward
We only need to show metric properties hold on

21. Alternative definition III
measures how much is distorted by when embedded into

22. Alternative definition III
measures how much is distorted by when embedded into

23. Alternative definition III
measures how far is from being the inverse of

24. Generalized MDS
A. Bronstein, M. Bronstein & R. Kimmel, 2006

25. Discrete Gromov-Hausdorff distance
Two coupled GMDS problems
Can be cast as a constrained problem
Bronstein, Bronstein & Kimmel, 2006

26. Gromov-Hausdorff distance
Numerical example
Canonical forms
(MDS based on 500 points)
Gromov-Hausdorff distance
(GMDS based on 50 points)
Bronstein, Bronstein & Kimmel, 2006

27. Extrinsic similarity using Gromov-Hausdorff distance
Congruence
Euclidean isometry
ICP distance:
GH distance with Euclidean metric:
Connection between Euclidean GH and ICP distances:
Mémoli (2008)
Mémoli, 2008

28. Connection to canonical form distance

29. Correspondence again
A different representation for correspondence using indicator functions
defines a valid correspondence if

30. Lp Gromov-Hausdorff distance
We can give an alternative formulation of the Gromov-Hausdorff distance
Can we define an Lp version of the Gromov-Hausdorff distance by relaxing the above definition?

31. Measure coupling Let be probability measures defined on and
(a metric space with measure is called a metric measure or mm space)
A measure on is a coupling of and if
for all measurable sets
The measure can be considered as a relaxed version of the indicator function or as fuzzy correspondence
Mémoli, 2007

32. Gromov-Wasserstein distance
The relaxed version of the Gromov-Hausdorff distance is given by
and is referred to as Gromov-Wasserstein distance [Memoli 2007]
Mémoli, 2007

33. Earth Mover’s distance
Let be a metric space, and measures supported on
Define the coupling of
The Wasserstein or Earth Mover’s distance (EMD) is given by
EMD as optimal mass transport:
mass transported from to
distance traveled
Mémoli, 2007

34. The analogy Hausdorff Wasserstein Gromov-Hausdorff Gromov-Wasserstein
Distance between subsets
of a metric space
Distance between subsets
of a metric measure space
Gromov-Hausdorff
Gromov-Wasserstein
Distance between metric spaces
Distance between metric measure spaces
Mémoli, 2007