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Geometric analysis of bivariate signalsLIA Meeting – GIPSA-Lab, GrenobleNovember, 30th 2017
J. Flamant(1), P. Chainais (1), N. Le Bihan(2)
(1) CRIStAL, Univ. Lille and Centrale Lille(2) GIPSA-Lab, Grenoble
Bivariate signalsBivariate signalsx(t) = [u(t) v(t)]T ∈ R2, equivalently x(t) = u(t) + iv(t) ∈ C
e.g. seismic traces, wind velocities, polarimetric radar signals, etc.
A geometric signal processing framework for bivariate signals?
u
v
x(t)
Julien Flamant [email protected] Geometric analysis of bivariate signals 1/331/33
Monochromatic polarized signal representation
u
v
x(t)
χ
θ
φ
•
a cosχ
asin |χ|
⟲ χ > 0
⟳ χ < 0
Ellipse parameters• a ≥ 0 intensity• θ ∈ [−π/2, π/2] orientation• χ ∈ [−π/4, π/4] ellipticity• φ ∈ [0, 2π] phase
Vector representation (optics, seismology) Jones vector
x(t) =[Au cos(2πν0t+ Φu)Av cos(2πν0t+ Φv)
]F←→ X(ν) =
[Aue
iΦu
AveiΦv
]δν0(ν) + sym.
tan 2θ = 2AuAv
A2u −A2
v
cos(Φv −Φu) sin 2χ = 2AuAv
A2u +A2
v
sin(Φv −Φu)
Julien Flamant [email protected] Geometric analysis of bivariate signals 2/332/33
Monochromatic polarized signal representation
u
v
x(t)
χ
θ
φ
•
a cosχ
asin |χ|
⟲ χ > 0
⟳ χ < 0
Ellipse parameters• a ≥ 0 intensity• θ ∈ [−π/2, π/2] orientation• χ ∈ [−π/4, π/4] ellipticity• φ ∈ [0, 2π] phase
Vector representation (optics, seismology) Jones vector
x(t) =[Au cos(2πν0t+ Φu)Av cos(2πν0t+ Φv)
]F←→ X(ν) =
[Aue
iΦu
AveiΦv
]δν0(ν) + sym.
tan 2θ = 2AuAv
A2u −A2
v
cos(Φv −Φu) sin 2χ = 2AuAv
A2u +A2
v
sin(Φv −Φu)
Julien Flamant [email protected] Geometric analysis of bivariate signals 2/332/33
Monochromatic polarized signal representation
u
v
x(t)
χ
θ
φ
•
a cosχ
asin |χ|
⟲ χ > 0
⟳ χ < 0
Ellipse parameters• a ≥ 0 intensity• θ ∈ [−π/2, π/2] orientation• χ ∈ [−π/4, π/4] ellipticity• φ ∈ [0, 2π] phase
Complex representation (oceanography, SPTM) rotary components
x(t) = A+eiθ+ei2πν0t F←→ X(ν) = A+e
iθ+δν0(ν)+A−e
−iθ−e−i2πν0t +A−e−iθ−δ−ν0(ν)
θ = θ+ − θ−2
tanχ = A+ −A−A+ +A−
Julien Flamant [email protected] Geometric analysis of bivariate signals 3/333/33
Need for physically interpretable representations
Comments• no direct parametrization in terms of θ, χ• need for augmented representations [u(t), v(t)] or [x(t), x(t)]• positive and negative frequencies in complex representation
This work: a dedicated framework for bivariate signals
✓ directly interpretable ✓ theorems ✓ numerically efficient
dedicated framework←→
efficient, relevant generalization of ubiquitous signal processing tools
Julien Flamant [email protected] Geometric analysis of bivariate signals 4/334/33
Outline
Introduction
Proposed framework
Spectral analysis of bivariate signalsQuaternion power spectral densityDegree of polarizationExamples
Time-Frequency Analysis of bivariate signalsQuaternion embedding of bivariate signalsPolarization spectrogram
Conclusion and perspectives
Julien Flamant [email protected] Geometric analysis of bivariate signals 5/335/33
Introduction
Proposed framework
Spectral analysis of bivariate signalsQuaternion power spectral densityDegree of polarizationExamples
Time-Frequency Analysis of bivariate signalsQuaternion embedding of bivariate signalsPolarization spectrogram
Conclusion and perspectives
Julien Flamant [email protected] Geometric analysis of bivariate signals 6/336/33
Framework
2 key ingredients :{
QuaternionsQuaternion Fourier Transform (QFT)
Quaternionsnatural embedding of C→ quaternions H
4D algebra i2 = j2 = k2 = −1 B ij = k, ij = −ji B
complex subfields of H: Ci = Span {1, i}, Cj = Span {1, j}, . . .
Ingredient #1: Write a bivariate signal as
x(t) = u(t) + iv(t) ∈ Ci ⊂ H
Julien Flamant [email protected] Geometric analysis of bivariate signals 7/337/33
FrameworkIngredient #2: use a slightly different FT
Quaternion Fourier Transform (QFT)
X(ν) =∫x(t)︸︷︷︸∈Ci
e−j2πνt︸ ︷︷ ︸∈Cj
dt ∈ H
Monochromatic polarized signal
u
v
x(t)
χ
θ
φ
•
a cosχ
asin |χ|
⟲ χ > 0
⟳ χ < 0
x(t) = Ci
{aeiθe−kχej(2πν0t+φ)
}↕ QFT
X(ν) = aeiθe−kχejφδν0(ν) + sym.
polar form ↔ physical parameters
ACHA, 2017
Julien Flamant [email protected] Geometric analysis of bivariate signals 8/338/33
QFT propertiesEasy to compute
x(t) = u(t) + iv(t) QFT←−→ X(ν) = U(ν)︸ ︷︷ ︸1,j
+ iV (ν)︸ ︷︷ ︸i,k
For bivariate signals keep ν ≥ 0 only (i-Hermitian symmetry)
X(−ν) = −iX(ν)i, for x(t) ∈ Ci
2 invariants for finite energy signals (QFT Parseval theorem)∫ +∞
−∞|x(t)|2 dt =
∫ +∞
−∞|X(ν)|2 dν (energy)∫ +∞
−∞x(t)jx(t) dt =
∫ +∞
−∞X(ν)jX(ν)︸ ︷︷ ︸∈ span{i,j,k}
dν (geometry)
Julien Flamant [email protected] Geometric analysis of bivariate signals 9/339/33
Introduction
Proposed framework
Spectral analysis of bivariate signalsQuaternion power spectral densityDegree of polarizationExamples
Time-Frequency Analysis of bivariate signalsQuaternion embedding of bivariate signalsPolarization spectrogram
Conclusion and perspectives
Julien Flamant [email protected] Geometric analysis of bivariate signals 10/3310/33
Setting: stationary random bivariate signals
x(t) = u(t) + iv(t) is second order stationary (SOS)⇕
u(t) and v(t) are jointly SOS
The second-order moments thus satisfy:
x(t) is SOS
E [x(t)] = E [u(t)] + iE [v(t)] = m ∈ Ci, (m = 0)Ruu(t, τ) = E [u(t)u(t− τ)] = Ruu(τ),Rvv(t, τ) = E [v(t)v(t− τ)] = Rvv(τ),Ruv(t, τ) = E [u(t)v(t− τ)] = Ruv(τ),
Julien Flamant [email protected] Geometric analysis of bivariate signals 11/3311/33
Quaternion spectral density of random bivariate signalsHeuristic definitionCompute a truncated QFT to make x(t) of finite energy
XT (ν) = 1√T
∫ T
0x(t)e−j2πνtdt
QFT invariants: |XT (ν)|2 ∈ R+ XT (ν)jXT (ν) ∈ span{i, j,k}
Quaternion Power Spectral Density
Γxx(ν) = limT →∞
E[|XT (ν)|2
]︸ ︷︷ ︸
classical PSD
+ E[XT (ν)jXT (ν)
]︸ ︷︷ ︸
geometric PSD
Rigorous definition from spectral increments of x(t)based on QFT spectral representation theorem
IEEE TSP, 2017
Julien Flamant [email protected] Geometric analysis of bivariate signals 12/3312/33
Relation to Stokes parameters
PSD and Stokes parameters
Γxx(ν) = S0(ν) + iS3(ν) + jS1(ν) + kS2(ν)︸ ︷︷ ︸geometry/polarization
→ Frequency-dependent polarization description of bivariate signals
Poincaré sphere of polarization states• State Cartesian coordinates
Sα(ν)/S0(ν), α = 1, 2, 3
• Degree of polarization Φ
Φ(ν) = |iS3(ν) + jS1(ν) + kS2(ν)|S0(ν)
i, S3
S0
j, S1
S0
k, S2
S0
Φ
2θ
2χ
Julien Flamant [email protected] Geometric analysis of bivariate signals 13/3313/33
Degree of polarization
Φ(ν) = power of the polarized part at νtotal power at ν ∈ [0, 1]
Vocabulary• Φ(ν) = 0: unpolarized• 0 < Φ(ν) < 1: partially polarized• Φ(ν) = 1: fully polarizedDecomposition into Unpolarized and Polarized parts
Γxx(ν) = Γuxx(ν)︸ ︷︷ ︸
unpolarized part
+ Γpxx(ν)︸ ︷︷ ︸
polarized part
Φ(ν): balance between unpolarized and polarized parts.
Julien Flamant [email protected] Geometric analysis of bivariate signals 14/3314/33
Narrow-band partially polarized signal
quasi-monochromatic bivariate signal w/ constant polarization
ν > 0 θ(ν) = π/5 χ(ν) = π/8 Φ(ν) = 0.5
-0.2
0.0
0.2
u(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
v(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
u(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
v(t)
Julien Flamant [email protected] Geometric analysis of bivariate signals 15/3315/33
Narrow-band partially polarized signalUnpolarized part xu(t)
-0.2
0.0
0.2
uu(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
vu(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
uu(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
vu(t
)
Polarized part xp(t)
-0.2
0.0
0.2
up(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
vp(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
up(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
vp(t
)
Julien Flamant [email protected] Geometric analysis of bivariate signals 16/3316/33
Bivariate white Gaussian noise
Definition
w(t) = u(t)+iv(t) biv. WGN⇐⇒{u(t)v(t)
correlated univariate WGN
Quaternion power spectral density
Γww(ν) = σ2u + σ2
v︸ ︷︷ ︸total power
+ j(σ2u − σ2
v) + 2kρuvσuσv︸ ︷︷ ︸geometric part
Quick facts• PSD is constant• no i-component: x(t) is partially linearly polarized for all ν• unpolarized iff σu = σv and ρuv = 0• fully polarized iff |ρuv| = 1
Julien Flamant [email protected] Geometric analysis of bivariate signals 17/3317/33
Bivariate white Gaussian noise
Unpolarized/polarized decomposition example Φ = 0.8, θ = π/6
= +
= +
w[t]√1− Φ wu[t]
√Φeiθ wp[t]
bivariate WGN unpolarized WGN polarized WGN
Julien Flamant [email protected] Geometric analysis of bivariate signals 18/3318/33
Spectral analysis summary
A quaternion PSD for bivariate signals
Γxx(ν) = S0(ν)︸ ︷︷ ︸scalar
+ iS3(ν) + jS1(ν) + kS2(ν)︸ ︷︷ ︸vector of R3
• constructed from QFT invariants• geometric interpretation and Stokes parameters• identification of unpolarized and polarized partsAdditional results• conventional PSD estimators: periodogram, multitaper• quaternion autocovariance (QFT Wiener-Khintchine theorem)• bivariate fractional noise study
(stage M2R Jeanne Lefèvre)
Julien Flamant [email protected] Geometric analysis of bivariate signals 19/3319/33
Introduction
Proposed framework
Spectral analysis of bivariate signalsQuaternion power spectral densityDegree of polarizationExamples
Time-Frequency Analysis of bivariate signalsQuaternion embedding of bivariate signalsPolarization spectrogram
Conclusion and perspectives
Julien Flamant [email protected] Geometric analysis of bivariate signals 20/3320/33
Time-frequency analysis in the univariate setting (1)Fundamental property: Hermitian symmetry of FT of real signals
Analytic signal of a real signalOne to one corresp. between a real signal and its analytic signal
x(t) ∈ R←→ x+(t) ∈ C
a(t) cos[φ(t)]←→ a(t)eiφ(t)
a(t)
signal
... does not work when there are multiple components.Julien Flamant [email protected] Geometric analysis of bivariate signals 21/3321/33
Time-frequency analysis in the univariate setting (2)
Spectrogram → energy density in the time-frequency plane.
2000 4000 6000 8000 10000 12000 14000
-0.5
0
0.5
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time [s]
0
0.5
1
1.5
2
Fre
qu
en
cy [
Hz]
104
scalogram, Wigner-Ville distribution, ...
Julien Flamant [email protected] Geometric analysis of bivariate signals 22/3322/33
Framework
x(t) non-stationary bivariate signal
instantaneous or time-frequency polarization attributes
x(t) deterministic bivariate signal
Φx(·) = 1 S20(·) = S2
1(·) + S22(·) + S2
3(·)
Julien Flamant [email protected] Geometric analysis of bivariate signals 23/3323/33
Quaternion embedding of bivariate signalsx(t) = u(t) + iv(t) X(−ν) = −iX(ν)i (i-Hermitian symmetry)
Quaternion embeddingOne-to-one correspondence
bivariate signal←→ quaternion embeddingx(t) ∈ Ci ←→ x+(t) ∈ H
Polar form: instantaneous attributes
x+(t) = a(t)︸︷︷︸amplitude
× eiθ(t)e−kχ(t)︸ ︷︷ ︸geometry
× ejφ(t)︸ ︷︷ ︸phase
a(t) ≥ 0θ(t) ∈ [−π/2, π/2]χ(t) ∈ [−π/4, π/4]φ(t) ∈ [−π, π]
Canonical quadruplet
Julien Flamant [email protected] Geometric analysis of bivariate signals 24/3324/33
Physical interpretationBivariate signal structure
x(t) = Ci{x+(t)} = a(t)eiθ(t) [cosφ(t) cosχ(t) + i sinφ(t) sinχ(t)]
0
π/2
π
3π/2θ(t)
0
π/8
π/4χ(t)
[arb
. uni
ts]
ϕ ′(t)
(a) (b)
t
t
t
bivariate linear chirp w/ orientation and ellipticity modulationJulien Flamant [email protected] Geometric analysis of bivariate signals 25/33
25/33
Polarization spectrogrammulticomponent bivariate signals −→ generalization
Quaternion Short Term Fourier TransformExtend the STFT to the QFT setting
Sx(t, ν) =∫x(u)︸ ︷︷ ︸∈Ci
g(u− t)︸ ︷︷ ︸∈R
exp(−j2πνu)︸ ︷︷ ︸∈Cj
du
|Sx(t, ν)|2 → Time-frequency energy densitySx(t, ν)jSx(t, ν)→ Time-frequency Stokes parameters S1, S2, S3
Theorems{
inversionconservation: energy geometry/polarization
Julien Flamant [email protected] Geometric analysis of bivariate signals 26/3326/33
Polarization spectrogram: two linear chirps
(a) (b) (c)
(d) (e) (f)
t
t
rotating orientation, null ellipticity
constant orientation, reversing ellipticity
Julien Flamant [email protected] Geometric analysis of bivariate signals 27/3327/33
A real world example (1)Salomon Island (1991) earthquake data
Julien Flamant [email protected] Geometric analysis of bivariate signals 28/3328/33
A real world example (2)
0 500 1000 1500 2000
Time [s]
0
5
10Fr
equency
[1
0−
2 H
z]
S1
0 500 1000 1500 2000
S2
0 500 1000 1500 2000
S3
-1 0 1
0 500 1000 1500 2000
Time [s]
0
5
10
Frequency
[1
0−
2 H
z]
Time-Frequency energy density
0 500 1000 1500 2000
Instantaneous orientation
0 500 1000 1500 2000
Instantaneous ellipticity−π
2 0π2−π
4 0π4
(a) (b) (c)
(d) (e) (f)
t
t
Julien Flamant [email protected] Geometric analysis of bivariate signals 29/3329/33
Time-frequency summary
Novel and generic approach to time-frequency-polarization analysis• Quaternion embedding x(t) ∈ Ci ↔ x+(t) ∈ H• Time-frequency-polarization analysis (Q-STFT)
✓ novel representations ✓ theorems ✓ numerically efficient
• Time-scale-polarization analysis (Q-CWT)ACHA, 2017
Further developments (in progress)• Quaternion Wigner-Ville distribution• Cohen class• Extension to non-stationary random signals
Julien Flamant [email protected] Geometric analysis of bivariate signals 30/3330/33
Introduction
Proposed framework
Spectral analysis of bivariate signalsQuaternion power spectral densityDegree of polarizationExamples
Time-Frequency Analysis of bivariate signalsQuaternion embedding of bivariate signalsPolarization spectrogram
Conclusion and perspectives
Julien Flamant [email protected] Geometric analysis of bivariate signals 31/3331/33
Unifying framework for bivariate signals
ubiquitous SP tools ←→ relevant physical parameters
✓ geometric interpretations ✓ theorems ✓ numerically efficient
generic and nonparametric
Perspectives• Linear filtering theory for bivariate signals (soon)
physical interpretability spectral synthesis Wiener filtering• Extension to n-D bivariate signals• Extension to multivariate signals (PhD Jeanne Lefèvre)
Julien Flamant [email protected] Geometric analysis of bivariate signals 32/3332/33
BiSPy: a Python packagefor signal processing of bivariate signals
code – tutorials – documentation
github.com/jflamant/bispy/
Thank you for your attention
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Appendices
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Narrow-band partially polarized signal
quasi-monochromatic bivariate signal w/ constant polarization
ν > 0 θ(ν) = π/5 χ(ν) = π/8 Φ(ν) = 0.5
-0.2
0.0
0.2
u(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
v(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
u(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
v(t)
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Narrow-band partially polarized signalUnpolarized part xu(t)
-0.2
0.0
0.2
uu(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
vu(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
uu(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
vu(t
)
Polarized part xp(t)
-0.2
0.0
0.2
up(t)
0 200 400 600 800 1000
samples
-0.2
0.0
0.2
vp(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
up(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
vp(t
)
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Narrow-band partially polarized signalRotary components
-0.2
0.0
0.2
u+(t)
0 200 400 600 800 1000
-0.2
0.0
0.2
v+(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
u+(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
v+(t
)
-0.2
0.0
0.2
u−(t)
0 200 400 600 800 1000
-0.2
0.0
0.2
v−(t)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
u−(t)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
v−(t
)
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Time-scale analysis of bivariate signalsThe Q-STFT obeys the same limitations as in the classical setting−→ towards polarization wavelets?
Quaternion Continuous Wavelet TransformLet ψ ∈ L2(R,Cj) s.t. Ψ(ν) = 0 for ν < 0.
Wx(t, s) =∫x(u)︸ ︷︷ ︸∈Ci
1√sψ
(u− ts
)︸ ︷︷ ︸
∈Cj
dt
|Wx(t, s)|2 → Time-scale energy densityWx(t, s)jWx(t, s)→ Time-scale Stokes parameters S1, S2, S3
Theorems{
inversionconservation: energy geometry/polarization
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33
Polarization scalogram: two hyperbolic chirps
0 0.25 0.5 0.75 1
Time [s]
3
4
5
6
7
8
9
−lo
g2(s
)
S1
0 0.25 0.5 0.75 1
S2
0 0.25 0.5 0.75 1
S3
-1 0 1
0 0.25 0.5 0.75 1
Time [s]
3
4
5
6
7
8
9
−lo
g2(s
)
Time-scale energy density
0 0.25 0.5 0.75 1
Instantaneous orientation
0 0.25 0.5 0.75 1
Instantaneous ellipticity−π
2 0π2−π
4 0π4
(a) (b) (c)
(d) (e) (f)
t
t
Julien Flamant [email protected] Geometric analysis of bivariate signals 33/3333/33