TEXTURE MODELING BY GAUSSIAN FIELDS WITH PRESCRIBED LOCAL ORIENTATION Kévin Polisano /
[email protected]
joint work with Marianne Clausel Valérie Perrier Laurent Condat IEEE ICIP 2014 : International Conference on Image Processing. Lecture session CNIT Paris, October 27-30, 2014
Outline Introduction Motivation General probabilistic framework
Our new stochastic model Definition: Locally Anisotropic Fractional Brownian Field Notion of tangent field
Synthesis methods Tangent field simulation by a turning bands method LAFBF simulation via tangent field formulation
Numerical experiments Conclusion and future work Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
2
How to synthesize natural random textures ?
Mathematical model ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Mathematical model ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Roughness and regularity Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Roughness and regularity Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Roughness and regularity Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
How to synthesize natural random textures ?
Randomness Self-similarity Mathematical model ?
Orientation and anisotropy
Roughness and regularity
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
3
The basic component : Fractional Brownian Field (FBF) B H FBF with Hurst index 0 < H < 1 c
H
stationary increments : B (· + z) L
H
self-similar : B ( ·) = isotropic : B
H
H
[Mandelbrot,Van Ness,1968]
L
H
B (z) = B H (·)
B H (0)
B H (·)
L
R✓ = B H
The covariance is given by H
H
2H
Cov(B (x), B (y)) = cH (kxk
2H
+ kyk
kx
yk
2H
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
)
4
The basic component : Fractional Brownian Field (FBF) Harmonizable representation Z ix·⇠ e 1 c H B (x) = d W (⇠) H+1 R2 k⇠k
[Samorodnitsky,Taqqu,1997]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
5
The basic component : Fractional Brownian Field (FBF) Harmonizable representation Z ix·⇠ e 1 c H B (x) = d W (⇠) H+1 R2 k⇠k
[Samorodnitsky,Taqqu,1997]
complex Brownian measure
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
5
The basic component : Fractional Brownian Field (FBF) Harmonizable representation Z ix·⇠ e 1 c H B (x) = d W (⇠) H+1 R2 k⇠k roughness indicator
[Samorodnitsky,Taqqu,1997]
complex Brownian measure
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
5
The basic component : Fractional Brownian Field (FBF) Harmonizable representation Z ix·⇠ e 1 c H B (x) = d W (⇠) H+1 R2 k⇠k
[Samorodnitsky,Taqqu,1997]
complex Brownian measure
roughness indicator H=0.2
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
5
The basic component : Fractional Brownian Field (FBF) Harmonizable representation Z ix·⇠ e 1 c H B (x) = d W (⇠) H+1 R2 k⇠k
[Samorodnitsky,Taqqu,1997]
complex Brownian measure
roughness indicator H=0.2
H=0.7 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
5
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012]
Global Anisotropy Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Global Anisotropy Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Global Anisotropy Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
Global
Local
Anisotropy
orientation
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
Global
Local
Anisotropy
orientation
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Global
Local
Local
Anisotropy
orientation
roughness
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Local
Local
orientation
roughness
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Local roughness Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
General model : anisotropic self-similar Gaussian fields X (x) =
Z
R2
(e ix·⇠
c (⇠) 1)f 1/2 (x, ⇠)d W
f 1/2 (x, ⇠) = c(x, ⇠)k⇠k
spectral density
h(x,⇠) 1
[Mandelbrot,Van Ness,1968] c(x, ⇠) ⌘ 1 and h(x, ⇠) ⌘ H ) X = B H c(x, ⇠) ⌘ c(arg ⇠) and h(x, ⇠) ⌘ h(arg ⇠) ) X = AFBF [Bonami,Estrade,2003]
Example : elementary fields c(arg ⇠) = 1[
c(x, ⇠) ⌘ c(x, arg ⇠) and h(x, ⇠) ⌘ h(x)
↵,↵] (arg ⇠
↵0 )
[Bierme,Richard,Moisan,2012] [Polisano,Clausel,Perrier,Condat,2014]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
6
Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary Z field 1[ ↵,↵] (arg ⇠ ↵0 (x)) c H ix·⇠ B↵0 ,↵ (x) = (e 1) d W (⇠) H+1 k⇠k R2
[Polisano et al.,2014]
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
7
Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary Z field 1[ ↵,↵] (arg ⇠ ↵0 (x)) c H ix·⇠ B↵0 ,↵ (x) = (e 1) d W (⇠) H+1 k⇠k R2
[Polisano et al.,2014]
The orientation may varies spatially. ↵0 is now a differentiable function on R2 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
7
Locally Anisotropic Fractional Brownian Field (LAFBF) Definition: Our new Gaussian model LAFBF is a local version of the elementary Z field 1[ ↵,↵] (arg ⇠ ↵0 (x)) c H ix·⇠ B↵0 ,↵ (x) = (e 1) d W (⇠) H+1 k⇠k R2
[Polisano et al.,2014]
↵ Texture orientation
x0
V~x0
f 1/2 (x0 , ⇠) =
c↵0 ,↵ (x0 , arg ⇠) k⇠kH+1
The orientation may varies spatially. ↵0 is now a differentiable function on R2
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
7
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵=
⇡ 2
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
8
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.7 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
9
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.6 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
10
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.5 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
11
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.4 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
12
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.3 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
13
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.2 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
14
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.1 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
15
Elementary field ↵0 = 0
Texture orientation
x0
V~x0
↵
↵ = 0.05 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
16
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.7 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
17
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.6 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
18
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.5 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
19
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.4 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
20
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.3 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
21
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.2 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
22
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.1 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
23
Elementary field ↵0 =
⇡ 3
↵ Texture orientation
x0
V~x0
↵ = 0.05 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
24
Tangent field H B↵0 ,↵ (x)
=
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x)) c d W (⇠)
Tangent field. For a random field X locally asymptotically self-similar of order H, X (x0 + ⇢h) ⇢H
X (x0 )
L
! Y x0
⇢!0
Yx0 : tangent field of X at point x0 2 R
2
[Benassi,1997] [Falconer,2002]
Taylor’s expansion
Tangent field
Deterministic case
Stochastic case
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
25
Tangent field H B↵0 ,↵ (x)
=
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x)) c d W (⇠)
Tangent field. For a random field X locally asymptotically self-similar of order H, X (x0 + ⇢h) ⇢H
X (x0 )
L
! Y x0
⇢!0
Yx0 : tangent field of X at point x0 2 R
2
[Benassi,1997] [Falconer,2002]
Taylor’s expansion
Tangent field
Deterministic case
Stochastic case
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
25
Tangent field H B↵0 ,↵ (x)
=
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x)) c d W (⇠)
Theorem. The LAFBF B↵H0 ,↵ admits for tangent field Yx0 : Yx0 (x) =
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x0 )) c d W (⇠)
Yx0 elementary field with global orientation ↵0 (x0 )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
26
Tangent field H B↵0 ,↵ (x)
=
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x)) c d W (⇠)
Theorem. The LAFBF B↵H0 ,↵ admits for tangent field Yx0 : Yx0 (x) =
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x0 )) c d W (⇠)
constant Yx0 elementary field with global orientation ↵0 (x0 )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
26
Tangent field H B↵0 ,↵ (x)
=
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x)) c d W (⇠)
Theorem. The LAFBF B↵H0 ,↵ admits for tangent field Yx0 : Yx0 (x) =
Z
R2
(e
ix·⇠
1)
1[
↵,↵] (arg ⇠ k⇠kH+1
↵0 (x0 )) c d W (⇠)
constant Yx0 elementary field with global orientation ↵0 (x0 )
B↵H0 ,↵ (x0 ) ⇡ Yx0 (x = x0 ) Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
26
Simulation of tangent fields Continuous formulation. Variogram of Yx0 : [Bierme,Richard,Moisan,2012] Z 1 vYx0 (x) = |e ix·⇠ 1|2 f (x0 , ⇠)d⇠ 2 R2 Z ⇡/2 1 2H = (H) c↵0 ,↵ (x0 , ✓) |x · u(✓)| d✓ 2 ⇡/2 Z ⇡/2 = v˜✓ (x · u(✓))d✓ ⇡/2
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
27
Simulation of tangent fields Continuous formulation. Variogram of Yx0 : [Bierme,Richard,Moisan,2012] Z 1 vYx0 (x) = |e ix·⇠ 1|2 f (x0 , ⇠)d⇠ 2 R2 Z ⇡/2 1 2H = (H) c↵0 ,↵ (x0 , ✓) |x · u(✓)| d✓ 2 in polar ⇡/2 Z ⇡/2 coordinates = v˜✓ (x · u(✓))d✓ ⇡/2
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
27
Simulation of tangent fields Continuous formulation. Variogram of Yx0 : [Bierme,Richard,Moisan,2012] Z 1 vYx0 (x) = |e ix·⇠ 1|2 f (x0 , ⇠)d⇠ 2 R2 Z ⇡/2 1 2H = (H) c↵0 ,↵ (x0 , ✓) |x · u(✓)| d✓ 2 in polar ⇡/2 Z ⇡/2 coordinates = v˜✓ (x · u(✓))d✓ ⇡/2
1 v˜✓ = (H)c↵0 ,↵ (x0 , ✓)| · |2H 2 u(✓) = (cos ✓, sin ✓) ⇡ (H) = H (2H) sin(H⇡)
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
27
Simulation of tangent fields Continuous formulation. Variogram of Yx0 : [Bierme,Richard,Moisan,2012] Z 1 vYx0 (x) = |e ix·⇠ 1|2 f (x0 , ⇠)d⇠ 2 R2 Z ⇡/2 1 2H = (H) c↵0 ,↵ (x0 , ✓) |x · u(✓)| d✓ 2 in polar ⇡/2 Z ⇡/2 coordinates variogram of a fractional = v˜✓ (x · u(✓))d✓ brownian motion (FBM) of order H ⇡/2
1 v˜✓ = (H)c↵0 ,↵ (x0 , ✓)| · |2H 2 u(✓) = (cos ✓, sin ✓) ⇡ (H) = H (2H) sin(H⇡)
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
27
Simulation of tangent fields Continuous formulation. Variogram of Yx0 : [Bierme,Richard,Moisan,2012] Z 1 vYx0 (x) = |e ix·⇠ 1|2 f (x0 , ⇠)d⇠ 2 R2 Z ⇡/2 1 2H = (H) c↵0 ,↵ (x0 , ✓) |x · u(✓)| d✓ 2 in polar ⇡/2 Z ⇡/2 coordinates variogram of a fractional = v˜✓ (x · u(✓))d✓ brownian motion (FBM) of order H ⇡/2
Y x0 =
Infinite sum of independant rotating FBM of order H
1 v˜✓ = (H)c↵0 ,↵ (x0 , ✓)| · |2H 2 u(✓) = (cos ✓, sin ✓) ⇡ (H) = H (2H) sin(H⇡)
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
27
Simulation of tangent fields Discrete formulation.
[Bierme,Richard,Moisan,2012]
(✓i )16i6n are n bands orientations and
i
= ✓i+1
✓i
The turning band field is defined as n p X 1 [n] H Yx0 (x) = (H) 2 c (x , ✓ )B i ↵0 ,↵ 0 i i (x · u(✓i )) i=1
BiH are n independent FBM of order H
Good approximation provided max i
i
6"
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
28
Simulation of tangent fields Discrete formulation.
[Bierme,Richard,Moisan,2012]
(✓i )16i6n are n bands orientations and
i
= ✓i+1
✓i
The turning band field is defined as n p X 1 [n] H Yx0 (x) = (H) 2 c (x , ✓ )B i ↵0 ,↵ 0 i i (x · u(✓i )) i=1
BiH are n independent FBM of order H
Good approximation provided max i
i
6"
not equispaced Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
28
Simulation of tangent fields Simulation along particular bands. Discrete grid r
1
[Bierme,Richard,Moisan,2012]
Z2 \ [0, 1]2 with r = 2k
Choose (✓i ) such that tan ✓i =
pi qi
and max i
1, k 2 N? i
6✏
Then BiH (x · u(✓i )) becomes ⇢
✓
◆
k1 k2 L cos ✓i + sin ✓i ; 0 6 k1 , k2 6 r = r r ✓ ◆H cos ✓i {BiH (k1 qi + k2 pi ); 0 6 k1 , k2 6 r } rqi BiH
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
29
Simulation of tangent fields Simulation along particular bands. Discrete grid r
1
Z2 \ [0, 1]2 with r = 2k
Choose (✓i ) such that tan ✓i = Then BiH (x · u(✓i )) becomes ⇢
✓
[Bierme,Richard,Moisan,2012]
pi qi
and max i
1, k 2 N? i
6✏
Dynamic programming
◆
k1 k2 L cos ✓i + sin ✓i ; 0 6 k1 , k2 6 r = r r ✓ ◆H cos ✓i {BiH (k1 qi + k2 pi ); 0 6 k1 , k2 6 r } rqi BiH
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
29
Simulation of tangent fields Simulation along particular bands. Discrete grid r
1
Z2 \ [0, 1]2 with r = 2k
Choose (✓i ) such that tan ✓i = Then BiH (x · u(✓i )) becomes
self-similarity L
B H ( ·) =
H
⇢
B H (·)
✓
[Bierme,Richard,Moisan,2012]
pi qi
and max i
1, k 2 N? i
6✏
Dynamic programming
◆
k1 k2 L cos ✓i + sin ✓i ; 0 6 k1 , k2 6 r = r r ✓ ◆H cos ✓i {BiH (k1 qi + k2 pi ); 0 6 k1 , k2 6 r } rqi BiH
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
29
Simulation of tangent fields Simulation along particular bands. Discrete grid r
1
Z2 \ [0, 1]2 with r = 2k
Choose (✓i ) such that tan ✓i = Then BiH (x · u(✓i )) becomes
self-similarity L
B H ( ·) =
H
⇢
B H (·)
✓
[Bierme,Richard,Moisan,2012]
pi qi
and max i
1, k 2 N? i
6✏
Dynamic programming
◆
k1 k2 L cos ✓i + sin ✓i ; 0 6 k1 , k2 6 r = r r ✓ ◆H cos ✓i {BiH (k1 qi + k2 pi ); 0 6 k1 , k2 6 r } rqi BiH
equispaced
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
29
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X i=1
i c↵0 ,↵ ((k1 , k2 ), ✓i )
✓
cos ✓i rqi
◆H
BiH (k1 qi + k2 pi )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2
B↵H0 ,↵ ((k1 , k2 )) B↵H0 ,↵ (x0 ) ⇡ Yx0 (x = x0 ) = ✓ ◆ n H X p 1 cos ✓i H (H) 2 c ((k , k ), ✓ ) B i ↵0 ,↵ 1 2 i i (k1 qi + k2 pi ) rqi i=1
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X i=1
i c↵0 ,↵ ((k1 , k2 ), ✓i )
✓
cos ✓i rqi
◆H
BiH (k1 qi + k2 pi )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X i=1
i c↵0 ,↵ ((k1 , k2 ), ✓i )
✓
cos ✓i rqi
◆H
BiH (k1 qi + k2 pi )
↵
x0
V~x0
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X
i c↵0 ,↵ ((k1 , k2 ), ✓i )
i=1
↵
✓
cos ✓i rqi
◆H
BiH (k1 qi + k2 pi )
n turning bands ✓i x0
V~x0
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X
i c↵0 ,↵ ((k1 , k2 ), ✓i )
i=1
↵
n turning bands ✓i
✓
cos ✓i rqi
1[
BiH (k1 qi + k2 pi )
↵0 ((k1 , k2 ))) 6= 0 , ↵0 ((k1 , k2 ))| 6 ↵
↵,↵] (✓i
x0
V~x0
◆H
|✓i
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X
i c↵0 ,↵ ((k1 , k2 ), ✓i )
i=1
↵
n turning bands ✓i
✓
cos ✓i rqi
1[
BiH (k1 qi + k2 pi )
↵0 ((k1 , k2 ))) 6= 0 , ↵0 ((k1 , k2 ))| 6 ↵
↵,↵] (✓i
x0
V~x0
◆H
|✓i
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X
i c↵0 ,↵ ((k1 , k2 ), ✓i )
i=1
↵
n turning bands ✓i
✓
cos ✓i rqi
1[
V~x0
BiH (k1 qi + k2 pi )
↵0 ((k1 , k2 ))) 6= 0 , ↵0 ((k1 , k2 ))| 6 ↵
↵,↵] (✓i
x0
s d n a few b e n o c in the
◆H
|✓i
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Simulation of LAFBF using tangent fields
[Polisano et al.,2014]
Algorithm. For each pixel x0 = (k1 , k2 ) 2 J0, r K2 B↵H0 ,↵ ((k1 , k2 )) =
(H)
1 2
n p X
i c↵0 ,↵ ((k1 , k2 ), ✓i )
i=1
↵
n turning bands ✓i
✓
cos ✓i rqi
1[
V~x0
BiH (k1 qi + k2 pi )
↵0 ((k1 , k2 ))) 6= 0 , ↵0 ((k1 , k2 ))| 6 ↵
↵,↵] (✓i
x0
s d n a few b e n o c in the
◆H
|✓i
Complexity O(r 2 log n)
~ x = ↵0 (k1 , k2 ) V 0 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
30
Numerical experiments ~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y ) α Texture orientation
x0 V⃗x0
f 1/2 (x0 , ξ) =
cα0 ,α (x0 , arg ξ) ∥ξ∥H+1
c˜α0 ,α (x0 , arg ξ) (a)
(b)
(c)
Parameters r = 255 ↵ = 10 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
H = 0.2 1
✏ = 10
2
31
Numerical experiments ~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y ) α Texture orientation
x0 V⃗x0
f 1/2 (x0 , ξ) =
cα0 ,α (x0 , arg ξ) ∥ξ∥H+1
c˜α0 ,α (x0 , arg ξ) (a)
(b)
(c)
Texture with prescribed local orientation at each point x0 given by a vector field
~ x = u(↵0 (x0 )) V 0 Parameters r = 255 ↵ = 10 Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
H = 0.2 1
✏ = 10
2
31
Numerical experiments ~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y ) α Texture orientation
x0 V⃗x0
f 1/2 (x0 , ξ) =
cα0 ,α (x0 , arg ξ) ∥ξ∥H+1
c˜α0 ,α (x0 , arg ξ) (a)
(b)
(c)
Texture with prescribed local orientation at each point x0 given by a vector field
~ x = u(↵0 (x0 )) V 0
A zoom around a point x0 shows that locally a LAFBF behaves as an elementary field
Parameters r = 255 ↵ = 10
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
H = 0.2 1
✏ = 10
2
31
Numerical experiments ~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y ) α Texture orientation
x0 V⃗x0
f 1/2 (x0 , ξ) =
cα0 ,α (x0 , arg ξ) ∥ξ∥H+1
c˜α0 ,α (x0 , arg ξ) (a)
(b)
Texture with prescribed local orientation at each point x0 given by a vector field
~ x = u(↵0 (x0 )) V 0
(c)
Regularized version of the anisotropy function
A zoom around a point x0 shows that locally a LAFBF behaves as an elementary field
Parameters r = 255 ↵ = 10
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
H = 0.2 1
✏ = 10
2
31
Numerical experiments ~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y ) α Texture orientation
x0 V⃗x0
f 1/2 (x0 , ξ) =
cα0 ,α (x0 , arg ξ) ∥ξ∥H+1
c˜α0 ,α (x0 , arg ξ) (a)
(b)
(c) 1
0.9
0.8
Texture with prescribed local orientation at each point x0 given by a vector field
~ x = u(↵0 (x0 )) V 0
0.7
Regularized version of the anisotropy function
A zoom around a point x0 shows that locally a LAFBF behaves as an elementary field
0.6
0.5
0.4
0.3
0.2
0.1
0 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Parameters r = 255 ↵ = 10
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
H = 0.2 1
✏ = 10
2
31
Numerical experiments ~2 V (x,y ) = (cos(cos(36xy )), sin(cos(36xy )))
~3 V (x,y ) = rF (x, y )
F (x, y ) = (4x
2)e
(4x 2)2 (4y
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
2)2
32
Numerical experiments ~2 V (x,y ) = (cos(cos(36xy )), sin(cos(36xy )))
~3 V (x,y ) = rF (x, y )
F (x, y ) = (4x
2)e
(4x 2)2 (4y
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
2)2
32
Numerical experiments ~2 V (x,y ) = (cos(cos(36xy )), sin(cos(36xy )))
~3 V (x,y ) = rF (x, y )
F (x, y ) = (4x
2)e
(4x 2)2 (4y
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
2)2
32
Numerical experiments H=0.2
H=0.5
~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
33
Numerical experiments H=0.2
H=0.5
~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
33
Numerical experiments H=0.2
H=0.5
~ 1 = (cos( ⇡/2 + y ), sin( ⇡/2)) V (x,y )
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
33
Conclusion and future work Conclusion
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF. Simulations based on tangent field formulation and the turning bands method produce textures with prescribed local orientations.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF. Simulations based on tangent field formulation and the turning bands method produce textures with prescribed local orientations.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF. Simulations based on tangent field formulation and the turning bands method produce textures with prescribed local orientations. Future work Extensions of our model include Hurst index may vary spatially.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF. Simulations based on tangent field formulation and the turning bands method produce textures with prescribed local orientations. Future work Extensions of our model include Hurst index may vary spatially.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Conclusion and future work Conclusion Introduce a new stochastic model defined as a local version of an AFBF. Simulations based on tangent field formulation and the turning bands method produce textures with prescribed local orientations. Future work Extensions of our model include Hurst index may vary spatially.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
34
Bibliography Selected papers K. Polisano, M. Clausel, V. Perrier and L. Condat, ”Texture modeling by Gaussian fields with prescribed local orientation”, IEEE ICIP, 2014. A. Bonami and A. Estrade, ”Anisotropic analysis of some Gaussian models”, Journal of Fourier Analysis and Applications, vol. 9, no. 3, pp. 215–236, 2003. H. Bierme, L. Moisan, and F. Richard, “A turning-band method for the simulation of anisotropic fractional Brownian fields,” preprint MAP5 No. 2012-312012, 2012. K.J. Falconer, “Tangent fields and the local structure of random fields,” Journal of Theoretical Probability, vol. 15, no. 3, pp. 731–750, 2002.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
35
Questions ?
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
36
Questions ?
Thank you for your attention.
Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
36
Dynamic programming. The choice of the bands orientation (✓i )16i6n is governed by the computational cost of the BiH ’s within dynamic programming. 1 ✏ e consider the following set: Let the error fixed. Taking N = d tan ✏ ⇢ ✓ ◆ ⇡ p ⇡ 2 VN = (p, q) 2 Z / N 6 p 6 N, 1 6 q 6 N, p ^ q = 1, < arctan < 2 q 2 The aim is to find n pairs in the set VN which minimize the following global cost: s X C(⇥) = C(r(|pik | + qik )) k=1
(✓i+1 where C(`) is the cost of one FBM BiH in O(n log n), under the constraint max i Polisano et al. - Texture modeling by Gaussian field with prescribed local orientation
✓i ) 6 ✏
37
