A Spherical Harmonic Expansion of the Hilbert Transform on S 2. Oliver Fleischmann Cognitive Systems Group Kiel University.

Transcription

1 A Spherical Harmonic Expansion of the Hilbert Transform on S 2 Oliver Fleischmann Cognitive Systems Group Kiel University München 2010

2 Motivation Goal: Extract local structural information from signals on the two-sphere

3 Motivation Goal: Extract local structural information from signals on the two-sphere

4 Motivation Goal: Extract local structural information from signals on the two-sphere

5 Motivation Analytic and monogenic signal: extract local structural information like local phase, orientation and amplitude But: They act on signals in R and R 2 respectively Therefore: Develop a monogenic signal on the two-sphere S 2

6 Classical Hilbert Transform H[f](x) =f(x) 1 πx = 1 π P.V. + F[H[f]](u) = i sgn(u)f[f](u) f(x) y x dy A cos(kx) 1 πx = A sin(kx)

7 Classical Analytic Signal Complex signal representation encoding local amplitude and phase f a (x) =f(x)+ih[f](x) ϕ(x) = i Ae iϕ(x) A cos(ϕ(x)) A sin(ϕ(x))

8 Classical Analytic Signal f a (x) =f(x)+ih[f](x) Features Local Phase Local Amplitude ϕ(x) = arctan f(x) H[f](x) A(x) = f(x) 2 + H[f](x) 2

9 Riesz Transform in R n - Spatial Domain R i [f](x) =f(x) x i 2π x n+1 = 1 2π P.V. R 2 x i y i f(y)dy x y n+1 x 1 2π x 2+1 x 2 2π x 2+1

10 Riesz Transform in R n - Fourier Domain F[R i [f]](u) =i u i u F[f(x)] u 1 u = cos(θ) u 2 u = sin(θ)

11 Riesz Transform in R n = A cos(ϕ(x, y)) x 1 2π x 2+1 A cos(θ) sin(ϕ(x, y)) = A cos(ϕ(x, y)) x 2 2π x 2+1 A sin(θ) sin(ϕ(x, y))

12 Monogenic Signal (Sommer, Felsberg) Quaternion valued signal representation encoding local orientation, phase and amplitude f m (x) =f(x)+ir x1 [f](x)+jr x2 [f](x) x 2 θ x 1 +i +j A cos(ϕ(x, y)) A cos(θ) sin(ϕ(x, y)) A sin(θ) sin(ϕ(x, y))

13 Monogenic signal - Features in R 2 (Sommer, Felsberg) Local Orientation: θ = arctan R2 [f](x) R 1 [f](x) R2 [f](x) 2 + R 1 [f](x) 2 Local Phase: φ = arctan 2 f(x) Local Amplitude: A = R 2 [f](x) 2 + R 1 [f](x) 2 + f(x) 2

14 Monogenic Signal: Example Features Original Local Orientation Local Phase

15 Monogenic Signal: Example Features Original Local Orientation Local Phase

16 Monogenic Signal: Example Features Original Local Orientation Local Phase

17 Monogenic Signal: Example Features Original Local Orientation Local Phase

18 From the plane to the two-sphere What about a monogenic signal on? We need a Hilbert transform on the two sphere

19 Clifford Analysis R 0,n+1 : Clifford algebra over R 0,n+1 with signature (0,n) Dirac operator: x = n i=0 e i xi f monogenic x f =0

20 The Cauchy Transform C[f](x) = G E n (x y)f(y)n(y)ds(y), x G E n (x) = 1 A n+1 x x n+1, Cauchy kernel G R n+1 n(y): outward pointing normal at y ds surface element on G

21 The Cauchy Transform - Properties C[f](x) = G E n (x y)f(y)n(y)ds(y), x G n(y) given a function on the boundary G G G C[ ] generates a monogenic function in G, i.e. C[f](x) =0

22 The Cauchy Transform - Properties C[f](x) = 1 2 P[f](x)+1 2 Q[f](x) = 1 2 P[f](x)+1 2 P[H[f]](x) P[f](x): Poisson integral in G, P[f]x =0 Q[f](x): Conjugate Poisson integral in G, Q[f]x =0

23 The Cauchy Transform - Properties C[f](x) = 1 2 P[f](x)+1 2 Q[f](x) = 1 2 P[f](x)+1 2 P[H[f]](x) A monogenic signal on G arises as the non-tangential boundary value of the Cauchy transform in G lim x ξ P[f](x) =f(ξ) and lim x ξ Q[f](x) =H[f](ξ)

24 The Cauchy Transform - Properties A monogenic signal on G arises as the non-tangential boundary value of the Cauchy transform in G lim x ξ P[f](x) =f(ξ) and lim x ξ Q[f](x) =H[f](ξ) S 1 ξ x B 1

25 The Cauchy Transform - Properties A monogenic signal on G arises as the non-tangential boundary value of the Cauchy transform in G lim x ξ P[f](x) =f(ξ) and lim x ξ Q[f](x) =H[f](ξ) S 1 x ξ B 1

26 The Cauchy Transform - Properties The Analytic Signal and the Monogenic Signal are just special cases of the non-tangential boundary values of the Cauchy transform Analytic Signal G R 2 + G R Monogenic Signal Spherical Monogenic signal R 3 + R 2 B 2 S 2

27 The Hilbert Transform on S 2 Should we use the Cauchy or the Hilbert transform? Hilbert transform: Only useful results if the signal model is present (i.e. exactly one frequency) Cauchy transform: Embeds the monogenic signal in a linear scale space, the Poisson scale space Use the Cauchy transform for multiscale signal analysis to select certain frequencies

28 Example: Cauchy transform in R 2 + P x0 [f](x) P x0 [R x1 [f]](x) P x0 [R x2 [f]](x)

29 Example: Cauchy transform in R 2 + P x0 [f](x) P x0 [R x1 [f]](x) P x0 [R x2 [f]](x)

30 The Hilbert Transform on S 2 How to interpret the Hilbert/Cauchy transform on S 2? Remember: In the case of R n the Hilbert transform acts as a simple multiplier in the Fourier domain Fourier representation of the Hilbert transform on S 2?

31 The Cauchy transform on S 2 as a convolution First step: Interpret the Cauchy transform as a group convolution over SO(3) (Directional correlation) C[f](rη) = S 2 E(rη ω)ωf(r 1 (ρ)ω)ds(ω) =R(ρ)[h] f Local feature analysis, treat every point as the northpole η of the sphere and evaluate the Cauchy transform for the rotated function

32 Directional Correlation on S 2 - Fourier series R(ρ)[h] f = l N l l [R[h] l f] m,n Dl m,n(ρ) m= l n= l with lm,n [R[h] f] = ĥl,n ˆf l,m D l m,n(ρ): Wigner-D function, SO(3) basis functions ĥl,n : Spherical harmonic coefficients of the Cauchy kernel ˆf l,m : Spherical harmonic coefficients of the target function

33 Fourier Series on S 2 Every function f L 2 (S 2 ) can be expanded into a series of spherical harmonics l ˆf l,m Y l,m (θ, ϕ),f L 2 (S 2 ) f(ω) = l N m= l =( ( + i + + i ( ( 0.5 Y 3, 2 (ω) 2.1 Y 4,2 (ω)

34 Fourier Series on SO(3) Every function f L 2 (SO(3)) can be expanded into a series of Wigner-D functions f(ρ) = l N 2l +1 8π 2 l m= l l n= l ˆf l m,nd l m,n(ρ),f L 2 (SO(3)) D l m,n(ρ) =D l m,n(θ, ϕ, ψ) =e imψ d l m,n(cos θ)e inϕ

35 The Cauchy transform on S 2 - Fourier series C[f](x) =C[f](rξ) = 1 P r (ξ, ω)f(ω)ds(ω) 2A 3 S Q r (ξ, ω)f(ω)ds(ω) 2A 3 S 2 P r (x, ω) = 1 x 2 x ω 3 Q r (x, ω) = 1+ x 2 +2xω x ω 3 Find a Fourier series expansion of the Poisson- and the conjugate Poisson kernel in the unit ball

36 Poisson Kernel - Spherical Harmonic Coefficients P r (ξ, ω) = 1 x 2 x ω 3 = (2k + 1)r k P n (ξ, ω) k=0 Evaluate at x = rη = [0, 0,r] T,r (0, 1) [P r ] l,m = S 2 P r (rη, ω)y l,m (ω)ds(ω) = r l for m =0 0 else Lowpass behaviour, acting on the frequencies l

37 Poisson Transform - Example f(ω) P 0.9 [f](ω) P 0.8 [f](ω) fade conjugate kernels P 0.9 (ξ, ω) P 0.8 (ξ, ω)

38 Conjugate Poisson Kernel - SH coefficients Conjugate Poisson kernel splits into scalar an bivector parts Q r (x, ω) =Q (0) r (x, ω) 2rQ (1) r (x, ω)e 13 2rQ (2) r (x, ω)e 23 [Q (1) r ] = l,±1 4π l(l + 1) (2l + 1) rl (2) [Q r ] = i l,±1 4π l(l + 1) (2l + 1) rl

39 The Cauchy transform on S 2 P r [f](ξ) Q (1) r [f](ξ) Q (2) r [f](ξ)

40 The Cauchy transform on S 2 P r [f](ξ) Q (1) r [f](ξ) Q (2) r [f](ξ)

41 Kernel SH coeffients m m m l l [P r ] (1) l,m [Q r ] i[q (2) l,m r ] l,m l

42 Complete filter set R(ρ)[P r (1) ] f = l N R(ρ)[Q (1) r ] f = l N R(ρ)[Q (2) r ] f = l N l r l fl,m Dm,0 l (ρ) = m= l l N m= l l 4π l(l + 1) (2l + 1) rl fl,m Dm, 1 l (ρ) Dl m,1 (ρ) m= l l 4π l(l + 1) i (2l + 1) rl fl,m Dm, 1 l (ρ)+dl m,1 (ρ) m= l l r l fl,m Y l,m (θ, ϕ)

43 Complete filter set Filterset acts as a derivative operator on the Poisson equation solution R(ρ)[P r ] f =2R(ρ)[Q (0) r ] f = l N l m= l r l ˆfl,m Y l,m R(ρ)[Q (1) r ] f =2 l N l m= l r l ˆfl,m θ Y l,m R(ρ)[Q (2) r ] f =2 l N l m= l r l ˆfl,m 1 sin θ ϕ Y l,m

44 Features Local orientation in the tangent plane β = arctan 2(R(0, 0, 0)[Q (2) r ] f, R(0, 0, 0)[Q (1) r ] f) Local phase in the tangent plane φ = arctan 2( (R(0, 0, 0)[Q (1) r ] f) 2 +(R(0, 0, 0)[Q (2) r ] f) 2, R(0, 0, 0)[Q (0) ]f) r Local amplitude in the tangent plane A = (R(0, 0, 0)[Q (1) r ] f) 2 +(R(0, 0, 0)[Q (2) r ] f) 2 +(R(0, 0, 0)[Q (0) r ] f) 2

45 Examples Original Orientation Phase

46 Examples Original Orientation Phase

47 Examples Original Orientation Phase

48 Examples Original Orientation Phase

49 Conclusion The Cauchy transform on arbitrary closed surfaces with smooth boundary leads to Hilbert transforms in a Poisson scale space on that surface Monogenic signals for different surfaces like the two-sphere arise Nonetheless: Interpretation of the structural information depends on the surface