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## Signal Processing and Representation Theory

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**Outline:**• Algebra Review • Numbers • Groups • Vector Spaces • Inner Product Spaces • Orthogonal / Unitary Operators • Representation Theory**Algebra Review**Numbers (Reals) Real numbers, ℝ, are the set of numbers that we express in decimal notation, possibly with infinite, non-repeating, precision.**Algebra Review**Numbers (Reals) Example: =3.141592653589793238462643383279502884197… Completeness: If a sequence of real numbers gets progressively “tighter” then it must converge to a real number. Size: The size of a real number aℝ is the square root of its square norm:**Algebra Review**Numbers (Complexes) Complex numbers, ℂ, are the set of numbers that we express as a+ib, where a,bℝ and i= . Example: ei=cos+isin**Algebra Review**Numbers (Complexes) Let p(x)=xn+an-1xn-1+…+a1x1+a0 be a polynomial with aiℂ. Algebraic Closure: p(x) must have a root, x0 in ℂ: p(x0)=0.**Algebra Review**Numbers (Complexes) Conjugate: The conjugate of a complex number a+ib is: Size: The size of a real number a+ibℂ is the square root of its square norm:**Algebra Review**Groups A group G is a set with a composition rule + that takes two elements of the set and returns another element, satisfying: • Asscociativity: (a+b)+c=a+(b+c) for all a,b,cG. • Identity: There exists an identity element 0G such that 0+a=a+0=a for all aG. • Inverse: For every aG there exists an element -aG such that a+(-a)=0. If the group satisfies a+b=b+a for all a,bG, then the group is called commutative or abelian.**Algebra Review**Groups Examples: • The integers, under addition, are a commutative group. • The positive real numbers, under multiplication, are a commutative group. • The set of complex numbers without 0, under multiplication, are a commutative group. • Real/complex invertible matrices, under multiplication are a non-commutative group. • The rotation matrices, under multiplication, are a non-commutative group. (Except in 2D when they are commutative)**Algebra Review**(Real) Vector Spaces A real vector space is a set of objects that can be added together and scaled by real numbers. Formally: A real vector space V is a commutative group with a scaling operator: (a,v)→av, aℝ, vV,such that: • 1v=v for all vV. • a(v+w)=av+aw for all aℝ, v,wV. • (a+b)v=av+bv for all a,bℝ, vV. • (ab)v=a(bv) for all a,bℝ, vV.**Algebra Review**(Real) Vector Spaces Examples: • The set of n-dimensional arrays with real coefficients is a vector space. • The set of mxn matrices with real entries is a vector space. • The sets of real-valued functions defined in 1D, 2D, 3D,… are all vector spaces. • The sets of real-valued functions defined on the circle, disk, sphere, ball,… are all vector spaces. • Etc.**Algebra Review**(Complex) Vector Spaces A complex vector space is a set of objects that can be added together and scaled by complex numbers. Formally: A complex vector space V is a commutative group with a scaling operator: (a,v)→av, aℂ, vV,such that: • 1v=v for all vV. • a(v+w)=av+aw for all aℂ, v,wV. • (a+b)v=av+bv for all a,bℂ, vV. • (ab)v=a(bv) for all a,bℂ, vV.**Algebra Review**(Complex) Vector Spaces Examples: • The set of n-dimensional arrays with complex coefficients is a vector space. • The set of mxn matrices with complex entries is a vector space. • The sets of complex-valued functions defined in 1D, 2D, 3D,… are all vector spaces. • The sets of complex-valued functions defined on the circle, disk, sphere, ball,… are all vector spaces. • Etc.**Algebra Review**(Real) Inner Product Spaces A real inner product space is a real vector space V with a mapping V,V→ℝ that takes a pair of vectors and returns a real number, satisfying: • u,v+w= u,v+ u,w for all u,v,wV. • αu,v=αu,v for all u,vV and all αℝ. • u,v= v,u for all u,vV. • v,v0 for all vV, and v,v=0 if and only if v=0.**Algebra Review**(Real) Inner Product Spaces Examples: • The space of n-dimensional arrays with real coefficients is an inner product space.If v=(v1,…,vn) and w=(w1,…,wn) then: v,w=v1w1+…+vnwn • If M is a symmetric matrix (M=Mt) whose eigen-values are all positive, then the space of n-dimensional arrays with real coefficients is an inner product space.If v=(v1,…,vn) and w=(w1,…,wn) then: v,wM=vMwt**Algebra Review**(Real) Inner Product Spaces Examples: • The space of mxn matrices with real coefficients is an inner product space.If M and N are two mxn matrices then: M,N=Trace(MtN)**Algebra Review**(Real) Inner Product Spaces Examples: • The spaces of real-valued functions defined in 1D, 2D, 3D,… are real inner product space.If f and g are two functions in 1D, then: • The spaces of real-valued functions defined on the circle, disk, sphere, ball,… are real inner product spaces.If f and g are two functions defined on the circle, then:**Algebra Review**(Complex) Inner Product Spaces A complex inner product space is a complex vector space V with a mapping V,V→ℂ that takes a pair of vectors and returns a complex number, satisfying: • u,v+w= u,v+ u,w for all u,v,wV. • αu,v=αu,v for all u,vV and all αℝ. • for all u,vV. • v,v0 for all vV, and v,v=0 if and only if v=0.**Algebra Review**(Complex) Inner Product Spaces Examples: • The space of n-dimensional arrays with complex coefficients is an inner product space.If v=(v1,…,vn) and w=(w1,…,wn) then: • If M is a conjugate symmetric matrix ( ) whose eigen-values are all positive, then the space of n-dimensional arrays with complex coefficients is an inner product space.If v=(v1,…,vn) and w=(w1,…,wn) then: v,wM=vMwt**Algebra Review**(Complex) Inner Product Spaces Examples: • The space of mxn matrices with real coefficients is an inner product space.If M and N are two mxn matrices then:**Algebra Review**(Complex) Inner Product Spaces Examples: • The spaces of complex-valued functions defined in 1D, 2D, 3D,… are real inner product space.If f and g are two functions in 1D, then: • The spaces of real-valued functions defined on the circle, disk, sphere, ball,… are real inner product spaces.If f and g are two functions defined on the circle, then:**Algebra Review**Inner Product Spaces If V1,V2V, then V is the direct sum of subspaces V1, V2, written V=V1V2, if: • Every vector vV can be written uniquely as:for some vectors v1V1 and v2V2.**Algebra Review**Inner Product Spaces Example: If V is the vector space of 4-dimensional arrays, then V is the direct sum of the vector spaces V1,V2V where: • V1=(x1,x2,0,0) • V2=(0,0,x3,x4)**Algebra Review**Orthogonal / Unitary Operators If V is a real / complex inner product space, then a linear map A:V→V is orthogonal / unitary if it preserves the inner product: v,w= Av,Aw for all v,wV.**Algebra Review**Orthogonal / Unitary Operators Examples: • If V is the space of real, two-dimensional, vectors and A is any rotation or reflection, then A is orthogonal. A(v1) v1 v2 A(v2) A=**Algebra Review**Orthogonal / Unitary Operators Examples: • If V is the space of real, three-dimensional, vectors and A is any rotation or reflection, then A is orthogonal. A=**Algebra Review**Orthogonal / Unitary Operators Examples: • If V is the space of functions defined in 1D and A is any translation, then A is orthogonal. A=**Algebra Review**Orthogonal / Unitary Operators Examples: • If V is the space of functions defined on a circle and A is any rotation or reflection, then A is orthogonal. A=**Algebra Review**Orthogonal / Unitary Operators Examples: • If V is the space of functions defined on a sphere and A is any rotation or reflection, then A is orthogonal. A=**Outline:**• Algebra Review • Representation Theory • Orthogonal / Unitary Representations • Irreducible Representations • Why Do We Care?**Representation Theory**Orthogonal / Unitary Representation An orthogonal / unitary representation of a group G onto an inner product space V is a map that sends every element of G to an orthogonal / unitary transformation, subject to the conditions: • (0)v=v, for all vV, where 0 is the identity element. • (gh)v=(g) (h)v**Representation Theory**Orthogonal / Unitary Representation Examples: • If G is any group and V is any vector space, then:is an orthogonal / unitary representation. • If G is the group of rotations and reflections and V is any vector space, then:is an orthogonal / unitary representation.**Representation Theory**Orthogonal / Unitary Representation Examples: • If G is the group of nxn orthogonal / unitary matrices, and V is the space of n-dimensional arrays, then:is an orthogonal / unitary representation.**Representation Theory**Orthogonal / Unitary Representation Examples: • If G is the group of 2x2 rotation matrices, and V is the vector space of 4-dimensional real / complex arrays, then:is an orthogonal / unitary representation.**Representation Theory**Irreducible Representations A representation , of a group G onto a vector space V is irreducible if cannot be broken up into smaller representation spaces. That is, if there exist WV such that: (G)WW Then either W=V or W=.**Representation Theory**Irreducible Representations IfWV is a sub-representation of G, and W is the space of vectors perpendicular to W: v,w=0 for all vW and wW, then V=WW and W is also a sub-representation of V. For any gG, vW, and wW, we have: So if a representation is reducible, it can be broken up into the direct sum of two sub-representations.**Representation Theory**Irreducible Representations Examples: • If G is any group and V is any vector space with dimension larger than one, then:is not an irreducible representation.**Representation Theory**Irreducible Representations Examples: • If G is the group of 2x2 rotation matrices, and V is the vector space of 4-dimensional real / complex arrays, then:is not an irreducible representation since it maps the space W=(x1,x2,0,0) back into itself.**Representation Theory**Why do we care?**Representation Theory**Why we care In shape matching we have to deal with the fact that rotations do not change the shape of a model. =**Representation Theory**Exhaustive Search If vM is a spherical function representing model M and vn is a spherical function representing model N, we want to find the minimum over all rotations T of the equation:**Representation Theory**Exhaustive Search If V is the space of spherical functions then we can consider the representation of the group of rotations on this space. By decomposing V into a direct sum of its irreducible representations, we get a better framework for finding the best rotation.**Representation Theory**Exhaustive Search (Brute Force) Suppose that {v1,…,vn} is some orthogonal basis for V, then we can express the shape descriptors in terms of this basis: vM=a1v1+…+anvn vN=b1v1+…+bnvn**Representation Theory**Exhaustive Search (Brute Force) Then the dot-product of M and N at a rotation T is equal to:**Representation Theory**Exhaustive Search (Brute Force) So that the nxn cross-multiplications are needed: v1 T(v1) + + v2 T(v2) + + = = T(vN) vM … … + + T(vn) vn**Representation Theory**Exhaustive Search (w/ Rep. Theory) Now suppose that we can decompose V into a collection of one-dimensional representations. That is, there exists an orthogonal basis {w1,…,wn} of functions such that T(wi)wiℂ for all rotations T and hence: wi,T(wj)=0 for all i≠j.**Representation Theory**Exhaustive Search (w/ Rep. Theory) Then we can express the shape descriptors in terms of this basis: vM=α1w1+…+αnwn vN=β1w1+…+βnwn**Representation Theory**Exhaustive Search (w/ Rep. Theory) And the dot-product of M and N at a rotation T is equal to:**Representation Theory**Exhaustive Search (w/ Rep. Theory) So that only n multiplications are needed: w1 T(w1) + + w2 T(w2) + + = = T(vN) vM … … + + T(wn) wn