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Statistical Invariances and Geometry Modeling for the
Analysis of TEXtures (SIGMA-TEX)-Application to high
resolution Transmission Electron Microscopy (TEM)
Imaging
LISTIC Seminar Presentation
Zhangyun TAN1
Abdourrahmane M. ATTO1
Olivier ALATA2
Maxime MOREAUD3
1
LISTIC, University of Savoie Mont Blanc
2
Hubert Curien Laboratory, University Jean Monnet of Saint Etienne
3
IFP Energies nouvelles
July 9, 2015
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 1 / 38
Transmission Electron Microscopy (TEM)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 2 / 38
Transmission Electron Microscopy (TEM)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 3 / 38
Catalyst composition [8] and TEM image
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 4 / 38
Carbon fringes morphology characterization existed,
problem and objective
Carbon fringes morphology
characterization existed [9]
fringe length
distance between carbon
fringes
aligned carbon fringes
parallel carbon fringes
discrimination between
aligned and parallel fringes
Problem and Objective
How to apply mathematics
models to carbon fringes
morphology characterization?
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 5 / 38
Problem and Objective
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 6 / 38
Table of Contents
1 Mathematics modeling
Texture synthesis from model GFBF and AR
ARFBF modeling
Hurst parameter estimation
2-factor FBF modeling
2 Experimental Results
Hurst parameter estimation
Location of the pole (u∗
, v∗
)
3 Application to TEM images
Discussion
TEM image characterization procedure
Experimental Results for TEM images
4 Conclusion, Publication and Perspective
5 Reference
6 Question
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 7 / 38
Texture synthesis from k-factor GFBF (Generalized
Fractional Brownian Field) εHk
for different values of k
(see [5])
Poles in [0, π/2] × [0, π/2] and Hurst parameters in ]0, 0.5[ generated from
random variables distributed as Gaussian, Uniformly and Gamma respectively
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 8 / 38
Texture synthesis from AR (Auto Regressive) with known
order (M1, M2) (see [6])
First row: textures synthesis in the spatial domain
Second row: respective spectrum SAR detailled in the following part
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 9 / 38
ARFBF: AR model A = A(x, y) and its spectrum SAR (see
[7])
AR model A
A(x, y) = −
m1,m2∈D
βm1,m2
A(x − m1, y − m2) + E(x, y) (1)
DM1,M2
= {(m1, m2) ∈ Z2
, 0 ≤ m1 ≤ M1, 0 ≤ m2 ≤ M2, (m1, m2) = (0, 0)} (2)
Explanation
(x, y) ∈ Z2
D ⊂ Z2
: prediction support
E(x, y): white Gaussian noise
β(m1, m2): coefficients
(m1, m2): index
(M1, M2): order
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 10 / 38
ARFBF: AR model A = A(x, y) and its spectrum SAR (see
[7])
AR model A
A(x, y) = −
m1,m2∈D
βm1,m2
A(x − m1, y − m2) + E(x, y) (3)
(x, y) ∈ Z2
and D ⊂ Z2
: 2D prediction support
E(x, y): 2D white Gaussian noise and β(m1, m2): coefficients
(m1, m2): index and (M1, M2): order
Spectrum SAR
SAR (u, v) =
(σe)2
|1 +
m1,m2∈D
βm1,m2
e−i2πum1 e−i2πvm2 |2
(4)
(σe)2
: variance of E = E(x, y)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 11 / 38
ARFBF: FBF model F = FH(x, y) and its spectrum SFBF
(see [4])
Auto-correlation function RF
RF (x, y, s, t) =
σ2
2
((x2
+ y2
)H
+ (s2
+ t2
)H
− ((x − s)2
+ (y − t)2
)H
) (5)
Spectrum SFBF
SFBF (u, v) = ξ(H)
1
(u2 + v2)(H+1)
= ξ(H)
1
f
(2H+2)
(6)
ξ(H) = 2−(2H+1)
π2
σ2
sin(πH)Γ2(1+H) , f = (u2 + v2) and Γ: the gamma function and
Hurst parameter H: 0 < H < 1 and σ2
: variance of white Gaussian noise
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 12 / 38
ARFBF: ARFBF model Z = Z(x, y) and its spectrum
SARFBF
ARFBF model Z
The 2D ARFBF, hereafter denoted Z = Z(x, y), is defined as the convolution of
the AR field A and the FBF field F.
Z(x, y) = A ∗ FH (x, y) (7)
Hurst parameter H: 0 < H < 1
Spectrum SARFBF
SARFBF (u, v) = SAR (u, v)SFBF (u, v)
=
(σe)2
|1 + m∈D βme−i2π<x,y>|2
ξ(H)
f
(2H+2)
(8)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 13 / 38
ARFBF: Hurst parameter estimation
Literature and Proposition
In the literature, Hurst parameter estimation methods based on periodogram
for 1D fractional Brownian motion (fBm) (see [1], [2] and [3])
We propose two Hurst parameter estimation methods based on 2D wavelet
packet spectrum
New form of the equation 6
From equation 6, the spectrum of FBF has the form as follows:
SFBF (u, v) ∼
1
f α
(9)
α = 2 × H + 2 (10)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 14 / 38
ARFBF: Log − RDWP estimation method
Formula
Log − RDWP estimation method: the Log-Regression on Diagonal Wavelet
Packet spectrum estimation method
αLog−RDWP =
1
C
0<j<k≤N
log(SFBF (uj , uj )) − log(SFBF (uk , uk ))
log( fk ) − log( fj )
(11)
Explanation
C = N!
2(N−2)! : the number of all possible combinations of the log-ratios
SFBF : the spectrum estimated from method proposed in [4]
N: the number of considered 2D frequencies
0 < j ≤ N and j < k ≤ N
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 15 / 38
ARFBF: Log − RPWP estimation method and estimation
procedure
The first step
Log − RPWP estimation method: the Log-Regression on Polar representation of
Wavelet Packet spectrum
spectrum with polar coordinates WI :
WI (r, θ) = T(SI (u, v)) (12)
Explanation
SI (u, v): the spectrum estimated from method proposed in [4] of the input
image with Cartesian coordinates
T: the Cartesian-to-polar transform
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 16 / 38
ARFBF: Log − RPWP estimation method and estimation
procedure
The second step
Averages are done over the angles.
PI (ri ) =
1
M
j=1
M
WI (ri , j )with1 ≤ i ≤ N (13)
The third step
α is estimated by:
αLog−RPWP =
1
C
0<i<k≤N
log(PI (ri )) − log(PI (rk ))
log(rk ) − log(ri )
(14)
C = N!
2(N−2)! : the number of all possible combinations of the log-ratios
N: the number of considered radii
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 17 / 38
2-factor FBF: Modulated FBF
Modulated FBF
We define a random field Y by setting
YH (x, y) = FH (x, y)eiu∗
x
eiv∗
y
(15)
The field YH is an FBF FH modulated by a complex exponential wave associated
with a frequency point (u∗
, v∗
) in the grid [−π, π] ∗ [−π, π].
Auto-correlation
RYH
(x, y, s, t) = RFH
(x, y, s, t)eiu∗
(x−s
eiv∗
(y−t)
(16)
Spectrum SYH
SYH
= SFH (u − u∗
, v − v∗
) = ξ(H)
1
((u − u∗)2 + (v − v∗)2)H+1
(17)
SYH
has a singularity at the frequency point (u∗
, v∗
)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 18 / 38
2-factor FBF: 2-factor FBF M(x, y) and its spectrum
SM(u, v)
2-factor FBF M(x, y)
The 2-factor FBF, hereafter noted M(x, y), is defined as the convolution of the
FBF field FH1 and the modulated FBF YH2 .
M(x, y) = FH1 ∗ YH2 (x, y) (18)
Spectrum SM(u, v)
SM (u, v) = SFH1
(u, v)SYH2
(u, v) =
ξ(H1)
(u2 + v2)H1+1
ξ(H2)
((u − u∗)2 + (v − v∗)2)H2+1
(19)
the standard FBF FH1
has Hurst parameter H1 and a pole located at (0, 0)
the modulated FBF YH2
has Hurst parameter H2 and a pole located at
(u∗
, v∗
)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 19 / 38
H1 Hurst parameter estimation of standard FBF
Mean values of estimated α = 2H1 + 2 and variances calculated from 10
realizations of an FBF for α ∈ {2.4, 2.8, 3.2, 3.6} and for two different image sizes:
512 × 512 and 2048 × 2048
Table: Hurst parameter estimation of standard FBF
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 20 / 38
H2 Hurst parameter estimation of FBF modulated
Mean values of estimated α = 2H2 + 2 and variances calculated from 10
realizations of an FBF for α ∈ {2.4, 2.8, 3.2, 3.6} and for two different image sizes:
512 × 512 and 2048 × 2048
Table: Hurst parameter estimation of FBF modulated
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 21 / 38
Hurst parameter estimation: H1 and H2
Explanation
RDWPN and RPWPN mean that α is estimated by the Log − RDWP
method and Log − RPWP estimation method, respectively
N: the number of samples in the equations 11 and 14
The Daubechy filter for computing the WP spectrum
Comparison
Computing the Mean Square Error (MSE = Bias2
+ variance) using values in
these tables:
For the small size images, the Log − RPWP method estimates better than
the Log − RDWP method
For the large size images, the Log − RDWP method gives comparable results
to those of the Log − RPWP method.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 22 / 38
2-factor FBF: estimation the location of the pole (u∗
, v∗
)
Explanation
The pole localization is
computed from the
maximum of the power
spectral density estimated
using WP spectrum.
Mean values of estimated
(u∗
, v∗
) and their variances
computed from Monte-Carlo
simulations on 10 modulated
FBF realizations with image
size equal to 512 × 512.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 23 / 38
Discussion
The periodogram represents [−π, π] × [−π, π] frequency domain.
Observation
Spectra of TEM images mostly show
A significant peak at the zero
frequency corresponding to slow
grey-level variations
An exponential decay in the
neighborhood of this peak
Figure: TEM image and periodogram
Proposition
The peak at the zero frequency modeled by an FBF is not related to the active
phase we wish to model.
Thus, we propose to remove this peak in order to characterize the main
information (active phase).
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 24 / 38
Application: TEM image characterization procedure with
ARFBF model
The first step
Let us denote now one TEM image as I = I(x, y) and its wavelet packet spectrum
as SI .
Estimate the parameter H from SI and thus SFBF associated to I is derived
The second step
Remove the contribution of the FBF in I, the residual is modeled by an AR model
with spectrum defined as follows:
SAR (u, v) =
SARFBF (u, v)
SFBF (u, v)
(20)
SARFBF (u, v) = SI (u, v) (21)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 25 / 38
Application: TEM image characterization procedure with
2-factor FBF model
The first step
Let us denote now one TEM image as I = I(x, y) and its wavelet packet spectrum
as SI .
Estimate the parameter H1 from SI and thus SFBF associated to I is derived
The second step
Remove the contribution of the FBF in I, the residual is modeled by another FBF
model with spectrum defined as follows:
SYH2
(u, v) =
SI (u, v)
SFBFH1
(u, v)
(22)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 26 / 38
Application: experimental Results for TEM images from
ARFBF model
First row: initial image, periodogram and wavelet packet spectrum
Second row: residual part, periodogram and power spectral density
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 27 / 38
Application: experimental Results for TEM images from
ARFBF model
Explanation
Residual part: after removing the FBF contribution from the initial image
Periodogram representing [−π, π] × [−π, π] frequency domain
Power spectral density computed using the AR parameters estimated from
the residual part
Parameter estimation
Parameters of the AR part are estimated with classical Yule-Walker method
Parameter of the FBF part is estimated with log − RPWP estimation method
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 28 / 38
Application: experimental Results for TEM images from
2-factor FBF model
explanation
For each line, from left to
right
initial HRTEM image
WP spectrum
WP spectrum after
removing the
zero-pole FBF part
two Hurst parameters
estimated
Figure: TEM images texture analysis
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 29 / 38
Application: texture synthesis from 2-factor FBF
texture synthesis
Texture synthesis with parameters H1 = 3.2519 and H2 = 2.8401:
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 30 / 38
Conclusion and Perspective
Conclusion
Proposition of the ARFBF model
Proposition of the 2-factor FBF model
Proposition of two Hurst parameter estimation methods
Application these models to TEM image texture analysis
Perspective
Morphology method for characterizing TEM image texture
Extension of the FBF model to k-factor GFBF model
Extension of the AR to ARMA model
ARMAFBF model ...
Comparison the Hurst parameter estimation with 2D Hurst parameter
estimation existed
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 31 / 38
Publication
Publication
Z. Tan, A. M. Atto, O. Alata and M. Moreaud, ARFBF Model for Non
Stationary Random Fields and Application to HRTEM images, IEEE
International Conference on Image Processing, ICIP, September 27-30,
Qubec City, France, 2015.
Z. Tan, O. Alata, M. Moreaud and A. M. Atto, Convolution mixture of FBF
and modulated FBF and application to HRTEM images, GRETSI
Symposium on Signal and Image Processing, September 8-11, Lyon,
France, 2015.
A. M. Atto, Z. Tan, O. Alata and M. Moreaud, Non-Stationary Texture
Synthesis from Random Field Modeling, IEEE International Conference on
Image Processing, ICIP, October 27-30, Paris, France, 2014.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 32 / 38
Publication
Intern Research Report
Z. Tan, A. M. Atto, O. Alata and M. Moreaud, Analyse de modles
statistiques dans un contexte d’imagerie HRTEM, Intern Research Report,
IFP Energies Nouvelles, 2015.
Journal paper
resuming what has been done
completed with a morphology approach (results obtained will be presented as
following)
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 33 / 38
Perspective: morphology analysis of SAR = SARFBF
SFBF
First row: TEM image
Second row: respective spd calculated from parameters estimated of AR
Third row: respective lobe form
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 34 / 38
Reference I
B. Pesquet-Popescu
Stochastic fractal models for image processing
IEEE Signal Processing Magazine, 2002.
P. Flandrin
On the spectrum of fractional Brownian Motions
IEEE, 2005.
G. Rilling and all
Empirical Mode Decomposition, Fractional Gauss Noise and Hurst Exponent
Estimation
IEEE Transaction on Information Theory, 1989.
A.M. Atto and all
Wavelet Packets of fractional Brownian motion: Asymptotic Analysis and
Spectrum Estimation
IEEE Transactions on Information Theory, 2010.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 35 / 38
Reference II
A.M. Atto and all
Non-stationary Texture Synthesis from Random Field Modeling
IEEE ICIP, 2014.
O. Alata and all
Choice of a 2-D causal autoregressive texture model using information criteria
Pattern Recognition Letters, 2003.
O. Alata and all
Unsupervised Textured Image Segmentation using 2-D Quarter Plane
Autoregressive Support with Four Prediction Support
Pattern Recognition Letters, 2005.
M. Moreaud and all
A quantitative morphological analysis of nanostructured ceriasilica composite
catalysts
Journal of Microscopy, 2008.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 36 / 38
Reference III
M. Moreaud and all
Quantitative characterization of soot Nanostructure from HRTEM images
ICS13 conference, 2011.
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 37 / 38
Question
TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 38 / 38

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Presentation_Tan

  • 1. Statistical Invariances and Geometry Modeling for the Analysis of TEXtures (SIGMA-TEX)-Application to high resolution Transmission Electron Microscopy (TEM) Imaging LISTIC Seminar Presentation Zhangyun TAN1 Abdourrahmane M. ATTO1 Olivier ALATA2 Maxime MOREAUD3 1 LISTIC, University of Savoie Mont Blanc 2 Hubert Curien Laboratory, University Jean Monnet of Saint Etienne 3 IFP Energies nouvelles July 9, 2015 TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 1 / 38
  • 2. Transmission Electron Microscopy (TEM) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 2 / 38
  • 3. Transmission Electron Microscopy (TEM) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 3 / 38
  • 4. Catalyst composition [8] and TEM image TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 4 / 38
  • 5. Carbon fringes morphology characterization existed, problem and objective Carbon fringes morphology characterization existed [9] fringe length distance between carbon fringes aligned carbon fringes parallel carbon fringes discrimination between aligned and parallel fringes Problem and Objective How to apply mathematics models to carbon fringes morphology characterization? TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 5 / 38
  • 6. Problem and Objective TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 6 / 38
  • 7. Table of Contents 1 Mathematics modeling Texture synthesis from model GFBF and AR ARFBF modeling Hurst parameter estimation 2-factor FBF modeling 2 Experimental Results Hurst parameter estimation Location of the pole (u∗ , v∗ ) 3 Application to TEM images Discussion TEM image characterization procedure Experimental Results for TEM images 4 Conclusion, Publication and Perspective 5 Reference 6 Question TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 7 / 38
  • 8. Texture synthesis from k-factor GFBF (Generalized Fractional Brownian Field) εHk for different values of k (see [5]) Poles in [0, π/2] × [0, π/2] and Hurst parameters in ]0, 0.5[ generated from random variables distributed as Gaussian, Uniformly and Gamma respectively TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 8 / 38
  • 9. Texture synthesis from AR (Auto Regressive) with known order (M1, M2) (see [6]) First row: textures synthesis in the spatial domain Second row: respective spectrum SAR detailled in the following part TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 9 / 38
  • 10. ARFBF: AR model A = A(x, y) and its spectrum SAR (see [7]) AR model A A(x, y) = − m1,m2∈D βm1,m2 A(x − m1, y − m2) + E(x, y) (1) DM1,M2 = {(m1, m2) ∈ Z2 , 0 ≤ m1 ≤ M1, 0 ≤ m2 ≤ M2, (m1, m2) = (0, 0)} (2) Explanation (x, y) ∈ Z2 D ⊂ Z2 : prediction support E(x, y): white Gaussian noise β(m1, m2): coefficients (m1, m2): index (M1, M2): order TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 10 / 38
  • 11. ARFBF: AR model A = A(x, y) and its spectrum SAR (see [7]) AR model A A(x, y) = − m1,m2∈D βm1,m2 A(x − m1, y − m2) + E(x, y) (3) (x, y) ∈ Z2 and D ⊂ Z2 : 2D prediction support E(x, y): 2D white Gaussian noise and β(m1, m2): coefficients (m1, m2): index and (M1, M2): order Spectrum SAR SAR (u, v) = (σe)2 |1 + m1,m2∈D βm1,m2 e−i2πum1 e−i2πvm2 |2 (4) (σe)2 : variance of E = E(x, y) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 11 / 38
  • 12. ARFBF: FBF model F = FH(x, y) and its spectrum SFBF (see [4]) Auto-correlation function RF RF (x, y, s, t) = σ2 2 ((x2 + y2 )H + (s2 + t2 )H − ((x − s)2 + (y − t)2 )H ) (5) Spectrum SFBF SFBF (u, v) = ξ(H) 1 (u2 + v2)(H+1) = ξ(H) 1 f (2H+2) (6) ξ(H) = 2−(2H+1) π2 σ2 sin(πH)Γ2(1+H) , f = (u2 + v2) and Γ: the gamma function and Hurst parameter H: 0 < H < 1 and σ2 : variance of white Gaussian noise TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 12 / 38
  • 13. ARFBF: ARFBF model Z = Z(x, y) and its spectrum SARFBF ARFBF model Z The 2D ARFBF, hereafter denoted Z = Z(x, y), is defined as the convolution of the AR field A and the FBF field F. Z(x, y) = A ∗ FH (x, y) (7) Hurst parameter H: 0 < H < 1 Spectrum SARFBF SARFBF (u, v) = SAR (u, v)SFBF (u, v) = (σe)2 |1 + m∈D βme−i2π<x,y>|2 ξ(H) f (2H+2) (8) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 13 / 38
  • 14. ARFBF: Hurst parameter estimation Literature and Proposition In the literature, Hurst parameter estimation methods based on periodogram for 1D fractional Brownian motion (fBm) (see [1], [2] and [3]) We propose two Hurst parameter estimation methods based on 2D wavelet packet spectrum New form of the equation 6 From equation 6, the spectrum of FBF has the form as follows: SFBF (u, v) ∼ 1 f α (9) α = 2 × H + 2 (10) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 14 / 38
  • 15. ARFBF: Log − RDWP estimation method Formula Log − RDWP estimation method: the Log-Regression on Diagonal Wavelet Packet spectrum estimation method αLog−RDWP = 1 C 0<j<k≤N log(SFBF (uj , uj )) − log(SFBF (uk , uk )) log( fk ) − log( fj ) (11) Explanation C = N! 2(N−2)! : the number of all possible combinations of the log-ratios SFBF : the spectrum estimated from method proposed in [4] N: the number of considered 2D frequencies 0 < j ≤ N and j < k ≤ N TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 15 / 38
  • 16. ARFBF: Log − RPWP estimation method and estimation procedure The first step Log − RPWP estimation method: the Log-Regression on Polar representation of Wavelet Packet spectrum spectrum with polar coordinates WI : WI (r, θ) = T(SI (u, v)) (12) Explanation SI (u, v): the spectrum estimated from method proposed in [4] of the input image with Cartesian coordinates T: the Cartesian-to-polar transform TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 16 / 38
  • 17. ARFBF: Log − RPWP estimation method and estimation procedure The second step Averages are done over the angles. PI (ri ) = 1 M j=1 M WI (ri , j )with1 ≤ i ≤ N (13) The third step α is estimated by: αLog−RPWP = 1 C 0<i<k≤N log(PI (ri )) − log(PI (rk )) log(rk ) − log(ri ) (14) C = N! 2(N−2)! : the number of all possible combinations of the log-ratios N: the number of considered radii TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 17 / 38
  • 18. 2-factor FBF: Modulated FBF Modulated FBF We define a random field Y by setting YH (x, y) = FH (x, y)eiu∗ x eiv∗ y (15) The field YH is an FBF FH modulated by a complex exponential wave associated with a frequency point (u∗ , v∗ ) in the grid [−π, π] ∗ [−π, π]. Auto-correlation RYH (x, y, s, t) = RFH (x, y, s, t)eiu∗ (x−s eiv∗ (y−t) (16) Spectrum SYH SYH = SFH (u − u∗ , v − v∗ ) = ξ(H) 1 ((u − u∗)2 + (v − v∗)2)H+1 (17) SYH has a singularity at the frequency point (u∗ , v∗ ) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 18 / 38
  • 19. 2-factor FBF: 2-factor FBF M(x, y) and its spectrum SM(u, v) 2-factor FBF M(x, y) The 2-factor FBF, hereafter noted M(x, y), is defined as the convolution of the FBF field FH1 and the modulated FBF YH2 . M(x, y) = FH1 ∗ YH2 (x, y) (18) Spectrum SM(u, v) SM (u, v) = SFH1 (u, v)SYH2 (u, v) = ξ(H1) (u2 + v2)H1+1 ξ(H2) ((u − u∗)2 + (v − v∗)2)H2+1 (19) the standard FBF FH1 has Hurst parameter H1 and a pole located at (0, 0) the modulated FBF YH2 has Hurst parameter H2 and a pole located at (u∗ , v∗ ) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 19 / 38
  • 20. H1 Hurst parameter estimation of standard FBF Mean values of estimated α = 2H1 + 2 and variances calculated from 10 realizations of an FBF for α ∈ {2.4, 2.8, 3.2, 3.6} and for two different image sizes: 512 × 512 and 2048 × 2048 Table: Hurst parameter estimation of standard FBF TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 20 / 38
  • 21. H2 Hurst parameter estimation of FBF modulated Mean values of estimated α = 2H2 + 2 and variances calculated from 10 realizations of an FBF for α ∈ {2.4, 2.8, 3.2, 3.6} and for two different image sizes: 512 × 512 and 2048 × 2048 Table: Hurst parameter estimation of FBF modulated TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 21 / 38
  • 22. Hurst parameter estimation: H1 and H2 Explanation RDWPN and RPWPN mean that α is estimated by the Log − RDWP method and Log − RPWP estimation method, respectively N: the number of samples in the equations 11 and 14 The Daubechy filter for computing the WP spectrum Comparison Computing the Mean Square Error (MSE = Bias2 + variance) using values in these tables: For the small size images, the Log − RPWP method estimates better than the Log − RDWP method For the large size images, the Log − RDWP method gives comparable results to those of the Log − RPWP method. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 22 / 38
  • 23. 2-factor FBF: estimation the location of the pole (u∗ , v∗ ) Explanation The pole localization is computed from the maximum of the power spectral density estimated using WP spectrum. Mean values of estimated (u∗ , v∗ ) and their variances computed from Monte-Carlo simulations on 10 modulated FBF realizations with image size equal to 512 × 512. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 23 / 38
  • 24. Discussion The periodogram represents [−π, π] × [−π, π] frequency domain. Observation Spectra of TEM images mostly show A significant peak at the zero frequency corresponding to slow grey-level variations An exponential decay in the neighborhood of this peak Figure: TEM image and periodogram Proposition The peak at the zero frequency modeled by an FBF is not related to the active phase we wish to model. Thus, we propose to remove this peak in order to characterize the main information (active phase). TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 24 / 38
  • 25. Application: TEM image characterization procedure with ARFBF model The first step Let us denote now one TEM image as I = I(x, y) and its wavelet packet spectrum as SI . Estimate the parameter H from SI and thus SFBF associated to I is derived The second step Remove the contribution of the FBF in I, the residual is modeled by an AR model with spectrum defined as follows: SAR (u, v) = SARFBF (u, v) SFBF (u, v) (20) SARFBF (u, v) = SI (u, v) (21) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 25 / 38
  • 26. Application: TEM image characterization procedure with 2-factor FBF model The first step Let us denote now one TEM image as I = I(x, y) and its wavelet packet spectrum as SI . Estimate the parameter H1 from SI and thus SFBF associated to I is derived The second step Remove the contribution of the FBF in I, the residual is modeled by another FBF model with spectrum defined as follows: SYH2 (u, v) = SI (u, v) SFBFH1 (u, v) (22) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 26 / 38
  • 27. Application: experimental Results for TEM images from ARFBF model First row: initial image, periodogram and wavelet packet spectrum Second row: residual part, periodogram and power spectral density TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 27 / 38
  • 28. Application: experimental Results for TEM images from ARFBF model Explanation Residual part: after removing the FBF contribution from the initial image Periodogram representing [−π, π] × [−π, π] frequency domain Power spectral density computed using the AR parameters estimated from the residual part Parameter estimation Parameters of the AR part are estimated with classical Yule-Walker method Parameter of the FBF part is estimated with log − RPWP estimation method TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 28 / 38
  • 29. Application: experimental Results for TEM images from 2-factor FBF model explanation For each line, from left to right initial HRTEM image WP spectrum WP spectrum after removing the zero-pole FBF part two Hurst parameters estimated Figure: TEM images texture analysis TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 29 / 38
  • 30. Application: texture synthesis from 2-factor FBF texture synthesis Texture synthesis with parameters H1 = 3.2519 and H2 = 2.8401: TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 30 / 38
  • 31. Conclusion and Perspective Conclusion Proposition of the ARFBF model Proposition of the 2-factor FBF model Proposition of two Hurst parameter estimation methods Application these models to TEM image texture analysis Perspective Morphology method for characterizing TEM image texture Extension of the FBF model to k-factor GFBF model Extension of the AR to ARMA model ARMAFBF model ... Comparison the Hurst parameter estimation with 2D Hurst parameter estimation existed TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 31 / 38
  • 32. Publication Publication Z. Tan, A. M. Atto, O. Alata and M. Moreaud, ARFBF Model for Non Stationary Random Fields and Application to HRTEM images, IEEE International Conference on Image Processing, ICIP, September 27-30, Qubec City, France, 2015. Z. Tan, O. Alata, M. Moreaud and A. M. Atto, Convolution mixture of FBF and modulated FBF and application to HRTEM images, GRETSI Symposium on Signal and Image Processing, September 8-11, Lyon, France, 2015. A. M. Atto, Z. Tan, O. Alata and M. Moreaud, Non-Stationary Texture Synthesis from Random Field Modeling, IEEE International Conference on Image Processing, ICIP, October 27-30, Paris, France, 2014. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 32 / 38
  • 33. Publication Intern Research Report Z. Tan, A. M. Atto, O. Alata and M. Moreaud, Analyse de modles statistiques dans un contexte d’imagerie HRTEM, Intern Research Report, IFP Energies Nouvelles, 2015. Journal paper resuming what has been done completed with a morphology approach (results obtained will be presented as following) TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 33 / 38
  • 34. Perspective: morphology analysis of SAR = SARFBF SFBF First row: TEM image Second row: respective spd calculated from parameters estimated of AR Third row: respective lobe form TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 34 / 38
  • 35. Reference I B. Pesquet-Popescu Stochastic fractal models for image processing IEEE Signal Processing Magazine, 2002. P. Flandrin On the spectrum of fractional Brownian Motions IEEE, 2005. G. Rilling and all Empirical Mode Decomposition, Fractional Gauss Noise and Hurst Exponent Estimation IEEE Transaction on Information Theory, 1989. A.M. Atto and all Wavelet Packets of fractional Brownian motion: Asymptotic Analysis and Spectrum Estimation IEEE Transactions on Information Theory, 2010. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 35 / 38
  • 36. Reference II A.M. Atto and all Non-stationary Texture Synthesis from Random Field Modeling IEEE ICIP, 2014. O. Alata and all Choice of a 2-D causal autoregressive texture model using information criteria Pattern Recognition Letters, 2003. O. Alata and all Unsupervised Textured Image Segmentation using 2-D Quarter Plane Autoregressive Support with Four Prediction Support Pattern Recognition Letters, 2005. M. Moreaud and all A quantitative morphological analysis of nanostructured ceriasilica composite catalysts Journal of Microscopy, 2008. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 36 / 38
  • 37. Reference III M. Moreaud and all Quantitative characterization of soot Nanostructure from HRTEM images ICS13 conference, 2011. TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 37 / 38
  • 38. Question TAN (LISTIC) SIGMA-TEX-MET Seminar Presentation 38 / 38