1 al geometry of non-rigid shapes Multidimensional scaling Numerical geometry of non-rigid shapes Multidimensional scaling Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved
Dec 22, 2015
1Numerical geometry of non-rigid shapes Multidimensional scaling
Numerical geometry of non-rigid shapes
Multidimensional scaling
Alexander Bronstein, Michael Bronstein, Ron Kimmel© 2007 All rights reserved
2Numerical geometry of non-rigid shapes Multidimensional scaling
How to measure shape similarity?
Extrinsic similarity Intrinsic similarity
Given two shapes and represented as discrete samples
and , compute their similarity
3Numerical geometry of non-rigid shapes Multidimensional scaling
Rigid shape similarity: congruence
Degrees of freedom: Euclidean transformations (rotation+translation)
Classical solution: iterative closest point (ICP) algorithm
Rigid (extrinsic) similarity
Hausdorff distance as a measure of congruence between point clouds
(other distances are usually preferred)
ROTATION MATRIX
TRANSLATION VECTOR
Y. Chen, G. Medioni, 1991P. J. Besl, N. D. McKay, 1992
4Numerical geometry of non-rigid shapes Multidimensional scaling
ICP in fairy tales
Cinderella measuring the glass slipper
Image: Disney
5Numerical geometry of non-rigid shapes Multidimensional scaling
Congruence is not a good criterion for similarity of non-rigid shapes
Geodesic distances are invariant to isometric deformations and can be
easily computed using FMM
Naïve approach: directly compare matrices of geodesic distances
Non-rigid (intrinsic) similarity
Problem: arbitrary ordering of points (permutation of rows and columns)
degrees of freedom
6Numerical geometry of non-rigid shapes Multidimensional scaling
Isometric embedding
Represent the intrinsic geometry in a Euclidean space by isometrically
embedding it into
Treat the resulting images (canonical forms) as rigid shapes, using ICP
or other rigid similarity algorithms
Isometric embedding “undoes” the degrees of freedom
Shape Canonical form
A. Elad, R. Kimmel, CVPR, 2001
7Numerical geometry of non-rigid shapes Multidimensional scaling
Mapmaking problem
Earth (sphere) Planar map
A
BB
A
8Numerical geometry of non-rigid shapes Multidimensional scaling
Embedding error
Theorema Egregium: a sphere has positive
Gaussian curvature, the plane has zero
Gaussian curvature, therefore, they are not
isometric.
Every cartographer knows: impossible to create a distance-preserving
planar map of the Earth!
Does isometric embedding into higher-dimensional spaces exist?
9Numerical geometry of non-rigid shapes Multidimensional scaling
Linial’s example
A
C D
1 1
2
AC D
1 1
1 1
2C D
B
A
B
C
D
B
A
C D
1 1
B
1
Conclusion: generally, isometric embedding does not exist!
10Numerical geometry of non-rigid shapes Multidimensional scaling
Find an embedding that distorts the distances the least
Stress function is a measure of distortion
Minimum distortion embedding
Multidimensional scaling (MDS) problem
where
11Numerical geometry of non-rigid shapes Multidimensional scaling
Examples of canonical forms
Canonical forms
Near-isometric deformations of a shape
A. Elad, R. Kimmel, CVPR, 2001
12Numerical geometry of non-rigid shapes Multidimensional scaling
- an matrix of coordinates in the embedding space
- an constant matrix with values
Matrix expression of the L2-stress
- an matrix-valued function
13Numerical geometry of non-rigid shapes Multidimensional scaling
variables
Non-convex non-linear optimization problem
Using convex optimization techniques is liable to local convergence
Optimum defined up to Euclidean transformation
MDS problem
14Numerical geometry of non-rigid shapes Multidimensional scaling
Instead of , minimize a convex majorizing function
satisfying
Iterative majorization
Start with some and iteratively update
15Numerical geometry of non-rigid shapes Multidimensional scaling
SMACOF algorithm
Majorize the stress by a convex quadratic function
Analytic expression for the minimum of :
SMACOF (Scaling by Minimizing a COnvex Function)
16Numerical geometry of non-rigid shapes Multidimensional scaling
Equivalent to constant-step gradient descent
SMACOF algorithm (cont)
Guarantees monotonically decreasing sequence of stress values
No guarantee of global convergence
Iteration cost:
17Numerical geometry of non-rigid shapes Multidimensional scaling
Application: face recognition
Facial expressions are approximate isometries of the facial surface
Identity = intrinsic geometry
Expression = extrinsic geometry
A. M. Bronstein et al., IJCV, 2005
18Numerical geometry of non-rigid shapes Multidimensional scaling
How to canonize a person?
3D surface acquisition
Smoothing CanonizationCropping
A. M. Bronstein et al., IJCV, 2005
19Numerical geometry of non-rigid shapes Multidimensional scaling
Application: face recognition
Canonical forms
Facial expressions
A. M. Bronstein et al., IJCV, 2005
20Numerical geometry of non-rigid shapes Multidimensional scaling
-1000 -800 -600 -400 -200 0 200 400 600 800 1000-1000
-800
-600
-400
-200
0
200
400
600
800
1000
-1000 -800 -600 -400 -200 0 200 400 600 800 1000-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Rigid similarity Non-rigid similarity(canonical forms)
MichaelAlex
Application: face recognition
A. M. Bronstein et al., IJCV, 2005
21Numerical geometry of non-rigid shapes Multidimensional scaling
Multiresolution MDS: motivation
Coarse MDS problem (N=200)
Fine MDS problem (N~1000)
22Numerical geometry of non-rigid shapes Multidimensional scaling
Multiresolution MDS
Interpolate
Coarse grid solution
Fine grid solution
Fine grid initialization
Solve coarse grid problem
Coarse grid initialization
Solve fine grid problem
Bottom-up: solve coarse grid MDS problem to initialize fine grid
problem
Can be performed on multiple resolution levels
Reduce complexity (less fine-grid iterations)
Reduce the risk of local convergence (good initialization)
23Numerical geometry of non-rigid shapes Multidimensional scaling
Multigrid MDS
Interpolateresidual
Coarse grid solution
Fine grid initialization
Fine grid residual
Solve coarse grid problem
Coarse grid initialization
Top-down: start with a fine grid initialization
Improve the fine grid initialization by solving a coarse grid problem and
propagating the error
Decimate
Coarse grid residual
Improved solution
24Numerical geometry of non-rigid shapes Multidimensional scaling
Problem: the minima of fine and coarse grid problems do not coincide!
Correction
CORRECTION
Add correction term to coarse grid problem to compensate for
inconsistency
Choosing
guarantees that is a coarse grid solution
M. M. Bronstein et al., NLAA, 2006
25Numerical geometry of non-rigid shapes Multidimensional scaling
Another problem: the new coarse grid problem is unbounded!
Modified stress
Modified stress: add a quadratic penalty to the stress
thus resolving translation ambiguity (forcing to be centered at the
origin)
( can be made arbitrarily large by adding a constant to
without changing the stress)
M. M. Bronstein et al., NLAA, 2006
26Numerical geometry of non-rigid shapes Multidimensional scaling
Plugging everything together
Hierarchy of data
Interpolation and decimation operators to transfer variables and
residuals from one resolution level to another
Hierarchy of optimization problems
Relaxation: optimization algorithm used to improve solution
M. M. Bronstein et al., NLAA, 2006
27Numerical geometry of non-rigid shapes Multidimensional scaling
Decimate
Relax1X
Decimate
Solve coarsest grid problem
Interpolate and correct
RelaxRelax
Interpolate and correct
V-cycle
M. M. Bronstein et al., NLAA, 2006
28Numerical geometry of non-rigid shapes Multidimensional scaling
Convergence example
M. M. Bronstein et al., NLAA, 2006
Order of magnitude speedup, especially pronounced for large
Time (sec)
Str
ess
Convergence of SMACOF and MG MDS (N=2145)
29Numerical geometry of non-rigid shapes Multidimensional scaling
How to choose the embedding space?
A generic, non-Euclidean embedding space
Must result in small embedding error (good representation)
Convenient representation of points in (local or preferably global
parametrization)
Simple (preferably analytic) expression for distances
The isometry group is simple (few degrees of freedom)
30Numerical geometry of non-rigid shapes Multidimensional scaling
Possible choices
Schwartz et al. 1989:
Elad & Kimmel 2001:
Elad & Kimmel 2002:
BBK 2005:
Walter & Ritter 2002:
Euclidean
Spherical
Hyperbolic
Problem: embedding error can be reduced, but not made zero!
31Numerical geometry of non-rigid shapes Multidimensional scaling
Generalized MDS
Embedding space = triangular mesh
Generalized stress
Generalized MDS (GMDS) problem
where is the image on triangular mesh
A. M. Bronstein et al., PNAS, 2006
32Numerical geometry of non-rigid shapes Multidimensional scaling
Difference 1: the distances have no analytic expression
Consequence 1: geodesic distance interpolation
Main differences
A. M. Bronstein et al., SIAM, 2006
Difference 2: points represented in local barycentric
coordinates
Consequence 2: optimization with a modified line search (unfolding)
33Numerical geometry of non-rigid shapes Multidimensional scaling
Distance interpolation
How to approximate the distances between
points ?
Precompute the pair-wise distances between all mesh
vertices using FMM
Find triangles and enclosing
Interpolate from known
distances
Interpolate from
A. M. Bronstein et al., SIAM, 2006
34Numerical geometry of non-rigid shapes Multidimensional scaling
Modified line search: unfolding
Optimization on triangular mesh requires displacing a point along a
ray
(line search)
Line search in barycentric coordinates requires unfolding
Result: polylinear path
A. M. Bronstein et al., SIAM, 2006
35Numerical geometry of non-rigid shapes Multidimensional scaling
Geodesic distances are intrinsic descriptors of non-rigid shapes
invariant to isometric deformations
MDS is an efficient method for representing and comparing intrinsic
invariants
Multiresolution and multigrid methods can yield a significant
convergence speedup and reduce the risk on local convergence
Generalized MDS allows avoiding the embedding error by embedding
one surface into another
Conclusions so far