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The 2D-Wold decomposition for textures

The texture field, $\{y(n,m)\}$ is decomposed as follows :

\begin{displaymath}y(n,m)=w(n,m)+h(n,m)+\sum_{(\alpha,\beta )\in O}e_{(\alpha,\beta)}(n,m)

$\{w(n,m)\}$ : random component (w.r.t. granularity)
$\{h(n,m)\}$ : harmonic component (w.r.t. repetitiveness)
$\sum_{(\alpha,\beta )\in O}e_{(\alpha,\beta)}(n,m)$ : generalised evanescent component (w.r.t directionality)

The random component can be written as :


$\{u(n,m)\}$ is the 2D innovation which is a white noise with variance $\sigma

The harmonic component is given by :


Cp,Dpare mutually orthogonal random variables

The evanescent component can be expressed :

e_{(\alpha,\beta)}(n,m) & = & \sum_{i=1}^...
...,\beta)}}{\alpha ^{2}+\beta ^{2}}(n\beta+m\alpha))

si,sj,tk,tl are 1-D random proceses mutually orthogonal for all $i,j,k,l,i\neq j,k\neq l$

Radu Stoica