signal processing and representation theory lecture 3

Signal Processingand

Representation Theory

Lecture 3

Outline:• Review• Spherical Harmonics• Rotation Invariance• Correlation and Wigner-D Functions


ReviewGiven a representation of a group G onto an inner product space V, decomposing V into the direct sum of irreducible sub-representations:

V=V1…Vnmakes it easier to:

– Compute the correlation between two vectors: fewer multiplications are needed

– Obtain G-invariant information: more transformation invariant norms can be obtained


ReviewIn the case that the group G is commutative, the irreducible sub-representations Vi are all one-complex-dimensional, (Schur’s Lemma).

Example:

If V is the space of functions on a circle, represented by n-dimensional arrays, and G is the group of 2D rotations:

– Correlation can be done in O(n log n) time (using the FFT)

– We can obtain n/2-dimensional, rotation invariant descriptors


What happens when the group G is not commutative?

Example:

If V is the space of functions on a sphere and G is the group of 3D rotations:

– How quickly can we correlate?

– How much rotation invariant information can we get?


Spherical Harmonic DecompositionGoal:

Find the irreducible sub-representations of the group of 3D rotation acting on the space of spherical functions.


Spherical Harmonic DecompositionPreliminaries:

If f is a function defined in 3D, we can get a function on the unit sphere by looking at the restriction of f to points with norm 1.



A polynomial p(x,y,z) is homogenous of degree d if it is the linear sum of monomials of degree d:

d

d

ddd

dddd

dd

dddd

dd

za

zyaxa

zyaxyayxaxa

yaxyayxaxazyxp

0,

10,11,1

10,1

21,1

22,1

11,1

0,01

1,01

1,0,0),,(



We can think of the space of homogenous polynomials of degree d in x, y, and z as:

where Pd(x,y) is the space of homogenous polynomials of degreed d in x and y.

ddddd zyxPzyxPzyxPyxPzyxP ),(),(),(),(),,( 0

111



If we let Pd(x,y,z) be the set of homogenous polynomials of degree d, then Pd(x,y,z) is a vector-space of dimension:

2

)1()1(1

2

0

ddi

d

i


Spherical Harmonic DecompositionObservation:

If M is any 3x3 matrix, and p(x,y,z) is a homogenous polynomial of degree d:

then p(M(x,y,z)) is also a homogenous polynomial of degree d:

333231

232221

131211

mmm

mmm

mmm

M

d

j

jd

k

jkjdkkj zyxazyxp

0 0,),,(

jd

j

jd

k

kjdkkj zmymxmzmymxmzmymxmazyxMp )()()(),,( 333231

0 0232221131211,


Spherical Harmonic DecompositionIf V is the space of functions on the sphere, we can consider the sub-space of functions on the sphere that are restrictions of homogenous polynomials of degree d.

Since a rotation will map a homogenous polynomial of degree d back to a homogenous polynomial of degree d, these sub-spaces are sub-representations.


Spherical Harmonic DecompositionIn general, the space of homogenous polynomials of degree d has dimension (d+1)+(d)+(d-1)+…+1:

d

j

jd

k

jkjdkkj zyxazyxp

0 0,),,(

Representation TheorySpherical Harmonic DecompositionIf (x,y,z) is a point on the sphere, we know that this point satisfies:

Thus, if q(x,y,z)Pd(x,y,z), then even though in general, the polynomial:

is a homogenous polynomial of degree d+2, its restriction to the sphere is actually a homogenous polynomial of degree d.

1222 zyx

))(,,( 222 zyxzyxq


Spherical Harmonic Decomposition

So, while the sub-spaces Pd(x,y,z) are sub-representations, they are not irreducible as Pd-2(x,y,z)Pd(x,y,z).

To get the irreducible sub-representations, we look at the spaces:

),,(),,( 2 zyxPzyxPV ddd


Spherical Harmonic DecompositionAnd the dimension of these sub-representations is:

12

)1()1(1

)1()1(

),,(dim),,(dimdim

2

0

2

0

2

00

2

d

iidd

ii

zyxPzyxPV

d

i

d

i

d

i

d

i

ddd


Spherical Harmonic DecompositionThe spherical harmonics of frequency d are an orthonormal basis for the space of functions Vd.

If we represent a point on a sphere in terms of its angle of elevation and azimuth:

with 0π and 0 <2π …

sinsin,cos,sincos,


Spherical Harmonic DecompositionThe spherical harmonics are functions Ylm, with l0 and -lml spanning the sub-representations Vl:

Span

Span

Span

),(),,(,),,(),,(

),(),,(),,(

),(

11

11

01

111

000

kk

kk

kk

kkk YYYYV

YYYV

YV


Spherical Harmonic DecompositionFact:

If we have a function defined on the sphere, sampled on a regular nxn grid of angles of elevation and azimuth, the forward and inverse spherical harmonic transforms can be computed in O(n2 log2n).

Like the FFT, the fast spherical harmonic transform can be thought of as a change of basis, and a brute force method would take O(n4) time.


What are the spherical harmonics Ylm(,)?


What are the spherical harmonics Ylm?Conceptually:

The Ylm are the different homogenous polynomials of degree l:

4

1,0

0 Y

)(8

3,

8

3,

)(8

3,

11

01

11

izxY

yY

izxY

222

12

22202

12

222

)(32

15,

)(8

15,

)2(16

5,

)(8

15,

)(32

15,

izxY

iyzxyY

zyxY

iyzxyY

izxY


What are the spherical harmonics Ylm?Technically:

Where the Plm are the associated Legendre polynomials:

Where the Pl are the Legendre polynomials:

imml

ml eP

mlmll

Y cos)!(

)!(

4

12,

)(1)1(2/2 zPdzd

zzP lm

mmmm

l

dttttzi

zP nl

12/12212

1


What are the spherical harmonics Ylm?Functionally:

The Ylm are the eigen-values of the Laplacian operator:

),(),(2 ff


What are the spherical harmonics Ylm?Visually:

The Ylm are spherical functions whose number of lobes get larger as the frequency, l, gets bigger:

l=1

l=2

l=3

l=0


What are the spherical harmonics Ylm?What is important about the spherical harmonics is that they are an orthonormal basis for the (2d+1)-dimensional sub-representations, Vd, of the group of 3D rotations acting on the space of spherical functions.


Sub-Representations


InvarianceGiven a spherical function f, we can obtain a rotation invariant representation by expressing f in terms of its spherical harmonic decomposition:

where each flVl:

l

lm

ml

mll Yaf ),(),(

0

),(),(l

lff


InvarianceWe can then obtain a rotation invariant representation by storing the size of each fl independently:

where:221212 l

lll

ll

lll aaaaf

,,,,)( 10 lffff


Invariance

Spherical Harmonic Decomposition

+ += +


Invariance

+ += +

+ + +

Constant 1st Order 2nd Order 3rd Order


Invariance

+ + +

Constant 1st Order 2nd Order 3rd Order

Ψ


InvarianceLimitations:

By storing only the energy in the different frequencies, we discard information that does not depend on the pose of the model:

– Inter-frequency information

– Intra-frequency information

+


InvarianceInter-Frequency information:

22.5o90o

=

= +


InvarianceIntra-Frequency information:


Invariance

…

……O(n2)

O(n)


Wigner-D FunctionsThe Wigner-D functions are an orthogonal basis of complex-valued functions defined on the space of rotations:

with l0 and -lm,m’l.

'', ,)( ml

ml

lmm YRYRD

Representation TheoryWigner-D FunctionsFact:If we are given a function defined on the group of 3D rotations, sampled on a regular nxnxn grid of Euler angles, the forward and inverse spherical harmonic transforms can be computed in O(n4) time.

Like the FFT and the FST, the fast Wigner-D transform can be thought of as a change of basis, and a brute force method would take O(n6) time.


MotivationGiven two spherical functions f and g we would like to compute the distance between f and g at every rotation. To do this, we need to be able to compute the correlation:

Corr(f,g,R)=f,R(g)at every rotation R.


CorrelationIf we express f and g in terms of their spherical harmonic decompositions:

00

),(),(),(),(l

l

lm

ml

ml

l

l

lm

ml

ml YbgYaf

0 ',',

'

0 ',

''

0',

'

''

''

''

0'

'

''

''

''

0

0'

'

''

''

''

0

)(

,

,

,

,)(,

l

l

lmm

lmm

ml

ml

l

l

lmm

ml

ml

ml

ml

ll

l

lm

l

lm

ml

ml

ml

ml

l

m

lm

ml

ml

l

m

lm

ml

ml

l

m

lm

ml

ml

l

m

lm

ml

ml

RDba

YRYba

YRYba

YRbYa

YbRYagRf


CorrelationThen the correlation of f with g at a rotation R is given by:

0 ',

',' )()(,

l

l

lmm

lmm

ml

ml RDbagRf


CorrelationSo that we get an expression for the correlation of f with g as some linear combination of the Wigner-D functions:


CorrelationThe complexity of correlating two spherical functions sampled on a regular nxn grid is:

– Forward spherical harmonic transform: O(n2 log2n)

00

),(),(),(),(l

l

lm

ml

ml

l

l

lm

ml

ml YbgYaf




– Multiplying frequency terms: O(n3)

0 ',

'' ,)(,l

l

lmm

ml

ml

ml

ml YRYbagRf





– Inverse Wigner-D transform: O(n4))(,)(

0, ',',

' gRfRDbal

l

lmm

lmm

ml

ml





– Inverse Wigner-D transform: O(n4)

Total complexity of correlation is: O(n4)

(Note that a brute force approach would take O(n5): For each of O(n3) rotations we would have to perform an O(n2) dot-product computation.)

signal processing and representation theory lecture 3

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