Starting point : Multicomponent signals (1) L s ( t ) = a ( t ) - - PowerPoint PPT Presentation

▶
starting point multicomponent signals 1
SMART_READER_LITE
LIVE PREVIEW

Starting point : Multicomponent signals (1) L s ( t ) = a ( t ) - - PowerPoint PPT Presentation

Feb 28, 2023 338 likes •470 views

Starting point : Multicomponent signals (1) L s ( t ) = a ( t ) cos( ( t )) , t R Transform ee de Riesz multi- echelles et =1 a limage 1 Applications ` - Decomposition problem : extraction of the di ff erent

slide-1
SLIDE 1

Transform´ ee de Riesz multi-´ echelles et Applications ` a l’image 1

Val´ erie Perrier

Laboratoire Jean Kuntzmann Universit´ e de Grenoble-Alpes

Collaborateurs : Marianne Clausel, Sylvain Meignen, K´ evin Polisano (LJK), Laurent Desbat (TIMC-Imag) and Thomas Oberlin (IRIT, Toulouse)

  • 1. Journ´

ee ” Temps-Fr´ equence et Non-Stationnarit´ e” , Marseille, 19 juin 2015

1

Starting point : Multicomponent signals (1)

s(t) =

L

  • ℓ=1

aℓ(t) cos(ϕℓ(t)) , t ∈ R

  • Decomposition problem : extraction of the different components (IMFℓ).
  • Demodulation problem for a mode : estimation of the instantaneous

amplitudes aℓ(t), phases ϕℓ(t), and frequencies ϕ′

ℓ(t).

4 6 8 10 x 10

−3

−0.2 −0.1 0.1 time (s) 2 4 6 8 10 x 10

−3

−0.1 −0.05 0.05 0.1 0.15 IMF 1 IMF 2 IMF 3

Bat echolocation call signal

Starting point : Multicomponent signals (2)

s(t) =

L

  • ℓ=1

aℓ(t) cos(ϕℓ(t)) , t ∈ R

  • Decomposition problem : extraction of the different components.
  • Demodulation problem for a mode : estimation of the instantaneous

amplitudes aℓ(t), phases ϕℓ(t), and frequencies ϕ′

ℓ(t).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −10 10 t s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 5 t s1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 2 t s2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 2 t s3

0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 time frequency (Hz) 0.5 1 1.5 2 2.5 3 s3 s2 s1

Amplitude

Academic multicomponent signal

Decomposition/demodulation of signals in AM-FM modes 2

Multicomponent signal : s(t) =

L

  • ℓ=1

aℓ(t) cos(ϕℓ(t))

  • aℓ(t) cos(ϕℓ(t)) : Intrinsec Mode Function (IMFℓ), (decomposition pb).
  • aℓ : amplitude, ϕ′

ℓ : instantaneous frequency (demodulation pb).

The problem of finding the aℓ, ϕℓ is ill-posed in general. Under suitable assumptions (separation of modes in Fourier domain, slowly variations of a′

ℓ, ϕ′ ℓ..), several methods

have been developed in the 90th by the wavelet community, based on reallocation techniques in a time-frequency representation :

  • Reassignment method [Auger-Flandrin 1995],
  • Squeezing method [Daubechies-Maes 1996],
  • Wavelet ridges [Carmona-Hwang-Torr´

esani 1997, 1999]. Another point of view :

  • Empirical Mode Decomposition (EMD) and Hilbert-Huang Transform

(HHT) [Huang et al 1998] Review : SPM 2013, [Auger et al, SPM 2013]

2. ANR Astres 2013-2016 (coord. P. Flandrin)

slide-2
SLIDE 2

Motivation : Image decomposition/demodulation AM-FM

f (x) = a(x) cos(ϕ(x)) + f1(x) , x ∈ R2

  • Decomposition problem : extraction of the different components.
  • Demodulation problem for a mode : estimation of the local

amplitude a(x), phase ϕ(x), frequency ∇ϕ(x).

Motivation : anisotropic texture analysis

Texture with prescribed orientation Locally parallel textures [Polisano et al 2014] [Maurel-Aujol-Peyre 2011]

Outline

1

Definitions Hilbert transform and Analytic signal Riesz transform and Monogenic signal

2

Computation of the Riesz Transform via the Fourier domain via the Radon domain In the direct space via a multiscale decomposition

3

Applications Local orientations from local Radon data Decomposition/demodulation of Multicomponent Images

4

Conclusion

Outline

1

Definitions Hilbert transform and Analytic signal Riesz transform and Monogenic signal

2

Computation of the Riesz Transform via the Fourier domain via the Radon domain In the direct space via a multiscale decomposition

3

Applications Local orientations from local Radon data Decomposition/demodulation of Multicomponent Images

4

Conclusion

slide-3
SLIDE 3

Hilbert transform and Analytic signal

  • 1D : Hilbert Transform. Let f : R → R

Hf in time domain : Hf (t) = 1 π vp 1 t

  • ∗ f
  • (t) = lim

ε→0

  • 1

π

  • |t−s|>ε

f (s) t − s ds

  • for a.e. t ∈ R.

Hf in Fourier domain : Hf (ξ) = −i

ξ |ξ|

f (ξ) = −i sgn(ξ) f (ξ)

  • Analytic signal (complex) : F(t) = f (t) + i Hf (t)

(ˆ F = 0 on R−) AM-FM analysis : F(t) = A(t) eiϕ(t)

  • Instantaneous amplitude : A(t) = |F(t)|
  • Instantaneous frequency : ω(t) = ϕ′(t)
  • Example : f (t) = A cos(ωt). Then H(f ) = A sin(ωt) and F(t) = Aeiωt

− → A(t) = A, ϕ(t) = ωt

Riesz transform and Monogenic Signal [Felsberg-Sommer 2001]

  • 2D (or n-D) : Riesz Transform ⃗

Rf =

  • R1f

R2f

  • ⃗

Rf in space domain : Rif (x) = lim

ε→0+

  • 1

π

  • ∥x−y∥>ε

(xi − yi) ∥x − y∥3 f (y) dy

  • ⃗

Rf in Fourier domain : Rif (ξ) = −i

ξi ∥ξ∥

f(ξ), for i = 1, 2.

  • Monogenic (quaternionique) signal : Mf =

f ⃗ Rf

  • = f + i R1f + j R2f

AM-FM analysis : Mf = A(x) eϕ(x)nθ(x)

  • Local amplitude : A(x) = |Mf (x)|
  • Local frequency : ω(x) = ∇ϕ(x)
  • Local orientation : θ(x) (nθ = cos θ i + sin θ j)

(link with the orientation of ∇ϕ(x) ?)

Monogenic signal

Mf = „ f ⃗ Rf « = A(x) eϕ(x)nθ(x) = A(x) @ cos(ϕ(x)) sin(ϕ(x)) cos(θ(x)) sin(ϕ(x)) sin(θ(x)) 1 A

  • Example : f (x) = A0 cos(k · x). Let k = (k1, k2), θ0 = Arctan( k2

k1 ).

Then Rf (x) = A0 sin(k · x) cos θ0 sin(k · x) sin θ0

  • = A0

k |k| sin(k · x) and Mf (x) = f (x) Rf (x)

  • = A0

⎛ ⎝ cos(k · x) sin(k · x) cos θ0 sin(k · x) sin θ0 ⎞ ⎠ = A0 e(k·x)(cos θ0 i+sin θ0 j) Finally A(x) = A0, ϕ(x) = k · x, θ(x) = θ0 = Arctan(k2/k1) .

Outline

1

Definitions Hilbert transform and Analytic signal Riesz transform and Monogenic signal

2

Computation of the Riesz Transform via the Fourier domain via the Radon domain In the direct space via a multiscale decomposition

3

Applications Local orientations from local Radon data Decomposition/demodulation of Multicomponent Images

4

Conclusion

slide-4
SLIDE 4

Computation of the Riesz Transform (1)

  • Via the Fourier domain :

Rif (ξ) = −i ξi

|ξ|

f(ξ), for i = 1, 2.

Shepp and Logan phantom first component R1f second component R2f

Fourier based Riesz computation : Involves a non local filtering (pb at frequency 0), Requires the knowledge of the whole image, Computed using FFT, complexity : O(N log2(N))

Computation of the Riesz Transform (2)

  • Via the Radon domain [Felsberg 2002]

Medical Scan : X-ray tomography Godfrey N. Hounsfield (ingenior in electronic) and Allan M. Cormack (mathematician) : Nobel Prize in Medicine 1979.

Scan and illustration of its principle : X-ray taken around the patient.

Computation of the Riesz Transform (2)

  • Radon Transform :

The Radon transform of function f (x) is measured on each detector of direction ⃗ θ = (cos θ, sin θ), corresponding to the mean of f along lines Lθ,s of direction ⃗ θ⊥ = (− sin θ, cos θ) :

Rf (θ, s) = Rθf (s) =

  • Lθ,s

f (x) dℓ = +∞

−∞

f (s⃗ θ + t⃗ θ⊥) dt

Computation of the Riesz Transform (2)

  • Inverse Radon transform : filtered back projection (FBP)

f (x) = R−1 (Rf (θ, s)) = π +∞

−∞

  • Rθf (ω)|ω| e2iπω(x.⃗

θ)dω

  • dθ

(where d Rθf denotes the 1D Fourier transform of Rθf .

Remark : involves the non local ramp filter |ω|.

  • Original Radon-based Riesz formula [Felsberg 2002], [Soulard-Carr´

e 2012]

⃗ Rf (x) = R−1 HR⃗

θf

⃗ θ

  • (x)

due to : ⃗ Rf (x) = π +∞

−∞

  • Rθf (ω)(−i)sign(ω)
  • |ω|e2iπω(x·⃗

θ)dω

  • ⃗

θ dθ Remark : involves two non local operators : the Hilbert transform H and the inverse Radon transform R−1.

slide-5
SLIDE 5

Computation of the Riesz Transform (2)

  • Local Radon-based Riesz formula [Desbat-P 2015] Since :

⃗ Rf (x) = π +∞

−∞

  • Rθf (ω)(−iω)e2iπω(x·⃗

θ)dω

  • ⃗

θ dθ Then : ⃗ Rf (x) = − 1 2π π ∂Rf ∂s

  • θ, x · ⃗

θ

  • ⃗

θ dθ

Non local Radon based Riesz Local Radon based Riesz

Computation of the Riesz Transform (2)

  • Interest : local Riesz transform from local Radon data

Phantom and ROI Radon data Truncated Radon data

Local (R1f , R2f ) with NO ERROR

Computation of the Riesz Transform - In direct space (3)

  • Pyramidale decomposition [Burt-Adelson 1983]
  • Multiscale Riesz transform : in each band a,

Rifa(ξ) = −i

ξi |ka|

fa(ξ) → Rif ∼

−1 2πka ∂fa ∂xi

→ Complexity : linear ! [Wahdwa-Rubinstein-Durand-Freeman 2014] for video

magnification

Computation of the Riesz Transform - In direct space (4)

Monogenic Wavelet Transform - [Olhede-Metikas 2009], [Unser-VanDeVille 2009]

  • 2D directional CWT

cf (a, b, α) =

  • R2 f (x) ψa,b,α(x) dx ,

ψa,b,α(x) = 1 aψ

  • r−α

x − b a

  • If ψ is isotropic, ψa,b,α = ψa,b,0 = ψa,b. Denote cf (a, b) = cf (a, b, 0).
  • Isotropic CWT of the monogenic signal F = Mf = f + R1f i + R2f j

cF(a, b) = (cf + cR1f i + cR2f j) (a, b)

  • Monogenic Wavelet Transform (ψ real isotropic)

c(M)

f

(a, b, α) =

  • R2 f (x) (Mψ)a,b,α(x) dx

cF(a, b) =

  • 1

−rα

  • c(M)

f

(a, b, α)

  • Example : F(x) = A e(k·x)nθ, cF(a, b) = a

ψ(ak)

  • Ae(k·b)nθ
slide-6
SLIDE 6

Outline

1

Definitions Hilbert transform and Analytic signal Riesz transform and Monogenic signal

2

Computation of the Riesz Transform via the Fourier domain via the Radon domain In the direct space via a multiscale decomposition

3

Applications Local orientations from local Radon data Decomposition/demodulation of Multicomponent Images

4

Conclusion

Local orientations from local Radon data

Psychedelic Lenna [Unser-Van De Ville 2009]

(a) (b) (c)

(a) Psychedelic Lenna image (of size 512 × 512) with the considered ROI (materialized by a white circle) and (b) corresponding sinogram (discretized Radon data : equiangular and equispaced 806 × 512 samples on [0, π)× the diagonal of the image). (c) Truncated Radon projections : only the lines passing through the ROI are measured.

Riesz transform via local Radon formula

(R1f , R2f ) from full (top) and truncated (bottom) Radon data

  • Orientations

Decomposition/demodulation of Multicomponent Images

  • Decomposition : Expand an image s(x) into oscillating modes :

s(x) =

L

  • ℓ=1

dℓ(x) + rℓ(x) dℓ is an Intrinsinc Mode Function (IMF) (ex : Bidimensional Empirical Mode Decomposition : [Nunes et al 03,

Linderhed 09, Damerval-Meignen-Perrier 05, ...])

  • Demodulation : dℓ(x) = Aℓ(x) eϕℓ(x)nθℓ (x) using the monogenic signal Mdℓ of

the mode dℓ. Different approaches :

  • 2D extension of the Hilbert-Huang Transform (HHT) : [Huang-Kunoth 2012]

[Schmitt-Pustelnik-Borgnat-Flandrin-Condat 2014]

  • 2D synchrosqueezing (2 steps simultaneously) : [Clausel-Oberlin-P. 2014]
slide-7
SLIDE 7

Principle of the 1D synchrosqueezed Wavelet Transform

[Daubechies-Lu-Wu, ACHA 2011] Multicomponent AM-FM signal f Wavelet Transform Wf (a, b) ωf (a, b) (to approximate ϕ′

ℓ(b))

SST= Sf (k, b)

Principles of the SST 1D - [Daubechies-Lu-Wu, ACHA 2011]

  • Example :

f (x) = A cos(ωx) The CWT of its analytic signal : F(x) = A eiωx : WF(a, b) = 1 √a +∞

−∞

F(x)ψ x − b a

  • dx = A√a ˆ

ψ(aω) eiωb Then ∂bWF(a, b) = iωWF(a, b) → ω = −i ∂bWF (a; b) WF (a; b) and F(b) = λψ √a0 WF (a0, b) where a0 = k0 ω (k0 peak wavenumber of ψ)

Principles of the SST 1D - [Daubechies-Lu-Wu, ACHA 2011]

  • Example :

f (x) = A cos(ωx) The CWT of its analytic signal : F(x) = A eiωx : WF(a, b) = 1 √a +∞

−∞

F(x)ψ x − b a

  • dx = A√a ˆ

ψ(aω) eiωb Then ∂bWF(a, b) = iωWF(a, b) → ω = −i ∂bWF (a; b) WF (a; b) and F(b) = λψ √a0 WF (a0, b) where a0 = k0 ω (k0 peak wavenumber of ψ)

  • Monocomponent complex signal : f (x) = A(x) exp(iϕ(x)),

with slowly varying A, ϕ. Candidate instantaneous frequency : ωF(a, b) = −i ∂bWF (a, b) WF (a, b) , when |WF(a; b)| > ε Estimate : |ωF(a, b) − ϕ′(b)| < ε

with suitable conditions (Cε) on A, ϕ.

Principles of the SST 1D

  • Multicomponent complex signal f (x) : superposition of several IMFs

assumed to be slowly varying and well separated in time-frequency domain : f (x) =

L

  • ℓ=1

Aℓ(x) eiϕℓ(x) Synchrosqueezed Wavelet Transform (SST) :

Sδ

f ,ε(b, k) =

Z

|Wf (a,b)|>ε

Wf (a, b)1 δ h „k − ωf (a, b) δ « da a3/2

Estimate : lim

δ→0

1 cψ

  • {k; |k−ϕ′

ℓ(b)|≤ε}

Sδ

f ,ε(b, k)dk = Aℓ(b)eiϕℓ(b) + O(ε)

2D extension of the analytic signal − → monogenic signal

slide-8
SLIDE 8

Example (Fourier) SST 1D [Oberlin-Meignen-Perrier 2014]

STFT t η 0.2 0.4 0.6 0.8 100 200 300 400 500 −1 1 Mode 1 −1 1 Mode 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −1 1 Mode 3 t FSST 0.5 Error 1 0.5 Error 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5 Error 3 t

Academic signal

Example (Fourier) SST 1D [Oberlin-Meignen-Perrier 2014] [Auger

et al, SPM 2013]

4 6 8 10 x 10

−3

−0.2 −0.1 0.1 time (s) 2 4 6 8 10 x 10

−3

−0.1 −0.05 0.05 0.1 0.15 IMF 1 IMF 2 IMF 3 STFT t η 5 10 x 10

−3

2000 4000 6000 8000 10000 SST t η 5 10 x 10

−3

2000 4000 6000 8000 10000 VSST t η 5 10 x 10

−3

2000 4000 6000 8000 10000 RSTFT t η 5 10 x 10

−3

2000 4000 6000 8000 10000 OSST t η 5 10 x 10

−3

2000 4000 6000 8000 10000 GAR t η 5 10 x 10

−3

2000 4000 6000 8000 10000

Bat echolocation call signal

Principle of 2D SST : WT of the Monogenic Signal

[Clausel-Oberlin-P. 2014]

F(x) = A e(k·x)nθ k = (k1, k2)

  • Isotropic wavelet transform of F :

cF(a, b) = a ψ(ak)

  • Ae(k·b)nθ
  • For i = 1, 2 :

∂bi cF(a, b) = kinθ

  • a

ψ(ak) Ae(k·b)nθ

  • Instantaneous frequency k and orientation nθ :

k1nθ = ∂b1cF(a, b) × (cF(a, b))−1 k2nθ = ∂b2cF(a, b) × (cF(a, b))−1

  • On the ”

ridge”a = a0 = |k0|

|k|

F(b) = λψa0 cF(a0, b)

Principle of 2D SST : WT of the Monogenic Signal

[Clausel-Oberlin-P. 2014]

F(x) = A e(k·x)nθ k = (k1, k2)

  • Isotropic wavelet transform of F :

cF(a, b) = a ψ(ak)

  • Ae(k·b)nθ
  • For i = 1, 2 :

∂bi cF(a, b) = kinθ

  • a

ψ(ak) Ae(k·b)nθ

  • Instantaneous frequency k and orientation nθ :

k1nθ = ∂b1cF(a, b) × (cF(a, b))−1 k2nθ = ∂b2cF(a, b) × (cF(a, b))−1

  • On the ”

ridge”a = a0 = |k0|

|k|

F(b) = λψa0 cF(a0, b)

slide-9
SLIDE 9

Intrinsic Monogenic Mode Function (IMMF)

  • Intrinsic Monogenic Mode Function (IMMF) with accuracy ε > 0 :

F(x) = A(x)eϕ(x)nθ(x) with nθ(x) = cos(θ(x)) i + sin(θ(x)) j A, ϕ, nθ slowly varying functions (|∇A(x)|, |∇θ(x)|, |∇2ϕ(x)| < ε|∇ϕ(x)|).

  • A : local amplitude of F
  • ϕ, nθ : local scalar phase and orientation of F
  • ∇ϕ : instantaneous frequency.
  • Candidate to approximate the instantaneous frequency :

Λ1(a, b) = ∂b1cF(a, b) × (cF(a, b))−1 Λ2(a, b) = ∂b2cF(a, b) × (cF(a, b))−1

  • Estimate : for i = 1, 2,

|Λi(a, b) − ∂biϕ(b)nθ(b)| ≤ ε where |cF(a, b)| > ε

Monogenic Synchrosqueered Wavelet Transform (MSST)

  • Multicomponent signal F(x) : superposition of IMMFs of accuracy ε, well

separated in the space-frequency domain : F(x) =

L

  • ℓ=1

Aℓ(x) eϕℓ(x)nθℓ(x)

  • MSST= local CWT-reconstruction at fixed point b, in the ε-vicinity of the

estimated instantaneous frequencies (Λ1, Λ2) :

Sδ

F,ε(b, k, n) =

Z

|cF (a,b)|>ε

cF(a, b) 1 δ2 h „k1 − Re(n Λ1(a, b)) δ « h „k2 − Re(n Λ2(a, b)) δ « da a2 (h ∈ C ∞

c

s.t. R h = 1)

  • ℓth-IMMF estimate ( ˆ

ψ compactly supported) : lim

δ→0

2π ˜ Cψ Z

S1

Z

{k; maxi |ki n−∂bi ϕℓ(b)nθℓ(b)|≤ε}

Sδ

f ,ε(b, k, n)dkdn = Aℓ(b)eϕℓ(b)nθℓ(b) + O(ε) .

Decomposition/demodulation of Multicomponent Images

x y x y x y x y

f (x) = f1(x) + f2(x) + f3(x)

100 200 100 200 300 8 16 32 64 128 x y Frequency (Hz) 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Frequency (Hz) y 0.2 0.4 0.6 0.8 128 64 32 16 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 200 100 200 300 8 16 32 64 128 x y Frequency (Hz) 4 6 8 10 12 14 16 18 20 Frequency (Hz) y 0.2 0.4 0.6 0.8 8 16 32 64 128 0.5 1 1.5 2 2.5 3

Monogenic WT (modulus 3D, 2D slice) Monogenic SST (modulus 3D, 2D slice)

Monogenic Synchrosqueezed Wavelet Transform of Images

Multicomponent signal : reconstruction of modes

x y x y x y x y

f (x) = f1(x) + f2(x) + f3(x) Reconstructed modes = f1(x) (MSE=0.05) + f2(x) (MSE= 0.06) + f3(x) (MSE=0.034) 8 < : f1(x1, x2) = e−10((x1−0.5)2+(x2−0.5)2)) sin(10π(x2

1 + x2 2 + 2(x1 + 0.2x2))

f2(x1, x2) = 1.2 sin(40π(x1 + x2)) f3(x1, x2) = cos(2π(70x1 + 20x2

1 + 50x2 − 20x2 2 − 41x1x2))

slide-10
SLIDE 10

Comparison with EMD and EEMB

x y

f (x) = d3(x) + d2(x) + d1(x) Reconstructed modes = d3(x) (MSE=0.86) + d2(x) (MSE= 0.74) + d1(x) (MSE=0.59)

Extraction of AM-FM modes from a real image

x y x y x y

f (x) = f1(x) + f2(x)

x y normalized MSE : 0.30954 x y normalized MSE : 0.12807

Reconstructed modes = f1(x) (MSE=0.31) + f2(x) (MSE= 0.13)

Extraction of AM-FM modes from a real image

Superposition of Lenna and a fingerprint. Extracted fingerprint by : 2D MSST (left), first mode of EMD (middle), first mode of EEMD (right).

Outline

1

Definitions Hilbert transform and Analytic signal Riesz transform and Monogenic signal

2

Computation of the Riesz Transform via the Fourier domain via the Radon domain In the direct space via a multiscale decomposition

3

Applications Local orientations from local Radon data Decomposition/demodulation of Multicomponent Images

4

Conclusion

slide-11
SLIDE 11

Conclusion

Riesz transform : easy way to compute local phases and orientations of images, alternative to oriented Gabor filters. − → Medical imaging applications, 3D, ... 2D generalization of the Synchrosqueezed Wavelet Method in the Monogenic Signal framework

Conclusion

Riesz transform : easy way to compute local phases and orientations of images, alternative to oriented Gabor filters. − → Medical imaging applications, 3D, ... 2D generalization of the Synchrosqueezed Wavelet Method in the Monogenic Signal framework

  • MSST allows to link local orientations to instantaneous frequency : new way

for detection and characterization of anisotropy in images (application to textures) − → problems still remain when the modes are not well separated in frequency (only in orientation)

Conclusion

Riesz transform : easy way to compute local phases and orientations of images, alternative to oriented Gabor filters. − → Medical imaging applications, 3D, ... 2D generalization of the Synchrosqueezed Wavelet Method in the Monogenic Signal framework

  • MSST allows to link local orientations to instantaneous frequency : new way

for detection and characterization of anisotropy in images (application to textures) − → problems still remain when the modes are not well separated in frequency (only in orientation)

  • M. Clausel, T. Oberlin, V. Perrier, The Monogenic Synchrosqueezed Wavelet

Transform : A tool for the Decomposition/Demodulation of AM-FM images, ACHA (2014).

  • L. Desbat, V. Perrier, On locality of Radon to Riesz transform, preprint, submitted.