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Signal Processingand
Representation Theory
Lecture 1
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Outline:• Algebra Review
– Numbers– Groups– Vector Spaces– Inner Product Spaces– Orthogonal / Unitary Operators
• Representation Theory
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Algebra ReviewNumbers (Reals)Real numbers, ℝ, are the set of numbers that we express in decimal notation, possibly with infinite, non-repeating, precision.
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Algebra ReviewNumbers (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: 2aa
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Algebra ReviewNumbers (Complexes)Complex numbers, , are the set of numbers that we ℂexpress as a+ib, where a,b and ℝ i= .
Example: ei=cos+isin
1
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Algebra ReviewNumbers (Complexes)Let p(x)=xn+an-1xn-1+…+a1x1+a0 be a polynomial with ai .ℂ
Algebraic Closure:p(x) must have a root, x0 in :ℂ
p(x0)=0.
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Algebra ReviewNumbers (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:
ibaiba
22)()( baibaibaiba
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Algebra ReviewGroupsA 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.
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Algebra ReviewGroupsExamples:
– 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)
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Algebra Review(Real) Vector SpacesA 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,a , ℝ vV, such that:
1. 1v=v for all vV.2. a(v+w)=av+aw for all a , ℝ v,wV.3. (a+b)v=av+bv for all a,b , ℝ vV.4. (ab)v=a(bv) for all a,b , ℝ vV.
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Algebra Review(Real) Vector SpacesExamples:• 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.
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Algebra Review(Complex) Vector SpacesA 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,a , ℂ vV, such that:
1. 1v=v for all vV.2. a(v+w)=av+aw for all a , ℂ v,wV.3. (a+b)v=av+bv for all a,b , ℂ vV.4. (ab)v=a(bv) for all a,b , ℂ vV.
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Algebra Review(Complex) Vector SpacesExamples:• 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.
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Algebra Review(Real) Inner Product SpacesA 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.
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Algebra Review(Real) Inner Product SpacesExamples:
– 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
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Algebra Review(Real) Inner Product SpacesExamples:
– 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)
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Algebra Review(Real) Inner Product SpacesExamples:
– 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:
dxxgxfgf )()(,
2
0)()(, dgfgf
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Algebra Review(Complex) Inner Product SpacesA 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.
uv,vu,
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Algebra Review(Complex) Inner Product SpacesExamples:
– 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
nn11 wv...wvwv, tMM
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Algebra Review(Complex) Inner Product SpacesExamples:
– The space of mxn matrices with real coefficients is an inner product space.If M and N are two mxn matrices then:
NMNM, tTrace
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Algebra Review(Complex) Inner Product SpacesExamples:
– 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:
dxxgxfgf )()(,
2
0)()(, dgfgf
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Algebra ReviewInner Product SpacesIf 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 v2V2.21 vvv
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Algebra ReviewInner Product SpacesExample: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)
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Algebra ReviewOrthogonal / Unitary OperatorsIf 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,Awfor all v,wV.
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Algebra ReviewOrthogonal / Unitary OperatorsExamples:
– If V is the space of real, two-dimensional, vectors and A is any rotation or reflection, then A is orthogonal.
A=
v2v1
A(v2)
A(v1)
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Algebra ReviewOrthogonal / Unitary OperatorsExamples:
– If V is the space of real, three-dimensional, vectors and A is any rotation or reflection, then A is orthogonal.
A=
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Algebra ReviewOrthogonal / Unitary OperatorsExamples:
– If V is the space of functions defined in 1D and A is any translation, then A is orthogonal.
A=
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Algebra ReviewOrthogonal / Unitary OperatorsExamples:
– If V is the space of functions defined on a circle and A is any rotation or reflection, then A is orthogonal.
A=
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Algebra ReviewOrthogonal / Unitary OperatorsExamples:
– If V is the space of functions defined on a sphere and A is any rotation or reflection, then A is orthogonal.
A=
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Outline:• Algebra Review• Representation Theory
– Orthogonal / Unitary Representations– Irreducible Representations– Why Do We Care?
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Representation TheoryOrthogonal / Unitary RepresentationAn 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:
1. (0)v=v, for all vV, where 0 is the identity element.2. (gh)v=(g) (h)v
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Representation TheoryOrthogonal / Unitary RepresentationExamples:
– 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.
vvg )(
vgvg )det()(
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Representation TheoryOrthogonal / Unitary RepresentationExamples:
– 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.
vgvg )(
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Representation TheoryOrthogonal / Unitary RepresentationExamples:
– 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.
),(),,(,,,)( 43214321 xxgxxgxxxxg
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Representation TheoryIrreducible RepresentationsA 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)WWThen either W=V or W=.
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Representation TheoryIrreducible RepresentationsIf WV is a sub-representation of G, and W is the space of vectors perpendicular to W:
v,w=0for 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.
wvgwggvgwgv ,,,0 11
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Representation TheoryIrreducible RepresentationsExamples:
– If G is any group and V is any vector space with dimension larger than one, then:
is not an irreducible representation.
vvg )(
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Representation TheoryIrreducible RepresentationsExamples:
– 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.
),(),,(,,,)( 43214321 xxgxxgxxxxg
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Representation Theory
Why do we care?
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Representation TheoryWhy we careIn shape matching we have to deal with the fact that rotations do not change the shape of a model.
=
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Representation TheoryExhaustive SearchIf 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:
NMNM
NMNM
vTvvvvTvvvT
,2),,(
22
2
D
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Representation TheoryExhaustive SearchIf 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.
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Representation TheoryExhaustive 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
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Representation TheoryExhaustive Search (Brute Force)Then the dot-product of M and N at a rotation T is equal to:
n
jijiji
n
jjj
n
iii
n
jjj
n
iiiNM
vTvba
vTbva
vbTvavTv
1,
11
11
,
,
,,
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n
jijijiNM vTvbavTv
1,
,,
Representation TheoryExhaustive Search (Brute Force)So that the nxn cross-multiplications are needed:
T(vn)
vM
v1
v2
vn
=
+
++
T(v1)
=
+
++
T(v2)T(vN)… …
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Representation TheoryExhaustive 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.
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Representation TheoryExhaustive Search (w/ Rep. Theory)Then we can express the shape descriptors in terms of this basis:
vM=α1w1+…+αnwn
vN=β1w1+…+βnwn
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Representation TheoryExhaustive Search (w/ Rep. Theory)And the dot-product of M and N at a rotation T is equal to:
n
iiiii
n
jijiji
n
jjj
n
iii
n
jjj
n
iiiNM
wTw
wTw
wTw
wTwvTv
1
1,
11
11
,
,
,
,,
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n
iiiiiNM wTwvTv
1
,,
Representation TheoryExhaustive Search (w/ Rep. Theory)So that only n multiplications are needed:
T(wn)
vM
w1
w2
wn
=
+
++
T(w1)
=
+
++
T(w2)T(vN)… …