geometric analysis of bivariate signals€¦ · geometric analysis of bivariate signals lia meeting...

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)


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−


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


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


Introduction

Proposed framework





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


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


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)


Introduction

Proposed framework





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),


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


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


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.


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)


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

)


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


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


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)


Introduction

Proposed framework





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, ...


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(·)


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


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


Polarization spectrogram: two linear chirps

(a) (b) (c)

(d) (e) (f)

t

t

rotating orientation, null ellipticity

constant orientation, reversing ellipticity


A real world example (1)Salomon Island (1991) earthquake data


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


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


Introduction

Proposed framework





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)


BiSPy: a Python packagefor signal processing of bivariate signals

code – tutorials – documentation

github.com/jflamant/bispy/

Thank you for your attention


Appendices


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)


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

)


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

)


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


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


geometric analysis of bivariate signals€¦ · geometric analysis of bivariate signals lia meeting...

Documents