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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
the function which represents this texture, we can write:
![$\displaystyle z=\sum_{k}u_{J,k}\phi_{J,k}+\sum_{j=-J}^{-1}\sum_{k}w_{j,k}\psi_{j,k}$](img37.png) |
(4.1) |
where
is the mother wavelet,
the scaling function and
the order of the decomposition.
Thus, we consider that a texture is characterized by the sequence:
![$\displaystyle \left( (\vert u_{J,k}\vert^2, k\in {\mathbb{Z}}), (\vert w_{j,k}\vert^2, k\in {\mathbb{Z}}, -J\leq j \leq -1)\right)$](img41.png) |
(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:
![$\displaystyle p_{X}(x)=A \exp \left( -\left(\frac{x}{\alpha}\right)^{\beta}\right) \mathbb{I}_{x\geq 0}$](img42.png) |
(4.3) |
For the energy, we get, thanks to (4.3) (with
):
![$\displaystyle p_{X^2}(y)=\frac{A}{2\sqrt{y}} \exp \left(-\left(\frac{\sqrt{y}}{\alpha}\right)^{\beta}\right)\mathbb{I}_{y\geq 0}$](img44.png) |
(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)) |
![\includegraphics[scale=0.6]{graphetheorie31param1.eps}](img45.png) |
Histogram of the energy distribution in a sub-band |
![\includegraphics[scale=0.6]{belhisto31param1.PS}](img46.png) |
|
Next: . Complete functional
Up: Supervised classification for textured
Previous: . About wavelets
Jean-Francois Aujol
2002-12-03