Signal Processing and Representation Theory

1
Signal Processingand Representation Theory

• Lecture 3

2

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

3
Representation Theory

• Review
• Given a representation ? of a group G onto an
  inner product space V, decomposing V into the
  direct sum of irreducible sub-representations
• VV1??Vn
• makes it easier to
• Compute the correlation between two vectors
  fewer multiplications are needed
• Obtain G-invariant information more
  transformation invariant norms can be obtained

4
Representation Theory

• Review
• In the case that the group G is commutative, the
  irreducible sub-representations Vi are all
  one-complex-dimensional, (Schurs 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

5
Representation Theory

• 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?

6

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

7
Representation Theory

• Spherical Harmonic Decomposition
• Goal
• Find the irreducible sub-representations of the
  group of 3D rotation acting on the space of
  spherical functions.

8
Representation Theory

• Spherical Harmonic Decomposition
• Preliminaries
• 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.

9
Representation Theory

• Spherical Harmonic Decomposition
• Preliminaries
• A polynomial p(x,y,z) is homogenous of degree d
  if it is the linear sum of monomials of degree d

10
Representation Theory

• Spherical Harmonic Decomposition
• Preliminaries
• We can think of the space of homogenous
  polynomials of degree d in x, y, and z
  aswhere Pd(x,y) is the space of homogenous
  polynomials of degreed d in x and y.

11
Representation Theory

• Spherical Harmonic Decomposition
• Preliminaries
• 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

12
Representation Theory

• Spherical Harmonic Decomposition
• Observation
• 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

13
Representation Theory

• Spherical Harmonic Decomposition
• If 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
  .

14
Representation Theory

• Spherical Harmonic Decomposition
• In general, the space of homogenous polynomials
  of degree d has dimension (d1)(d)(d-1)1

15
Representation Theory

• Spherical Harmonic Decomposition
• If (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 d2, its
  restriction to the sphere is actually a
  homogenous polynomial of degree d.

16
Representation Theory

• 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

17
Representation Theory

• Spherical Harmonic Decomposition
• And the dimension of these sub-representations is

18
Representation Theory

• Spherical Harmonic Decomposition
• The 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 azimuthwith 0???p
  and 0?? lt2p

19
Representation Theory

• Spherical Harmonic Decomposition
• The spherical harmonics are functions Ylm, with
  l?0 and -l?m?l spanning the sub-representations
  Vl

20
Representation Theory

• Spherical Harmonic Decomposition
• Fact
• 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.

21
Representation Theory

• What are the spherical harmonics Ylm(?,?)?

22
Representation Theory

• What are the spherical harmonics Ylm?
• Conceptually
• The Ylm are the different homogenous polynomials
  of degree l

23
Representation Theory

• What are the spherical harmonics Ylm?
• Technically
• Where the Plm are the associated Legendre
  polynomials
• Where the Pl are the Legendre polynomials

24
Representation Theory

• What are the spherical harmonics Ylm?
• Functionally
• The Ylm are the eigen-values of the Laplacian
  operator

25
Representation Theory

• What are the spherical harmonics Ylm?
• Visually
• The Ylm are spherical functions whose number of
  lobes get larger as the frequency, l, gets bigger

26
Representation Theory

• What are the spherical harmonics Ylm?
• What is important about the spherical harmonics
  is that they are an orthonormal basis for the
  (2d1)-dimensional sub-representations, Vd, of
  the group of 3D rotations acting on the space of
  spherical functions.

27
Representation Theory

• Sub-Representations

28
Representation Theory

• Sub-Representations

29
Representation Theory

• Sub-Representations

30
Representation Theory

• Sub-Representations

31

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

32
Representation Theory

• Invariance
• Given a spherical function f, we can obtain a
  rotation invariant representation by expressing f
  in terms of its spherical harmonic decomposition
• where each fl?Vl

33
Representation Theory

• Invariance
• We can then obtain a rotation invariant
  representation by storing the size of each fl
  independently
• where

34
Representation Theory

• Invariance

Spherical Harmonic Decomposition
35
Representation Theory

• Invariance

Constant
1st Order
2nd Order
3rd Order



36
Representation Theory

• Invariance

?
Constant
1st Order
2nd Order
3rd Order



37
Representation Theory

• Invariance
• Limitations
• 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

38
Representation Theory

• Invariance
• Inter-Frequency information

22.5o
90o


39
Representation Theory

• Invariance
• Intra-Frequency information

40
Representation Theory

• Invariance

O(n)


O(n2)
41
Representation Theory

• Invariance

O(n)


O(n2)
42
Representation Theory

• Invariance

O(n)


O(n2)
43
Representation Theory

• Invariance

O(n)


O(n2)
44
Representation Theory

• Invariance

O(n)


O(n2)
45

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

46
Representation Theory

• Wigner-D Functions
• The Wigner-D functions are an orthogonal basis of
  complex-valued functions defined on the space of
  rotationswith l?0 and -l?m,m?l.

47
Representation Theory

• Wigner-D Functions
• Fact
• 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.

48
Representation Theory

• Motivation
• Given 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.

49
Representation Theory

• Correlation
• If we express f and g in terms of their spherical
  harmonic decompositions

50
Representation Theory

• Correlation
• Then the correlation of f with g at a rotation R
  is given by

51
Representation Theory

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

52
Representation Theory

• Correlation
• The complexity of correlating two spherical
  functions sampled on a regular nxn grid is
• Forward spherical harmonic transform O(n2 log2n)

53
Representation Theory

• Correlation
• The complexity of correlating two spherical
  functions sampled on a regular nxn grid is
• Forward spherical harmonic transform O(n2
  log2n)
• Multiplying frequency terms O(n3)

54
Representation Theory

• Correlation
• The complexity of correlating two spherical
  functions sampled on a regular nxn grid is
• Forward spherical harmonic transform O(n2
  log2n)
• Multiplying frequency terms O(n3)
• Inverse Wigner-D transform O(n4)

55
Representation Theory

• Correlation
• The complexity of correlating two spherical
  functions sampled on a regular nxn grid is
• Forward spherical harmonic transform O(n2
  log2n)
• Multiplying frequency terms O(n3)
• 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.)