SQUARE-ROOT WAVELET DENSITIES AND SHAPE ANALYSIS

Anand Rangarajan, Center for Vision, Graphics and Medical Imaging (CVGMI), University of Florida, Gainesville

Walking the straight and narrow….on a sphere

Square-root densities

Square-root densities

∑ ∑∞

≥

+=k kjj

kjkjkjkj xxxp,

,,,,0

00)()()( ψβφα

Shape is a point on hyperspheredue to Fisher-Rao geometry

Wavelets

Wavelet Representations

Wavelets can approximate any f∊ℒ2, i.e.

Only work with compactly supported, orthogonal basis families: Haar, Daubechies, Symlets, Coiflets

∑ ∑∞

≥

+=k kjj

kjkjkjkj xxxf,

,,,,0

00)()()( ψβφα

Translation index Resolution level

Father Mother

Expand , Not !

Expand in multi-resolution basis:

Integrability constraints:

Estimate coefficients using a constrained maximum likelihood objective:

p p

∑ ∑∞

≥

+=k kjj

kjkjkjkj xxxp,

,,,,0

00)()()( ψβφα

1),(,

2,

2,,,

0

00=+= ∑ ∑

∞

≥k kjjkjkjkjkjh βαβα

{ }0 , ,where ,j k j kα β=Θ

( )

−++Θ−=Θ ∑ ∑∏

∞

≥=1)|(log)(

,

2,

2,

1

2

0

0k kjj

kjkj

N

iixp βαλL

{ } IH 4=LEAsymptotic Hessian of negative log likelihood

Objective is convex

2D Density Estimation

Density WDE KDE

Basis ISE Fixed BW ISE

Variable BW ISE

Bimodal SYM7 6.773E-03 1.752E-02 8.114E-03

Trimodal COIF2 6.439E-03 6.621E-03 1.037E-02

Kurtotic COIF4 6.739E-03 8.050E-03 7.470E-03

Quadrimodal COIF5 3.977E-04 1.516E-03 3.098E-03

Skewed SYM10 4.561E-03 8.166E-03 5.102E-03

Peter and Rangarajan, IEEE T-IP, 2008

Shape L’Âne Rouge: Sliding Wavelets

How Do We Select the Number of Levels? In the wavelet expansion of we need set j0

(starting level) and j1 (ending level)

Balasubramanian [32] proposed geometric approach by analyzing the posterior of a model class

The model selection criterion (razor) is

p

∑ ∑>

+=k

j

kjjkjkjkjkj xxxp

1

0

00,

,,,, )()()( ψβφα

)()|()()(

)|(Ep

dEpppEp ∫ ΘΘΘ

=M

M

ΘΘ

+ΘΘ++Θ−= ∫ )ˆ(det)ˆ(~det

ln21)(detln)

2ln(

2)ˆ|(ln)(

ij

ijij g

gdgNkEpR

πM

ML fit Scales with parameters and samples.

Volume of model class manifold

Ratio of expected Fisher to empirical Fisher

+Θ−=

Θ )()(ln)ˆ|(ln)(

ˆ MM

MVVEpR

Total volume of manifold

Volume of distinguishable distributions around ML

Connections to MDL Volume around MLE

Last term of razor disappears

This simplification leads to

∞→→

ΘΘ

=Θ Ngg

Gij

ij ,1)ˆ(~det)ˆ(det

)(

∫ ΘΘ++Θ−==⇒ dgNkEpMDLR ij )(detln)2

ln(2

)ˆ|(ln)(~π

M

21

2

ˆ )ˆ(~det)ˆ(det2)(

ΘΘ

=Θ

ij

ij

k

gg

NV πM

Geometric Intuition

Space of distributions

The razor prefers

these.

Counting volumes

saupto50Color

MDL for Wavelet Densities on the Hypersphere

Space of distributions

Intuition Behind Shrinking Surface Area

Volume gets pushed into corners as dimensions increase.

In 100 dimensions diagonal of unit length for sphere is only 10% of way to the cube diagonal.

d Vs/Vc

1 1

2 .785

3 .524

4 .308

5 .164

6 .08

Nested Subspaces Lead to Simpler Model Selection

Hypersphere dimensionality remains the same with MRA

It is sufficient to search over j0, using only scaling functions for density estimation.

MDL is invariant to MRA, however sparsity not considered.

k 2k

2k= +

2k

4k

4k= =+ +

Other Model Selection Criteria

Two-term MDL (MDL2) (Rissanen 1978)

Akaike Information Criterion (AIC) (Akaike 1973)

Bayesian Information Criterion (BIC) (Schwarz 1978)

Also compared to other distance measures Hellinger divergence (HELL) Mean Squared Error (MSE) L1

+Θ−=

π2ln

2)ˆ|(ln2 NkEpMDL

kEpAIC 2)ˆ|(ln2 +Θ−=

( )NkEpBIC ln2)ˆ|(ln2 +Θ−=

1D Model Selection with CoifletsDensity COIF1 (j0) COIF2 (j0)

MDL3 MDL2 AIC BIC MSE HELL L1 MDL3 MDL2 AIC BIC MSE HELL L1

Gaussian

0 0 1 0 1 1 1 -1 -1 0 -1 0 0 0Skewed Uni. 1 1 1 1 2 1 1 0 0 1 0 1 0 1Str. Skewed Uni. 2 2 3 2 4 3 3 2 2 2 2 4 2 3Kurtotic Uni. 2 2 2 1 4 2 2 2 2 2 2 2 2 2Outlier

2 2 3 2 5 3 4 2 2 2 2 4 2 4Bimodal

1 0 1 0 2 1 1 0 0 0 0 1 0 1Sep. Bimodal 1 1 2 1 2 1 2 1 1 1 1 1 1 1Skewed Bimodal 1 1 1 1 2 2 2 1 1 1 1 1 1 1Trimodal

1 1 1 1 1 1 1 1 1 1 1 1 2 1Claw

2 2 2 2 2 2 2 2 2 2 2 2 2 2Dbl. Claw

1 0 1 0 2 1 1 0 0 0 0 1 0 1Asym. Claw 2 1 2 1 3 2 3 2 1 2 1 3 2 3Asym. Dbl. Claw 1 1 1 0 2 1 2 0 0 2 0 2 2 2

MDL3 vs. BIC and MSE

BIC

, j0=

0

MD

L3 , j 0=

1

MSE

, j0=

4

MD

L3 , j 0=

2

Part III Summary

Simplified geometry of allows us to compute the model volume term of MDL in closed form.

Misspecified models can be avoided by assuring we have enough samples relative to the number of coefficients in the wavelet density expansion.

Leveraged the nested property of the hypersphere to restrict the parameter search space to only scaling function start levels.

MDL for WDE provides a geometrically motivated way to select the decomposition levels for wavelet

densities.

p

Shape L’Âne RougeA red donkey solves Klotski

Shape L’Âne Rouge: Sliding Wavelets

Geometry of Shape Matching

Wavelet density estimationPoint set representation

Shape isa point on hypersphere

Or Geodesic Distance

( ) )(cos, 211

21 ΘΘ= − TppD

Fast Shape Similarity Using Hellinger Divergence

( )( )21

2

2121

22

)|()|()||(

ΘΘ−=

Θ−Θ= ∫T

dppppD xxx

Localized Alignment Via

Local shape differences will cause coefficients to shift.

Permutations ⇒ Translations Slide coefficients back into alignment.

Sliding

T

00000

31000

31000

3100

T

00

31000

31000

3100000

Penalize Excessive Sliding

Location operator, , gives centroid of each (j,k) basis. Sliding cost equal to square of Euclidean distance.

),( kjr

Sliding Objective

Objective minimizes over penalized permutation assignments

Solve via linear assignment using cost matrix

where Θiis vectorized list of ith shape’s coefficients and D is the matrix of distances between basis locations.

( )

+−= ∑∑

> kjjkjkj

kjkjkjE

,

)2()(,

)1(,

,

)2()(,

)1(,

0000 ππ ββααπ

( ) ( ) ( ) ( )

−+−+ ∑∑

kjkjkjkjkjkj

,

2

,

200 ),(,),(,

0

ππλ rrrr

DC T λ+ΘΘ= 21

Location operator

Permutation

Penalty Weight

Effects of λ Peter and Rangarajan, CVPR 2008

Recognition Results on MPEG-7 DB

All recognition rates are based on MPEG-7 bulls-eye criterion.

D2 shape distributions (Osada et al.) only at 59.3%.

Summary

The geometry associated with the wavelet representation allows us to represent densities as points on a unit hypersphere.

For the first time, non-rigid alignment can be addressed using linear assignment framework.

Advantages of our method: no topological restrictions, very little pre-processing, closed-form metric.

Sliding wavelets provide a fast and accurate method of shape matching

p