and calculus of shapes Alexander amp Michael Bronstein 20062010 toscacstechnionacilbook VIPS Advanced School on Numerical Geometry of NonRigid Shapes University of Verona April 2010 ID: 234439
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Slide1
Invariant correspondence
and calculus of shapes
© Alexander & Michael Bronstein, 2006-2010tosca.cs.technion.ac.il/book
VIPS Advanced School onNumerical Geometry of Non-Rigid Shapes University of Verona, April 2010Slide2
“Natural” correspondence?
?Slide3
Correspondence
accurate
‘
‘
‘
‘
makes sense
‘
‘
‘
‘
beautiful
‘
‘
‘
‘
Geometric
Semantic
AestheticSlide4
Correspondence
Correspondence is not a well-defined problem
!
Chances to solve it with geometric tools are slim.
If objects are sufficiently
similar
, we have better chances.
Correspondence between
deformations
of the
same
object.Slide5
Invariant correspondence
Ingredients:
Class of
shapesClass of deformations
Correspondence procedure which given two shapes
returns a map
Correspondence procedure is -
invariant
if it
commutes
with
i.e., for every and every ,Slide6Slide7
Invariant similarity (reminder)
Ingredients:
Class of shapes
Class of deformationsDistance
Distance is -
invariant
if for every and everySlide8
Closest point correspondence
between , parametrized by
Its distortion
Minimize distortion over all possible congruencesRigid similarity
Class of deformations:
congruences
Congruence-invariant (rigid)
similarity
:Slide9
Rigid correspondence
Class of deformations: congruences
Congruence-invariant similarity:
Congruence-invariant correspondence:
RIGID SIMILARITY
RIGID CORRESPONDENCE
INVARIANT SIMILARITY
INVARIANT CORRESPONDENCESlide10
Representation procedure is -
invariant
if it translates into an isometry in , i.e., for every and , there exists such that
Invariant representation (canonical forms)
Ingredients:Class of shapes
Class of
deformations
Embedding space
and its
isometry group
Representation procedure
which given a shape
returns an embeddingSlide11
INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITYSlide12
Invariant parametrization
Ingredients:
Class of shapes
Class of deformationsParametrization space and its isometry group
Parametrization procedure which given a shape
returns a chart
Parametrization procedure is -
invariant
if it
commutes
with
up to an
isometry
in , i.e., for every and ,
there exists such thatSlide13Slide14
INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCESlide15
Representation errors
Invariant similarity / correspondence is reduced to finding isometry
in embedding / parametrization space.Such isometry does not exist and invariance holds approximately
Given parametrization domains and , instead of
isometry
find a
least distorting mapping
.
Correspondence isSlide16
Dirichlet energy
Minimize Dirchlet energy functional
Equivalent to solving the
Laplace equation
Boundary conditions
Solution (minimizer of Dirichlet energy) is a
harmonic function
.
N. Litke, M. Droske, M. Rumpf, P. Schroeder,
SGP
, 2005Slide17
Dirichlet energy
Caveat: Dirichlet functional is not invariant
Not parametrization-independent
Solution: use intrinsic quantitiesFrobenius norm becomes
Hilbert-Schmidt norm
Intrinsic area element
Intrinsic Dirichlet energy functional
N. Litke, M. Droske, M. Rumpf, P. Schroeder,
SGP
, 2005Slide18
The harmony of harmonic maps
Intrinsic Dirichlet energy functional
is the
Cauchy-Green deformation tensorDescribes square of local change in distances
Minimizer is a harmonic map.
N. Litke, M. Droske, M. Rumpf, P. Schroeder,
SGP
, 2005Slide19
Physical interpretation
METAL MOULD
RUBBER SURFACE
= ELASTIC ENERGY CONTAINED IN THE RUBBERSlide20
Minimum-distortion correspondence
Ingredients:
Class of shapesClass of deformations
Distortion function which given a correspondence between two shapes assigns to it
a non-negative number Minimum-distortion correspondence
procedureSlide21
Minimum-distortion correspondence
Correspondence procedure is -
invariant
if distortion is
-invariant, i.e., for every , and ,Slide22
Minimum-distortion correspondence
CONGRUENCES
CONFORMAL
ISOMETRIES
Dirichlet energy
Quadratic stress
Euclidean normSlide23
Minimum distortion correspondenceSlide24
Intrinsic symmetries
create distinct isometry-invariant minimum-
distortion correspondences, i.e., for every
Uniqueness & symmetry
The converse in
not true
, i.e. there might exist two distinct
minimum-distortion correspondences such that
for every Slide25
Partial correspondenceSlide26
Measure coupling
Let be probability measures defined on and
The measure can be considered as a 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
)Slide27
Intrinsic similarity
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 spacesSlide28
Minimum-distortion correspondence
Mémoli, 2007
Gromov-Hausdorff
Minimum-distortion correspondence between metric spaces
Gromov-Wasserstein
Minimum-distortion fuzzy correspondence between
metric measure spacesSlide29
TIME
Reference
Transferred texture
Texture transferSlide30
Virtual body paintingSlide31
Texture substitution
I’m
Alice
.I’m Bob.
I’m
Alice
’s texture
on
Bob
’s geometrySlide32
=
How to add two dogs?
+
1
2
1
2
C A L C U L U S O F S H A P E SSlide33
Addition
creates displacement
Affine calculus in a linear space
Subtraction
creates direction
Affine combination
spans subspace
Convex combination
( )
spans polytopesSlide34
Affine calculus of functions
Affine space of functions
Subtraction
Addition
Affine combination
Possible because functions share a
common domainSlide35
Affine calculus of shapes
?
A. Bronstein, M. Bronstein, R. Kimmel,
IEEE TVCG
, 2006Slide36
Temporal super-resolution
TIMESlide37
Motion-compensated interpolationSlide38
Metamorphing
100%
Alice
100%Bob
75% Alice
25%
Bob
50%
Alice
50%
Bob
75%
Alice
50%
BobSlide39
Face caricaturization
0
1
1.5
EXAGGERATED
EXPRESSIONSlide40
Affine calculus of shapesSlide41
What happened?
SHAPE SPACE IS NON-EUCLIDEAN!Slide42
Shape space
Shape space is an abstract manifold
Deformation fields of a shape are vectors in tangent space Our affine calculus is valid only locally
Global affine calculus can be constructed by defining trajectories confined to the manifold
AdditionCombinationSlide43
Choice of trajectory
Equip tangent space with an inner product
Riemannian metric on Select to be a minimal geodesic
Addition: initial value
problemCombination: boundary value
problemSlide44
Choice of metric
Deformation field of is called
Killing field if for every
Infinitesimal displacement byKilling field is metric preserving and are
isometricCongruence is always a Killing field
Non-trivial
Killing field
may not exist
Slide45
Choice of metric
Inner product on
Induces
norm
measures
deviation
of from Killing field
– defined modulo
congruence
Add
stiffening termSlide46
Minimum-distortion trajectory
Geodesic
trajectoryShapes along are
as isometric as possible to Guaranteeing no self-intersections is an open problem