1 numerical geometry of non-rigid shapes invariant shape similarity invariant shape similarity ©...
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1Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Invariant shape similarity
© Alexander & Michael Bronstein, 2006-2009© Michael Bronstein, 2010tosca.cs.technion.ac.il/book
048921 Advanced topics in visionProcessing and Analysis of Geometric Shapes
EE Technion, Spring 2010
2Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Invariant similarity
SIMILARITY
TRANSFORMATION
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Equivalence
Equal
Congruent Isometric
4Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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
5Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Equivalence
6Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Equivalence
All deformations of the
human shape are “the same”
7Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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
8Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Similarity
A monkey shape is
more similar to a
deformation of a
monkey shape…
…than to a
human shape
9Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Isometry-invariant distance
Non-negative function satisfying for all
Corollary: is a metric on the quotient space
Similarity: and are -isometric;
and are -isometric
(In particular, if and only if )
Symmetry:
Triangle inequality:
Given discretized shapes and
sampled with radius
Consistency to sampling:
10Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Canonical forms distance
Compute Hausdorff distance over all isometries in
Minimum-distortion
embedding
Minimum-distortion
embedding
No fixed embedding space will give distortion-less canonical forms
11Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Gromov-Hausdorff distance
Isometric
embedding
Isometric
embedding
Mikhail Gromov
Gromov-Hausdorff distance: include
into minimization
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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 spacesGromov, 1981
13Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Alternative definition I (metric coupling)
is the disjoint union of and
the (semi-) metric satisfies and
where
Mémoli, 2008
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Optimization over translates into finding the values of
Mémoli, 2008
A lot of constraints!
Alternative definition I (metric coupling)
15Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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
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Correspondence distortion
The distortion of correspondence is defined as
17Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Alternative definition II (correspondence distortion)
1. Show that for any there exists with
Proof sketch
Since , by definition of , and are subspaces of
some such that
Let
By triangle inequality, for
18Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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 ).
19Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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
20Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Alternative definition III
measures how much is distorted by when embedded into
21Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
measures how much is distorted by when embedded into
Alternative definition III
22Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
measures how far is from being the inverse of
Alternative definition III
23Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Generalized MDS
A. Bronstein, M. Bronstein & R. Kimmel, 2006
24Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Difficulties
How to represent points on ?
Global parametrization is not always available.
Some local representation is required in general case.
No more closed-form expression for .
Metric needs to be approximated.
Minimization algorithm.
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Local representation
is sampled at and represented as a
triangular mesh .
Any point falls into one of the triangles .
Within the triangle, it can be represented as convex combination
of triangle vertices ,
Barycentric coordinates .
We will need to handle discrete indices in minimization algorithm.
26Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Geodesic distances
Distance terms can be precomputed, since are
fixed.
How to compute distance terms ?
No more closed-form expression.
Cannot be precomputed, since are minimization variables.
can fall anywhere on the mesh.
Precompute for all .
Approximate
for any .
27Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Geodesic distance approximation
Approximation from .
First order accurate:
Consistent with data:
Symmetric:
Smoothness: is and a closed-form expression
for its derivatives is available to minimization algorithm.
Might be only at some points or along some lines.
Efficiently computed: constant complexity independent of .
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Geodesic distance approximation
Compute for .
falls into triangle and is represented as
Particular case:
Hence, we can precompute distances
How to compute from ?
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Geodesic distance approximation
We have already encountered this problem in fast marching.
Wavefront arrives at triangle vertex at time .
When does it arrive to ?
Adopt planar wavefront model.
Distance map is linear in the triangle (hence, linear in )
Solve for coefficients and obtain a linear interpolant
30Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Geodesic distance approximation
General case: falls into triangle and is
represented
as
Apply previous steps in triangle to obtain
Apply once again in triangle to obtain
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A four-step dance
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Minimization algorithm
How to minimize the generalized stress?
Particular case: L2 stress
Fix all and all except for some .
Stress as a function of only becomes quadratic
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Quadratic stress
Quadratic stress
is positive semi-definite.
is convex in (but not necessarily in
together).
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Quadratic stress
Closed-form solution for minimizer of
Problem: solution might be outside the
triangle.
Solution: find constrained minimizer
Closed-form solution still exists.
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Minimization algorithm
Initialize
For each
Fix and compute gradient
Select corresponding to maximum .
Compute minimizer
If constraints are active
translate to adjacent triangle.
Iterate until convergence…
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How to move to adjacent triangles?
Three cases
All : inside triangle.
: on edge opposite to .
: on vertex .
inside on edge on vertex
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Point on edge
on edge opposite to .
If edge is not shared by any other triangle
we are on the boundary – no translation.
Otherwise, express the point as in triangle .
contains same values as .
May be permuted due to different vertex
ordering in .
Complication: is not on the edge.
Evaluate gradient in .
If points inside triangle, update to .
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Point on vertex
on vertex .
For each triangle sharing vertex
Express point as in .
Evaluate gradient in .
Reject triangles with pointing
outside.
Select triangle with maximum
.
Update to .
39Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
MDS vs GMDS
Stress
Generalized MDSMDS
Generalized stress
Analytic expression for
Nonconvex problem
Variables: Euclidean
coordinates
of the points
must be interpolated
Nonconvex problem
Variables: points on in
barycentric coordinates
40Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Multiresolution
Stress is non convex – many small local minima.
Straightforward minimization gives poor results.
How to initialize GMDS?
Multiresolution:
Create a hierarchy of grids in ,
Each grid comprises
Sampling:
Geodesic distance matrix:
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Multiresolution
Initialize at the coarsest resolution in .
For
Starting at initialization , solve the GMDS problem
Interpolate solution to next resolution level
Return .
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GMDS
Interpolation
GMDS
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Multiresolution encore
So far, we created a hierarchy of embedded spaces .
One step further: create a hierarchy of embedding spaces
.
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Labeling problem
Torresani, Kolmogorov, Rother 2008Wang, B 2010
Build a graph with vertices and edges
Label each vertex
Minimum distortion correspondence = graph labeling problem
Efficient solvers with good global convergence properties
Complexity:
Hierarchical solution complexity can be lowered to
45Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
MATLAB® intermezzoGMDS
46Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Discrete Gromov-Hausdorff distance
Two coupled GMDS problems
Can be cast as a constrained problem
Bronstein, Bronstein & Kimmel, 2006
47Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Numerical example
Canonical forms (MDS based on 500 points)
Gromov-Hausdorff distance(GMDS based on 50 points)
Bronstein, Bronstein & Kimmel, 2006
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Extrinsic similarity using Gromov-Hausdorff distance
Mémoli (2008)
Connection between Euclidean GH and ICP distances:
Congruence Euclidean isometry
EXTRINSIC SIMILARITY
ICP distance: GH distance with Euclidean
metric:
Mémoli, 2008
49Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Connection to canonical form distance
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Correspondence again
A different representation for correspondence using indicator functions
defines a valid correspondence if
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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?
52Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
Measure coupling
Let be probability measures defined on and
The measure can be considered as a relaxed version of the indicator
function or as fuzzy correspondence
A measure on is a coupling of and if
for all measurable sets
Mémoli, 2007
(a metric space with measure is called a metric measure or mm space)
53Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
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
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Earth Mover’s distance
Let be a metric space, and measures supported
on
EMD as optimal mass transport:
mass transported from to
distance traveled
Mémoli, 2007
The Wasserstein or Earth Mover’s distance (EMD) is given by
Define the coupling of
55Numerical Geometry of Non-Rigid Shapes Invariant shape similarity
The analogy
Hausdorff
Mémoli, 2007
Distance between subsets
of a metric space .
Gromov-Hausdorff
Distance between metric spaces
Wasserstein
Distance between subsets
of a metric measure
space .
Gromov-Wasserstein
Distance between metric
measure spaces