A method comprising characterizing the anisotropy of the texture of a digital image, wherein characterizing comprises:
- a) acquiring a digital image formed from pixels, each pixel being associated with a light intensity and with a position in space d, where d is a natural integer higher than or equal to two;
- b) automatically converting) the acquired image in order to obtain a converted image Ij,k, the conversion comprising applying a modification Tj,k of the image that makes each pixel of the acquired image rotate from one position to the next by an angle αj about a point or an axis and that enlarges or shrinks the image by a factor γk, where αj=arg(ujk) and γk=|ujk|2, ujk being a vector that completely characterizes the modification Tj,k, the indices j and k uniquely identifying the angle αj and the factor γk, respectively,
- then, for each converted image, calculating a K-increment Vj,k[m] for each pixel of position m of the converted image, this K-increment being calculated by applying a convolution kernel v by means of the following formula:
- V
- j
- ,
- k
-
- [
- m
- ]
- =
- ∑
- p
- ∈
- [
- 0
- ,
- L
- ]
- d
-
- v
-
- [
- p
- ]
- ·
- Z
-
- [
- m
- -
- T
- j
- ,
- k
- ·
- p
- ]
- where:
- the product Tj,k·p corresponds to the application of the modification Tj,k to the pixel that initially had the position p in the image I;
- the convolution kernel v performs linear filtering and possesses a characteristic polynomial Qv(z) and a finite support [0,L]d, v[p] being the value of the convolution kernel v for the position p, the characteristic polynomial Qv(z) being defined by the following formula:
- ∀
- z
- ∈
- d
- ,
- Q
- v
-
- (
- z
- )
- =
- ∑
- p
- ∈
- [
- 0
- ,
- L
- ]
- d
-
-
- v
-
- [
- p
- ]
-
- z
- p
- and meeting the following condition:
- ∀a∈[0,K]d such that |a|≤K then
- ∂
-
- a
-
-
- Q
- v
- ∂
- z
- 1
- a
- 1
-
-
- …
-
-
- ∂
- z
- d
- a
- d
-
- (
- 1
- ,
- …
-
- ,
- 1
- )
- =
- 0
- where:
- L is an acquired vector of [0, N]d that parametrizes the kernel v, N is a vector belonging to d that codes the size of the image and the components of which are strictly positive natural integers;
- the constant K is a nonzero acquired natural integer;
- z is a vector of components z1, z2, . . . , zd;
- zp designates the monomial z1p1*z2p2* . . . *zdpd;
- ∂|a|Qv/∂z1a1 . . . ∂zdad is the partial derivative of the polynomial Qv(z) with respect to the components of the vector z, the symbol ∂ziai indicating a differentiation of the polynomial Qv(z) of order ai with respect to the variable zi, where zi is the i-th component of the vector z and ai is the i-th component of the vector a, i being an integer index higher than or equal to 0 and lower than or equal to d;
- the step b) being executed with nj different angles αj and, for each angle αj, with at least two different factors γk, nj being an integer higher than or equal to 2 so as to obtain at least four different converted images Ij,k;
- c) for each different converted image Ij,k, calculating a p-variation Wj,k of this converted image from said calculated K-increments;
- d) estimating the terms βj of the following statistical regression:
- log(|Wj,k|)=log(|ujk|2)*H+βj+εj,k, where:
- H is the Hurst exponent of the acquired image;
- εj,k is an error term of the regression the statistical properties of which are predetermined;
- wherein the method further comprises:
- e) estimating the scalar coefficients τm of an even function τ(θ) defined in [0;
- 2π] that minimizes the following criterion C:
- C
- =
- ∑
- j
- =
- 1
- n
- j
-
-
- (
- β
- j
- -
- τ
- *
- Γ
-
- (
- α
- j
- )
- )
- 2
- where:
- βj are the terms estimated in step d), τ(θ) is the function defined by the following relationship for any angle θ belonging to [0;
- 2π]:
- τ
-
- (
- θ
- )
- =
- ∑
- m
- =
- 0
- M
-
-
- τ
- m
-
- f
- m
-
- (
- θ
- )
- where:
- M is a constant integer number higher than an acquired datum, τm are the scalar coefficients of the function τ(θ), fm(θ) are the functions of a basis of π-periodic functions defined in the interval [0;
- 2π], Γ(θ) is the function defined by the following relationship:
- Γ
-
- (
- θ
- )
- =
- ∫
- ℝ
- +
-
-
- v
- ^
-
- (
- ρθ
- )
-
- 2
-
- ρ
- -
- 2
-
- H
- -
- 1
-
- d
-
- ρ
- where:
- {circumflex over (V)} is the discrete Fourier transform of the kernel v,
- H is the Hurst exponent of the acquired image, ρ is the variable of integration,
- the symbol “*” designates the circular convolution of the functions τ(θ) and Γ(θ), f) then, calculating, depending on the estimate of the scalar coefficients τm, an anisotropy index that characterizes the anisotropy of the image, this index varying monotonically as a function of the statistical dispersion of the values of the function τ(θ) for θ varying between 0 and π.