Hölder regularity of anisotropic Gaussian fields’ "projections" Geoffrey Boutard Laboratory Paul Painlevé - Lille 1 University of Science and Technology
July 2, 2014
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
1 / 19
1
Introduction and motivations
2
A main result of Bonami and Estrade
3
Some improvements version of it
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
2 / 19
Introduction and motivations
1
Introduction and motivations
2
A main result of Bonami and Estrade
3
Some improvements version of it
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
3 / 19
Introduction and motivations
One example of application in medicine
How to study the roughness of a 2D-picture modeled by a stochastic field X = {X (t1 , t2 ) : (t1 , t2 ) ∈ R2 } ?
→ Directional Average Method: we focus on 1D-signals obtained by averaging the image along some lines. Application : resistance of a material, detection of osteoporosis,...
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
4 / 19
Introduction and motivations
P for projection Let us define precisely those 1D-signals. One denotes by ϕ : R → R an arbitrary compactly supported continuous function. It will play the same role as a weight (even if it takes negative values); this is why we impose, Z ϕ(x)dx = 1. (1) R
Then, taking the ϕ-weighted average of the image X along (let us say) the vertical axis, one obtains a 1D-signal denoted by Pϕ X and defined by, Z ∀t ∈ R, Pϕ X (t) =
(2)
X (t, s)ϕ(s) ds. R
→ The notation Pϕ X is not innocent: Pϕ X can be viewed somehow as a projection and is called the projection of X with weight ϕ. G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
5 / 19
Introduction and motivations
Definition of the model we will focus on A classical model in image processing is {BH (t) : t ∈ R2 }, the isotropic fractional Brownian field (fBf) of Hurst parameter H ∈ (0, 1). It is defined by 2
Z
∀t ∈ R , BH (t) =
R2
e it·ξ − 1 c dW (ξ), |ξ|H+1
(3)
c is the Fourier transform of the white noise dW . where dW Yet such a modeling does not fit with anisotropic images. A natural method to introduce anisotropy in the fBf model consists in extending it in the following way, Z e it·ξ − 1 c(ξ), ∀t ∈ R2 , Bhaniso (t) = dW (4) h(ξ/|ξ|)+1 R2 |ξ| where h is a function defined on the unit sphere S 1 of the frequency domain R2 . G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
6 / 19
Introduction and motivations
We will focus on a quite more general model: real-valued Gaussian fields with stationary increments X = {X (t); t ∈ Rd } whose harmonizable representation is given by, ∀t ∈ Rd , X (t) =
Z
c(ξ), e it·ξ − 1 fb(ξ) dW
(5)
Rd
where fb is a non-negative function called the spectral density of X and satisfying, Z min 1, |ξ|2 fb(ξ)2 dξ < +∞.
(6)
Rd
Remark: even if be d can be an arbitrary positive integer, for the sake of convenience, in the sequel, we will take d = 2 or d = 1.
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
7 / 19
Introduction and motivations
What is the effect of the "projection" on Hölder regularity ?
Remark: the 1D-signal obtained by averaging over all the lines orthogonal to the horizontal axis has an harmonizable representation, and its spectral density is given by the square root of the function T (fb) defined by, Z ∀ξ1 ∈ R, T (fb)(ξ1 ) = |ϕ(ξ b 2 )|2 fb(ξ1 , ξ2 )2 dξ2 . (7) R
Then, since it will be the rate of vanishing at infinity of the spectral density fb which will determine the Hölder regularity of X , it is natural to ask us: Question Is there a link between the Hölder regularity of the field X and the Hölder regularity of the "projected" process Pϕ,X ?
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
8 / 19
A main result of Bonami and Estrade
1
Introduction and motivations
2
A main result of Bonami and Estrade
3
Some improvements version of it
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
9 / 19
A main result of Bonami and Estrade
A main result of Bonami and Estrade A main result: Bonami, Estrade (2003) One has on an event of probability one Ω∗ (fb, ϕ) depending on fb and ϕ such that, Hölder regularity of Pϕ X 1 = Hölder regularity of X + . 2 This result has been obtained under some conditions on the rate of vanishing at infinity of fb in a cone containing the horizontal axis. The strategy of their proof relies on the Hölder continuity theorem of Kolmogorov and Čentsov; it consists in showing that the incremental variance of Pϕ X satisfies, E |Pϕ X (s 0 ) − Pϕ X (s 00 )|2 ≈ |s 0 − s 00 |2(βX +1/2) ,
(8)
where βX denotes the Hölder regularity of X . G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
10 / 19
A main result of Bonami and Estrade
A natural question Is it possible to show that the result of Bonami and Estrade remains true on a universal event of probability one non-depending on fb and ϕ?
Main result of our talk Under some conditions on fb and ϕ, our strategy to provide a positive answer to this question relies on wavelet methods.
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
11 / 19
Some improvements version of it
1
Introduction and motivations
2
A main result of Bonami and Estrade
3
Some improvements version of it
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
12 / 19
Some improvements version of it
Wavelets basis of L2 (R2 ) : some basical recalls
The Lemarié-Meyer wavelet basis of L2 (R), LM = {ψj,k : j ∈ Z, k ∈ Z}, is obtained by translation and dilatation of a mother wavelet ψ belonging to the Schwartz space S(R): that is, one has for all t ∈ R, ψj,k (t) = 2j/2 ψ(2j t − k).
(9)
b the Fourier transform of ψ; it is worth noticing that it is a We denote by ψ, compactly supported function vanishing in a neighhorhood of the origin. An othonormal basis of L2 (R2 ) can be obtained as the tensor product of LM with itself: LM ⊗ LM = {ψj1 ,k1 ⊗ ψj2 ,k2 ; j1 , j2 , k1 , k2 ∈ Z}, where ∀(t1 , t2 ) ∈ R2 , ψj1 ,k1 ⊗ ψj2 ,k2 (t1 , t2 ) = ψj1 ,k1 (t1 )ψj2 ,k2 (t2 ).
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
(10)
13 / 19
Some improvements version of it
Construction of a wavelet series representation for X Preliminary work: for any fixed (t1 , t2 ) ∈ R2 , the kernel function (ξ1 , ξ2 ) 7−→ Fb(t1 , t2 ; ξ1 , ξ2 ) = e i(t1 ξ1 +t2 ξ2 ) − 1 fb(ξ1 , ξ2 )
(11)
d ⊗ LM, d the orthonormal basis of belongs to L2 (R2 ), by expanding it in LM 2 2 L (R ) obtained as the image of LM ⊗ LM by Fourier transform, one obtains that F (t1 , t2 ; ·, ·) = (12) X Ψj1 ,j2 (2j1 t1 − k1 , 2j2 t2 − k2 ) − Ψj1 ,j2 (−k1 , −k2 ) ψbj1 ,k1 ⊗ ψbj2 ,k2 , (j1 ,j2 ,k1 ,k2 )
where the series converge in L2 (R2 ) and where for all (x1 , x2 ) ∈ R2 , Z (j1 +j2 )/2 b 1 )ψ(ξ b 2 ) dξ1 dξ2 Ψj1 ,j2 (x1 , x2 ) = 2 e i(x1 ξ1 +x2 ξ2 ) fb(2j1 ξ1 , 2j2 ξ2 )ψ(ξ
(13)
R2
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
14 / 19
Some improvements version of it
Wavelet series representation for the field X Next, using the isometry property of Wiener integral one obtains the following wavelet decomposition: for all t ∈ R2 , in L2 (Ω),
(14)
X (t1 , t2 ) = X
j1
j2
Ψj1 ,j2 (2 t1 − k1 , 2 t2 − k2 ) − Ψj1 ,j2 (−k1 , −k2 ) εj1 ,j2 ,k1 ,k2 .
(j1 ,j2 ,k1 ,k2 )∈Z2 ×Z2
where the εj1 ,j2 ,k1 ,k2 are independent N (0, 1) random variables defined by Z c(ξ1 , ξ2 ). εj ,j ,k ,k = ψbj ,k ⊗ ψbj ,k (ξ1 , ξ2 )dW 1 2
1
2
R2
1
1
2
2
(15)
One of the main advantage of this wavelet representation comes from the fact that it isolates the randomness of the field X in the sequence of random variables εj1 ,j2 ,k1 ,k2 which does not depend on the spectral density fb nor the weight function ϕ G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
15 / 19
Some improvements version of it
A result about the Hölder regularity of X Let suppose that, fb has continuous partial derivatives of any order α ∈ {0, 1, 2}2 , there are constants (a1 , a2 ) ∈ R2 , a0 ∈ (0, 1), A > 0 and C > 0 such that ∀(α1 , α2 ) ∈ {0, 1, 2}2 , ∀|ξ| ≥ A, −(a1 +1/2+α1 )/2 1 + |ξ2 |2 −(a2 +1/2+α2 )/2 , |∂ α fb(ξ1 , ξ2 )| ≤ C 1 + |ξ1 |2
(16)
2
∀α ∈ {0, 1, 2} , ∀|ξ| ≤ A, |∂ α fb(ξ)| ≤ C |ξ|−a
0
−1−α1 −α2
(17)
.
Then, denoting by [·] the floor function, almost surely, X admits partial derivatives of order (β1 , β2 ) ∈ {0, 1, . . . , [a1 ]} × {0, 1, . . . , [a2 ]}, for all ε > 0, for all compact set K of R2 , there exists a positive constant C > 0 such that, for all (t10 , t20 ), (t100 , t200 ) in K , |∂ ([a1 ],[a2 ]) X (t10 , t20 ) − ∂ ([a1 ],[a2 ]) X (t100 , t200 )| ≤ C (|t10 − t100 |a1 −[a1 ]−ε + |t20 − t200 |a2 −[a2 ]−ε ).
(18) G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
16 / 19
Some improvements version of it
A result about the Hölder regularity Pϕ X Let suppose now that, fb has continuous partial derivatives of any order α ∈ {0, 1, 2}2 , there are constants (a1 , a2 ) ∈ R2 , a0 ∈ (0, 1), A > 0 and C > 0 such that ∀(α1 , α2 ) ∈ {0, 1, 2}2 , ∀|ξ| ≥ A, −(a1 +α1 )/2 1 + |ξ2 |2 −(a2 +α2 )/2 |(ξ1 , ξ2 )|−1 , (19) |∂ α fb(ξ1 , ξ2 )| ≤ C 1 + |ξ1 |2 ∀α ∈ {0, 1, 2}2 , ∀|ξ| ≤ A, |∂ α fb(ξ)| ≤ C |ξ|−a
0
−1−|α|
(20)
Then, almost surely, Pϕ X can be derivated [a1 + 1/2] times, for all ε > 0, for all compact set K of R, there exists a positive constant C > 0, such that for all t10 , t100 in K , |Pϕ X ([a1 +1/2]) (t10 ) − Pϕ X ([a1 +1/2]) (t100 )| ≤ C |t10 − t100 |a1 +1/2−[a1 +1/2]−ε . (21) G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
17 / 19
Some improvements version of it
References
A. Ayache, S. Jaffard. M.S. Taqqu. Wavelet construction of Generalized Multifractal processes (2007), A. Ayache, M.S. Taqqu. Multifractal Processes with Random Exponent (2005), A. Ayache, Y. Xiao. Asymptotic properties and Hausdorff Dimensions of Fractional Brownian Sheets (2005), A. Bonami, A. Estrade. Anisotropic Analysis of Some Gausian Models (2003), R. Jennane, R. Harba, E. Perrin, A. Bonami, A. Estrade. Analyse de champs browniens fractionnaires anisotropes.
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
18 / 19
Some improvements version of it
Thank you for your attention !
G. Boutard (University Lille 1)
Hölder regularity
July 2, 2014
19 / 19
