# Methods for Improving Image Quality for Contour and Textures Analysis Using New Wavelet Methods
## 2 Materials and Methods
### Wiener Filter in the Field Transformed  
If the Wiener filter is not scalar, the calculations required to estimate the useful signal are very laborious. The approach of Wiener filtration in the transformed domain, when working with an orthogonal transformation, has the advantage that in this case we can work with a scalar filter, or which can be approximated by a scalar filter. At the same time, Gaussian additive white noise retains its second-order statistical properties when an orthogonal transformation is used, which means that the noise autocorrelation matrix can be expressed by \(\text{R}_{\text{nn}}\ =\ \alpha^{2}\text{I}_{\text{N}}\), \(\sigma\) are noise dispersion, and I\({}_{\text{N}}\) the N-order unit matrix. The matrix corresponding to the Wiener filter G\({}_{\text{wiener}}\ =\ \text{G}_{\text{w}}\) is given by:  
\[\text{G}_{w}\ =\ \text{R}_{\text{xx}}\left(\text{R}_{\text{xx}}+\sigma^{2} \text{I}_{\text{N}}\right)^{-1} \tag{12}\]  
where R\({}_{\text{xx}}\) is the autocorrelation matrix of the signal. The development of the matrix R\({}_{\text{xx}}\) according to the eigenvalues and eigenvectors is given by:  
\[\text{R}_{\text{xx}}\ =\ \text{U}\text{A}\text{U}^{\text{T}}\ =\ \sum_{k\ =\ 1}^{\text{N}_{k}}\lambda_{k}\mu_{k}\mu_{k}^{\text{T}} \tag{13}\]  
where u\({}_{k}\) and \(\lambda_{k}\) are the eigenvectors and the eigenvalues of the matrix R\({}_{\text{xx}}\), N\({}_{\text{S}}\)\(<\) N, respectively, being the number of eigenvalues. The matrix U consists of the eigenvectors of the matrix R\({}_{\text{xx}}\), being a matrix of dimensions N\(\times\) N\({}_{\text{s}}\), U \(=\ [\text{u}_{1},\text{u}_{2},\dots,\text{u}_{\text{N}_{k}}]\). The matrix \(\Lambda\) is a diagonal matrix containing eigenvalues, \(\Lambda\ =\ \text{diag}[\lambda_{1},\lambda_{2},\dots,\lambda_{\text{N}_{k}}]\). The matrix U defines the Karhunen-Loewe transformation for the signal \(x\) and, as a result, decoratesthe input data, respectively, the matrix \(\rm R_{xx}\), concentrating the signal energy in the smallest possible subspace. In terms of development according to eigenvalues and eigenvectors, the Wiener filter can be expressed as:  
\[\rm G_{w}\ =\ \rm U\widetilde{\Lambda}U^{T}\ =\ \sum_{k\ =\ 1}^{N_{s}}\frac{ \lambda_{k}}{\lambda_{k}+\sigma^{2}}u_{k}u_{k}^{T} \tag{14}\]  
where  
\[\rm\widetilde{\Lambda}\ =\ \rm diag\bigg{[}\frac{\lambda_{1}}{\lambda_{1}+ \sigma^{2}},\frac{\lambda_{2}}{\lambda_{2}+\sigma^{2}},\ldots,\frac{\lambda_{ N_{s}}}{\lambda_{N_{s}}+\sigma^{2}}\bigg{]} \tag{15}\]  
This formulation indicates that the Wiener filter in the transformed domain first performs the Karhunen-Loewe transformation of the data by, \(\rm U^{T}\) then processes, using the optimal weighting matrix \(\rm\widetilde{\Lambda}\), individually each coefficient. Finally, the data thus processed are returned to the primary space by \(\rm U\). From the wavelet transformation that approximates the deceleration of the input data and the concentration of the signal energy in a small number of coefficients, it can be used as an approximation of the Karhunen-Loewe transformation, i.e., the wavelet development can approximate the development of a wide class of signals. Additionally, the procedure of hard truncation of the wavelet coefficients, by retaining only a fraction of all the development coefficients, implicitly presupposes this approximation. It is considered that the procedure of hard truncation of the wavelet coefficients is supposed to be retained following the operation of comparison with \(\lambda\) a first threshold \(\rm N_{\lambda}\ <\rm N\) (as a result of an operation of rearrangement of the coefficients, since the wavelet transformation unlike other transformations such as Fourier, discrete cosine or even Karhunen-Loewe does not generate decreasing spectra).