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