A Riemannian Fourier Transform via Spin Representations

28/08/2013
Auteurs : Michel Berthier
OAI : oai:www.see.asso.fr:2552:4890
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A Riemannian Fourier Transform via Spin Representations

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A Riemannian Fourier Transform via Spin Representations Geometric Science of Information 2013 T. Batard - M. Berthier - Outline of the talk The Fourier transform for multidimensional signals - Examples Three simple ideas The Riemannian Fourier transform via spin representations Applications to filtering The Fourier transform for multidimensional signals The problem : How to define a Fourier transform for a signal φ : Rd −→ Rn that does not reduce to componentwise Fourier transforms and that takes into account the (local) geometry of the graph associated to the signal ? Framework of the talk : the signal φ : Ω ⊂ R2 −→ Rn is a grey-level image, n = 1, or a color image, n = 3. In the latter case, we want to deal with the full color information in a really non marginal way. Many already existing propositions (without geometric considerations) : • T. Ell and S.J. Sangwine transform : Fµφ(U) = ∫ R2 φ(X)exp(−µ⟨X, U⟩)dX (1) where φ : R2 −→ H0 is a color image and µ is a pure unitary quaternion encoding the grey axis. Fµφ = A∥ exp[µθ∥] + A⊥ exp[µθ⊥]ν (2) where ν is a unitary quaternion orthogonal to µ. Allows to define an am- plitude and a phase in the chrominance and in the luminance. The Fourier transform for multidimensional signals The problem : How to define a Fourier transform for a signal φ : Rd −→ Rn that does not reduce to componentwise Fourier transforms and that takes into account the (local) geometry of the graph associated to the signal ? Framework of the talk : the signal φ : Ω ⊂ R2 −→ Rn is a grey-level image, n = 1, or a color image, n = 3. In the latter case, we want to deal with the full color information in a really non marginal way. Many already existing propositions (without geometric considerations) : • T. Bülow transform : Fijφ(U) = ∫ R2 exp(−2iπx1u1)φ(X)exp(−2jπu2x2)dX (1) where φ : R2 −→ R. Fijφ(U) = Fccφ(U) − iFscφ(U) − jFcsφ(U) + kFssφ(U) (2) Allows to analyse the symetries of the signal with respect to the x and y variables. The Fourier transform for multidimensional signals The problem : How to define a Fourier transform for a signal φ : Rd −→ Rn that does not reduce to componentwise Fourier transforms and that takes into account the (local) geometry of the graph associated to the signal ? Framework of the talk : the signal φ : Ω ⊂ R2 −→ Rn is a grey-level image, n = 1, or a color image, n = 3. In the latter case, we want to deal with the full color information in a really non marginal way. Many already existing propositions (without geometric considerations) : • M. Felsberg transform : Fe1e2e3 φ(U) = ∫ R2 exp(−2πe1e2e3⟨U, X⟩)φ(X)dX (1) where φ(X) = φ(x1e1 + x2e2) = φ(x1, x2)e3 is a real valued function defined on R2 (a grey level image). The coefficient e1e2e3 is the pseudos- calar of the Clifford algebra R3,0. This transform is well adapted to the monogenic signal. The Fourier transform for multidimensional signals The problem : How to define a Fourier transform for a signal φ : Rd −→ Rn that does not reduce to componentwise Fourier transforms and that takes into account the (local) geometry of the graph associated to the signal ? Framework of the talk : the signal φ : Ω ⊂ R2 −→ Rn is a grey-level image, n = 1, or a color image, n = 3. In the latter case, we want to deal with the full color information in a really non marginal way. Many already existing propositions (without geometric considerations) : • F. Brackx et al. transform : F±φ(U) = ( 1 √ 2π )n ∫ Rn exp( i π 2 ΓU) × exp(−i⟨U, X⟩)φ(X)dX (1) where ΓU is the angular Dirac operator. For φ : R2 −→ R0,2 ⊗ C F±φ(U) = 1 2π ∫ R2 exp(±U ∧ X)φ(X)dX (2) where exp(±U ∧ X) is a bivector. Three simple ideas ..1 The abstract Fourier transform is defined through the action of a group. • Shift theorem : Fφα(u) = e2iπαu Fφ(u) (3) where φα(x) = φ(x + α). Here, the involved group is the group of trans- lations of R. The action is given by (α, x) −→ x + α := τα(x) (4) The mapping (group morphism) χu : τα −→ e2iπuα = χu(α) ∈ S1 (5) is a so-called character of the group (R, +). The Fourier transform reads Fφ(u) = ∫ R χu(−x)φ(x)dx (6) .. Spinor Fourier Three simple ideas ..1 The abstract Fourier transform is defined through the action of a group. • More precisely : – By means of χu, every element of the group is represented as a unit complex number that acts by multiplication on the values of the function. Every u gives a representation and the Fourier transform is defined on the set of representations. – If the group G is abelian, we only deal with the group morphisms from G to S1 (characters). Three simple ideas ..1 The abstract Fourier transform is defined through the action of a group. • Some transforms : – G = (Rn, +) : we recover the usual Fourier transform. – G = SO(2, R) : this corresponds to the theory of Fourier series. – G = Z/nZ : we obtain the discrete Fourier transform. – In the non abelian case one has to deal with the equivalence classes of unitary irreducible representations (Pontryagin dual). Some of these irreducible representations are infinite dimensional. Applications to ge- neralized Fourier descriptors with the group of motions of the plane, to shearlets,... Three simple ideas ..1 The abstract Fourier transform is defined through the action of a group. • The problem : Find a good way to represent the group of translations (R2, +) in order to make it act naturally on the values (in Rn) of a multidimensional function Three simple ideas ..2 The vectors of Rn can be considered as generalized numbers. • Usual identifications : X = (x1, x2) ∈ R2 ↔ z = x1 + ix2 ∈ C (3) X = (x1, x2, x3, x4) ∈ R4 ↔ q = x1 + ix2 + jx3 + kx4 ∈ H (4) The fields C and H are the Clifford algebras R0,1 (of the vector space R with the quadratic form Q(x) = −x2) and R0,2 (of the vector space R2 with the quadratic form Q(x1, x2) = −x2 1 − x2 2). • Clifford algebras : the vector space Rn with the quadratic form Qp,q is embedded in an algebra Rp,q of dimension 2n that contains scalars, vectors and more generally multivectors such as the bivector u ∧ v = 1 2 (uv − vu) (5) Three simple ideas ..2 The vectors of Rn can be considered as generalized numbers. • The spin groups : the group Spin(n) is the group of elements of R0,n that are products x = n1n2 · · · n2k (3) of an even number of unit vectors of Rn. • Some identifications : Spin(2) ≃ S1 (4) Spin(3) ≃ H1 (5) Spin(4) ≃ H1 × H1 (6) • Natural idea : replace the group morphisms from (R2, +) to S1 , the cha- racters, by group morphisms from (R2, +) to Spin(n), the spin characters. Three simple ideas ..2 The vectors of Rn can be considered as generalized numbers. • The problem : Compute the spin characters, i.e. the group morphisms from (R2, +) to Spin(n) Find meaningful representation spaces for the action of the spin characters Three simple ideas ..2 The vectors of Rn can be considered as generalized numbers. • Spin(3) characters : χu1,u2,B : (x1, x2) −→ exp 1 2 [ B A ( x1 x2 )] = exp 1 2 [(x1u1 + x2u2)B] (3) where A = (u1 u2) is the matrix of frequencies and B = ef with e and f two orthonormal vectors of R3. .. Spinor Fourier Three simple ideas ..2 The vectors of Rn can be considered as generalized numbers. • Spin(4) and Spin(6) characters : (x1, x2) −→ exp 1 2 [ (B1 B2) A ( x1 x2 )] (3) (x1, x2) −→ exp 1 2 [ (B1 B2 B3) A ( x1 x2 )] (4) where A is a 2 × 2, resp. 2 × 3, real matrix and Bi = eifi for i = 1, 2, resp. i = 1, 2, 3, with (e1, e2, f1, f2), resp. (e1, e2, e3, f1, f2, f3), an orthonormal basis of R4, resp. R6. Three simple ideas ..3 The spin characters are parametrized by bivectors. • Fundamental remark : the spin characters are as usual parametrized by frequencies, the entries of the matrix A. But they are also parametrized by bivectors, B, B1 and B2, B1, B2 and B3, depending on the context. • How to involve the geometry ? it seems natural to parametrize the spin characters by the bivector corresponding to the tangent plane of the image graph, more precisely by the field of bivectors corresponding to the fiber bundle of the image graph. Three simple ideas ..3 The spin characters are parametrized by bivectors. • Several possibilities for dealing with representation spaces for the action of the spin characters : – Using Spin(3) characters and the generalized Weierstrass representa- tion of surface (T. Friedrich) : in “Quaternion and Clifford Fourier Trans- form and Wavelets (E. Hitzer and S.J. Sangwine Eds), Trends in Mathe- matics, Birkhauser, 2013. – Using Spin(4) and Spin(6) characters and the so-called standard re- presentations of the spin groups : in IEEE Journal of Selected Topics in Signal Processing, Special Issue on Differential Geometry in Signal Pro- cessing, Vol 7, Issue 4, 2013. The Riemannian Fourier transform The spin representations of Spin(n) are defined through complex represen- tations of Clifford algebras. They do not “descend” to the orthogonal group SO(n, R) (since they send −1 to −Identity contrary to the standard represen- tations). These are the representations used in physics. The complex spin representation of Spin(3) is the group morphism ζ3 : Spin(3) −→ C(2) (5) obtained by restricting to Spin(3) ⊂ (R3,0 ⊗ C)0 a complex irreducible repre- sentation of R3,0. An color image is considered as a section .. Spinor Fourier σφ : (x1, x2) −→ 3∑ k=1 (0, φk (x1, x2)) ⊗ gk (6) of the spinor bundle PSpin(E3(Ω)) ×ζ3 C2 (7) where E3(Ω) = Ω × R3 and (g1, g2, g3) is the canonical basis of R3. The Riemannian Fourier transform Dealing with spinor bundles allows varying spin characters and the most natural choice for the field of bivectors B := B(x1, x2) which generalized the field of tangent planes is B = γ1g1g2 + γ2g1g3 + γ3g2g3 (8) with γ1 = 1 δ γ2 = √∑3 k=1 φ2 k,x2 δ γ3 = − √∑3 k=1 φ2 k,x1 δ δ = 1 + 2∑ j=1 3∑ k=1 φ2 k,xj (9) The operator B· acting on the sections of S(E3(Ω)), where · denotes the Clif- ford multiplication, is represented by the 2 × 2 complex matrix field B· = ( iγ1 −γ2 − iγ3 γ2 − iγ3 −iγ1 ) (10) Since B2 = −1 this operator has two eigenvalue fields i and −i. Consequently, every section σ of S(E3(Ω)) can be decomposed into σ = σB + + σB − where σB + = 1 2 (σ − iB · σ) σB − = 1 2 (σ + iB · σ) (11) The Riemannian Fourier transform The Riemannian Fourier transform of σφ is given by .. Usual Fourier FBσφ(u1, u2) = ∫ R2 χu1,u2,B(x1,x2)(−x1, −x2) · σφ(x1, x2)dx1dx2 (12) .. Spin characters .. Image section The decomposition of a section σφ associated to a color image leads to φ(x1, x2) = ∫ R2 3∑ k=1 [ φk+ (u1, u2)eu1,u2 (x1, x2) √ 1 − γ1 2 +φk− −1 (u1, u2)e−u1,−u2 (x1, x2) √ 1 + γ1 2 ] ⊗ gkdu1du2 (13) where φk+ = φk √ 1 − γ1 2 φk− = φk √ 1 + γ1 2 (14) Low-pass filtering Figure: Left : Original - Center : + Component - Right : - Component Low-pass filtering (a) + Component (b) Variance : 10000 (c) Variance : 1000 (d) Variance : 100 (e) Variance : 10 (f) Variance → 0 Figure: Low-pass filtering on the + component Low-pass filtering (a) - Component (b) Variance : 10000 (c) Variance : 1000 (d) Variance : 100 (e) Variance : 10 (f) Variance → 0 Figure: Low-pass filtering on the - component Thank you for your attention !