Axial Flip Invariance and Fast Exhaustive Searching with Wavelets

Matthew Bolitho

Outline

Goals Shape Descriptors

Invariance to rigid transformation Wavelets

The wavelet transform Haar basis functions Axial ambiguity with wavelets

Axial ambiguity Invariance Fast Exhaustive Searching

Wavelet based Shape Descriptor

Voxel based descriptorRasterise model into voxel gridApply Wavelet TransformSubset of information into feature vectorsCompare vectors

Shape Descriptor Goals

Concise to store Quick to compute Efficient to match Discriminating Invariant to transformations Invariant to deformations Insensitive to noise Insensitive to topology Robust to degeneracies

Project focus

Invariance to transformation Efficient matching

Scale, Translation, Rotation Invariance Invariance through normalisation Scale: scale voxel grid such that is just fits

the whole model Translation: set the origin of voxel grid to

be model center of mass Rotation: Principal Component Analysis

Principal Component Analysis

Align model to a canonical frame Calculate variance of points Eigen-values of covariance matrix map to

(x,y,z) axes in order of size

[1]

Axial Ambiguity

PCA has a problem Eigen-values are only defined up to sign In 3D, flip about x,y,z axes

[1]

Resolving the Ambiguity

Exhaustive search approachCompare all possible alignments (8 in 3D)Select alignment with minimal distance as

best match An invariant approach: make comparison

invariant to axial flip

The Wavelet Transform

Transforms a function to a new basis: Haar basis functions

Invertible Non-Lossy

[2]

Haar Basis Functions

Family of step functions i specifies frequency family j indexes family Orthogonal Orthonormal when scaled

by

Fast to compute Compute in-place

120

0

)2()(

15.0

5.00

0

1

1

)(

,

j

iji

j

i

jxx

otherwise

x

x

x

i2

Constant Function

+1

-1

01½

1)( xc

Family i=0

+1

-1

01½

x(x)

)()(0,0 xx

Family i=1

+1

-1

01½

x(x)

)12()(

)2()(

1,1

0,1

xx

xx

+1

-1

01½

x(x)

Family i=2+1

-1

01½

x(x)

)34()(

)24()(

)14()(

)4()(

3,2

2,2

1,2

0,2

xx

xx

xx

xx

+1

-1

01½

x(x)

+1

-1

01½

x(x)

+1

-1

01½

x(x)

Nomenclature

Adopt a more convenient indexing scheme

)(

)(

)(

)(

)(

)(

)(

)(

3,27

2,26

1,25

0,24

1,13

0,12

0,01

0

xf

xf

xf

xf

xf

xf

xf

xf c

i=2

i=1

i=0

Vector Basis

Basis functions can also be represented as a set of orthonormal basis vectors:

Wavelet transform of function g is:

4

3210

2105.05.0

2105.05.0

0215.05.0

0215.05.0

H

hhhh

g

ii

gHgW

hggW

)(

,)(

Example

Given a function Wavelet transform is

Aside: given function

5524g

02228

5524

)(

4

4

H

gHgW

22028)(

4255

hW

h

Resolving Axial Ambiguity

Exploit wavelets to get:Axial Flip Invariance

Make Wavelet Transform invariant to axial flip

Fast Exhaustive Search Reduce the complexity of exhaustively testing all

permutations of flip (recall: 8 in 3D)

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

00 ff

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

11 ff

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

32 ff

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

74 ff

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

65 ff

Wavelets and Axial Flip

Established a mapping for axial flip

f0 itself

f1 inverse of itself

Pairs inverse of each other

47

56

65

74

23

32

11

00

ff

ff

ff

ff

ff

ff

ff

ff

Invariance

Goal: Discard information that determines flip Goal: Not loose too much information

Use mapping to make wavelet transform invariant to flip f0 is already invariant | f1 | is invariant

Pairs are not, yet…

Invariance with pairs

For a pair

So, a+b and a-b behave like f1 and f0 under axial flip

Note: when a+b and a-b are known, a and b can be known – no loss of information; transform invertible

abba

)( ba

abba

ba

abba

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

3232 ffff

Observation

+1

-1

01½

x(x)+1

-1

01½

x(x)

3232 ffff

A New Basis

Redefine basis with a new mapping S( f )

Now all coefficients either map to themselves (+) or their inverse (-) under reflection

)(

)(

)(

)(

)(

)(

)(

)(

747

656

655

744

323

322

11

00

fff

fff

fff

fff

fff

fff

ff

ff

Invariance

New basis defines reflections with change in sign of half the coefficients

Invariance: Store f0, f3, f6, f7

Store absolute value of f1, f2, f4, f5, …

)(),(,)(,)(),(,)(,)(),( 76543210 fSfSfSfSfSfSfSfSS I

Invariance Example

Given g and h from previous example

Perform wavelet transform:

Transform basis:

5524g 4255h

02228)( gW 22028)( hW

02202228))(( gWS I

222228

22022028))((

hWS I

Invariance Evaluation

AdvantagesOnly perform single comparison

DisadvantageDiscards sign of half the coefficients

may hurt ability to discriminate

Exhaustive Searching

Rather than making comparison invariant, perform it a number of times:

R is the set of all possible axial reflections

Good Idea: If possible reduce this comparison cost fast exhaustive searching

RhgRhg 22)(min

Fast Exhaustive searching

Distance between g and h, R(g) and h:

Recall gi , hi : sign according to axial reflection

iiii

ii

hghg

hg

hg

2

)(22

2

2

0

iiii

ii

hgRhgR

hgR

hgR

)(2)(

))((

)(

22

2

2

1

22)(gRg

Fast Exhaustive searching iiii hghg 222

0 iiii hgRhg )(2221

7766554433221100 hghghghghghghghghg ii

77665544

33221100

)()()()(

)()()()()(

hgRhgRhgRhgR

hgRhgRhgRhgRhgR ii

Recall the mapping of R(gi) gi, thus:

7766554433221100)( hghghghghghghghghgR ii

Fast Exhaustive searching 5544221177663300 hghghghghghghghghg ii

5544221177663300

5544221177663300)(

hghghghghghghghg

hghghghghghghghghgR ii

Collect together terms to form:

55442211

77663300

hghghghggh

hghghghggh

ghghhgR

ghghhg

ii

ii

)(

Fast Exhaustive searching

Now, we can express andonly in terms of gi and hi

We can calculate both from the decomposition of the first, with minimal extra computation

2hg 2

)( hgR

ghghhg

hg

ii 222

2

0

ghghhg

hgR

ii 2

)(22

2

1

Fast Exhaustive search Example

Given g and h from previous examplesTransform basis:

222228

02202228))((

gWS

222228

22022028))((

hWS

Fast Exhaustive search Example

Calculate gh+ and gh- from S(W(g)) and S(W(h)):

Calculate norms:

4

66

2211

3300

hghggh

hghggh

0

)466(27070

2)(

16

)466(27070

2

222

222

ghghhghgR

ghghhghg

Fast search Evaluation

For minimal extra computation, all permutations of flip can be compared

No information is discardedc.f. invariance

Higher Dimensions

Both invariance and fast exhaustive search apply to higher dimensions

As dimensionality increases, invariance needs to discard more and more information In 2D, 4 flips In 3D, 8 flips

Applying Transforms in 2D

Transform rows S(W(r0)

S(W(r1)

r5

...

...

r15

c0

c1

...

c15

Applying Transforms in 2D

Transform columns S(W(r0)

S(W(r1)

r5

...

...

r15

S(W(c

0 ))

S(W(c

1 )) ...

c15

c5

...

Exhaustive Searching in 2D

In 1D we had gh+ and gh-

In 2D we will have gh++, gh+-, gh-+ and gh--

By applying both W(g) and S(g) in rows then columns, the 2D flip problem is reduced to two 1D flip problemsThis makes the cross multiplication easier

Cross multiplication

gh++, gh+-, gh-+ and gh--

are determined by cross multiplying the grid + * + = gh++

etc

+--+--++----++++

+ - - + - - + - - - - + + + ++

Exhaustive Searching in 2D

ghghghghgRg

ghghghghgRg

ghghghghgRg

ghghghghgg

xy

y

x

:)(

:)(

:)(

:

2222 hghg

In 3D

The extension into 3D is similar:8 flips8 gh terms8 ways to combine gh terms

Conclusion

Presented a way to overcome PCA alignment ambiguityWith minimal extra computationWith no loss of useful shape information

Conclusion II

PCA still has problems Instability: Small change in PCA alignment

can change voxel vote

Gaussian smoothing can distribute votes better

Future Work

Integrate into complete shape descriptorConcise to storeQuick to computeDiscriminatingRobustnessetc

Actual precision vs. recall results

References

[1] Misha Kazhdan: Alignment slides, October 25 2004 [2] Original teapot image from

http://www.plunk.org/~grantham/public/graphics.html