. Texture modelisation

We characterize textures through their wavelet decomposition.

Idea

We consider that a texture is characterized by the energy of its wavelets coefficients. If we note $z$ the function which represents this texture, we can write:

$$z=\sum_{k}u_{J,k}\phi_{J,k}+\sum_{j=-J}^{-1}\sum_{k}w_{j,k}\psi_{j,k}$$ (4.1)

where $\psi$ is the mother wavelet, $\phi$ the scaling function and $J$ the order of the decomposition. Thus, we consider that a texture is characterized by the sequence:

$$\left( (\vert u_{J,k}\vert^2, k\in {\mathbb{Z}}), (\vert w_{j,k}\vert^2, k\in {\mathbb{Z}}, -J\leq j \leq -1)\right)$$ (4.2)

Probability distribution of the energy

S.G. Mallat checked experimentally that the distribution of the modulus of the wavelets coefficients in a sub-band follows a generalized gaussian law of the form:

$$p_{X}(x)=A \exp \left( -\left(\frac{x}{\alpha}\right)^{\beta}\right) \mathbb{I}_{x\geq 0}$$ (4.3)

For the energy, we get, thanks to (4.3) (with $X\geq 0$):

$$p_{X^2}(y)=\frac{A}{2\sqrt{y}} \exp \left(-\left(\frac{\sqrt{y}}{\alpha}\right)^{\beta}\right)\mathbb{I}_{y\geq 0}$$ (4.4)

Experimentally, we have checked that the distribution of the energy inside a sub-band is well approximated by a law of this type (see figure 6).

Figure 6: Theoretical graph of the energy distribution in a sub-band (law (4.4)) and experimental histogram

Theoretical graph of the energy distribution in a sub-band (law (4.4))

Histogram of the energy distribution in a sub-band