Signal Processing and Representation Theory

Signal Processingand Representation Theory Lecture 3

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

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: V=V1…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

Representation Theory Review In 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

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?

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

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.

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:

Representation Theory Spherical Harmonic Decomposition Preliminaries: 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.

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:

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:

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.

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

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

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:

Representation Theory Spherical Harmonic Decomposition And the dimension of these sub-representations is:

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 azimuth:with 0π and 0 <2π …

Representation Theory Spherical Harmonic Decomposition The spherical harmonics are functions Ylm, with l0 and -lml spanning the sub-representations Vl:

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.

Representation Theory What are the spherical harmonics Ylm(,)?

Representation Theory What are the spherical harmonics Ylm? Conceptually: The Ylm are the different homogenous polynomials of degree l:

Representation Theory What are the spherical harmonics Ylm? Technically: Where the Plm are the associated Legendre polynomials: Where the Pl are the Legendre polynomials:

Representation Theory What are the spherical harmonics Ylm? Functionally: The Ylm are the eigen-values of the Laplacian operator:

l=0 l=1 l=2 l=3 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:

Representation Theory 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.

Representation Theory Sub-Representations

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:

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

Representation Theory Invariance = + + + Spherical Harmonic Decomposition

Representation Theory Invariance = + + + 3rd Order Constant 1st Order 2nd Order + + +

Representation Theory Invariance Ψ 3rd Order Constant 1st Order 2nd Order + + +

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

Representation Theory Invariance Inter-Frequency information: = + 22.5o 90o = +

Representation Theory Invariance Intra-Frequency information:

Representation Theory Invariance … O(n) … … O(n2)

Representation Theory Wigner-D Functions The Wigner-D functions are an orthogonal basis of complex-valued functions defined on the space of rotations:with l0 and -lm,m’l.

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.

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.

Representation Theory Correlation If we express f and g in terms of their spherical harmonic decompositions:

Representation Theory Correlation Then the correlation of f with g at a rotation R is given by:

Signal Processing and Representation Theory