Flattening via Multi- Dimensional Scaling Ron Kimmel ron Computer Science Department Geometric Image...

Flattening via Multi-Dimensional Scaling

Ron Kimmel

www.cs.technion.ac.il/~ron

Computer Science Department

Geometric Image Processing Lab

Technion-Israel Institute of Technology

On isometric surfaces and bending invariant signatures.

Multi-dimensional scaling techniques: Classical. Least squares. Fast.

Surface classification: experimental results.

Outline

Matching Surfaces

Problem: Given 2D surfaces, define a measure of their

similarity. Classical techniques:

Find a rigid transformation that maximizes some measure.

Match key points on the surface. Compare local or semi-differential invariants,

e.g. matching graphs.

Isometric surface matching via bending invariant signatures: Map the surface into a small Euclidean space, in which

isometric surfaces transform to similar (rigid) surfaces.

Advantages: Handle (somewhat) non-rigid objects. A global operation, does not rely on selected

key points or local invariants.

Bending Invariant Signatures

Bending invariant signatures Basic Concept

Input – a surface in 3D

Extract from [D] coordinates in an m dimensional Euclidean space via

Multi-Dimensional Scaling (MDS).

Output – A 2D surface embedded in in R (for some small

m)

i j

ij ij

ij

Compute geodesic distance matrix [D] between each pair of vertices

δ = geodesic_distance(vertex ,vertex )

[D] = δ

m

Fast Marching on Surfaces

Source point

Euclidean Distance

Geodesic Distance

Multi-Dimensional Scaling

MDS is a family of methods that map similarity measurements among objects, to points in a small dimensional Euclidean space.

The graphic display of the similarity measurements provided by MDS enables to explore the geometric structure of the data. ix

Stress =

Σ(δ - d )ij ij

2

Σδij

MDS Dissimilarity measures

coordinates in m-dimensional Euclidean Space

Stress Function

A simple example1 2 3 4 5 6 7 8 9 10

1. London 0

2. Stockholm

569 0

3. Lisbon 667 1212

0

4. Madrid 530 1043

201 0

5. Paris 141 617 596 431 0

6. Amsterdam

140 446 768 608 177 0

7. Berlin 357 325 923 740 340 218

0

8. Prague 396 423 882 690 337 272

114 0

9. Rome 569 787 714 516 436 519

472 364 0

10. Dublin 190 648 714 622 320 302

514 573 755

01 2 3 4 5 6 7 8 9 10

12.7 0 32.8 29.4 14.6 8.8 4.5 6.4 16.4 13.8

4.1 13.9 13.2 14.9 9.0 8.2 16.6 19.6 25.4 0

x

Y Stockholm

London

Dublin

Lisbon

Amsterdam

ParisPrague

Berlin

RomeMadrid

Stoc

khol

m

Lond

on

Dub

lin

Lisb

on

Am

ster

dam

Paris

Prag

ueBerlin

Rom

e

Mad

ridStockholm

London

Dublin

Lisbon Am

sterda

mParis

PragueBerlin

Rom

e

Madrid

Rotation

Reflection

Flattening via MDS Compute geodesic distances between pairs of

points. Construct a square distance matrix of

geodesic distances^2. Find the coordinates in the plane via multi-

dimensional scaling. The simplest is `classical scaling’.

Use the flattened coordinates for texturing the surface, while preserving the

texture features. Zigelman, Kimmel, Kiryati, IEEE TVCG 2002

Grossmann, Kiryati, Kimmel, IEEE TPAMI 2002

Bending invariant surface matchingElad (Elbaz), Kimmel CVPR 2001

00

0

Flattening

Distances - comparison

Texture Mapping

Fattening via MDS: 3D

OriginalFast

ClassicalLeast Squares

OriginalFast

Least Squares

Classical

Input Surfaces


Elad, Kimmel, CVPR’2001

?


?



3Original surfaces Canonical surfaces in R


00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CCCC

AAAA

D

DDD

BBBB

EEEE

FFFF

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CE

CC

B

A

DE

EE

B

C

A

B

D

B

F

DAD

F

A

FF

Bending Invariant Clustering

2nd moments based MDS for clustering

Original surfaces Canonical forms

*A=human body*B=hand*C=paper*D=hat*E=dog*F=giraffe


Classical Scaling Young et al.

1930

Given n points in , denote

Define coordinates vector

The Euclidean distance between 2 points:

22

1

22

1

222

2

2

jjii

k

l

lj

lj

li

li

k

l

lj

lijiij xxxxxxd

pppp

pp

T

Tp ],...,,[ 21 kiiii xxxkR

TpppP ],...,,[ 21 n

Define the `centering’ matrix where

Let the matrix

We have that and also

Thus,

Classical Scaling

Tn 11IJ 1

T

n

,...,, ][1 111

222

2

2

2

2

2

2

2

1

2

1

2

1

...

......

...

...

nnn ppp

ppp

ppp

Qnn

TTT

TT

PP00JJPPJJQJQJ

JPPQQJJDJ~~

22

)2(

0QJ 0JQQJ TT

Classical Scaling

The coordinates are related to by Thus the operation is also called `double centering’.Applying SVD, we can compute where and the coordinates can be extracted asIf we choose to take only part of the eignstructure,

then, our approximation minimizes the Frobenius norm

TPPJDJ~~

21

P~

P 1~

1

n

m

lm

li

li x

nxx

TUUJDJ 21

21UP ~

F

TUU ̂

IUUT

MDS

Matlab code for 2D flattening

);2(:,*)).2,2((

);1(:,*)).1,1((

);'',2,(],[

;***5.0

;/).()(

QLsqrtnewx

QLsqrtnewy

LMBeigsLQ

JDJB

nnonesneyeJ

Zigelman, Kimmel, Kiryati IEEE TVCG 2002

Classical Scaling

The eigenvalues are the 2nd order moments of the flattened surface, since by definition

all the cross 2nd order moments vanish by the unitarity of U, thus `flattening’.

iiii

n

m

im iiiix

TTUUUU :],[:],[:],[:],[~ 2121

1

2

Conclusions A method for bending invariant signatures Based on:

Fast marching on surfaces MDS LS/Classical/Fast

Results: Texture mapping Bending invariant signatures Classification of isometric surfaces.

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CCCC

AAAA

DDDD

BBBB

EEEE

FFFF

00.2

0.40.6

0.81

00.20.40.6

0.810

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CE

CC

B

ADE

EE

B

C

A

B

D

B

F

DA

D

F

A

FF

Least Squares MDS

Standard optimization approach to solve the minimization problem of the stress cost function.

Solved via ‘scaling by maximizing convex function’ (SAMCOF) algorithm.

Starting with a random solution and iteratively minimizing another stress function, which satisfies.

ƒ (x,z) ≥ ƒ (x) for x ≠ z

and ƒ (z,z) = ƒ (z)

The complexity is O(n ). Converges to the optimal solution.

2

Fast MDS

The fast MDS: heuristic efficient technique O(mn). Works recursively by generating a new dimension at each step, Providing m-dimensional coordinates after m recursion steps. Project the vertices on a selected ‘line’.

First, the algorithm selects the Farthest two vertices. Next, all other vertices are projected On that line using the cosine law. Next step is to project all items to An (n-1) hyper plane (H) that is Perpendicular to the line that Connects those vertices. Generate a new distance matrix. Repeat the last three steps m times.

Flattening via Multi- Dimensional Scaling Ron Kimmel ron Computer Science Department Geometric Image...

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