# Methods for Improving Image Quality for Contour and Textures Analysis Using New Wavelet Methods
## 2 Materials and Methods
### Reconstruction of Contours from Images Using Nonlinear Cellular Network (NCN) Algorithm

Table 4: Results obtained by the combined use of undecided wavelet transformation and Wiener filtration in the wavelet domain. Test images are 256 \(\times\) 256 pixels. A total of 16 circular displacements were used to calculate the translation invariant wavelet transform. The formulation of processing an image using differential equations allows the description under a similar form of complex processing of the same image. For example, if two distinct processes are described by the equations:  
\[\frac{\partial\mathrm{I}}{\partial t}\ =\ F_{1}[\mathrm{I}(\mathrm{x},\mathrm{y},t)]\ \ \textit{and}\ \frac{\partial\mathrm{I}}{\partial t}\ =\ F_{2}[\mathrm{I}(\mathrm{x}, \mathrm{y},t)] \tag{25}\]  
A combined processing of the distinct processes can be described by combining the two equations, namely:  
\[\frac{\partial\mathrm{I}}{\partial t}\ =\ F_{1}+\lambda F_{2} \tag{26}\]  
where \(\lambda\in\mathrm{R}^{+}\). If \(\mathrm{I}_{1}\) and \(\mathrm{I}_{2}\) result from minimizing the energies \(\mathrm{E}_{1}\) and \(\mathrm{E}_{1}\), the differential Equation (27) is solved by minimizing the energy \(\mathrm{E}_{1}+\lambda\mathrm{E}_{2}\). To minimize such energies in as short a time as possible, nonlinear cellular network can be used. For the use of cellular neural networks in minimizing energies in applications of processing images with grayscale, appropriate operators (templates) must be designed. The proposed method of improving the contours in the images assumes that the corresponding transfer functions depend on the local, or sometimes regional, level of intensity or contrast. When implementing the proposed method, we start from an energy function \(\mathrm{E}\), of general form:  
\[\mathrm{E}(\mathrm{I},\mathrm{G})\ =\ \iint\|\
abla\mathrm{I}\|^{2}dxdy+ \lambda|\mathrm{G}| \tag{27}\]  
In image analysis and processing an important problem is the reconstruction of an original image \(\Phi_{0}\), from the acquired degraded image \(\Phi\) (observed and degraded image). Two phenomena can contribute to image damage: one is related to the acquisition method (for example, the acquisition of a CT image by projection or blurring caused by motion), and the other is the inherent random noise that is associated with any useful signal. Consider an original grayscale image by \(\Phi_{0}(\mathrm{x},\mathrm{y})\), \(\Phi_{0}(\mathrm{x},\mathrm{y})\ :\mathrm{R}^{2}\rightarrow\mathrm{R}\), and \(\Omega\ =\ \left\{(\mathrm{x},\mathrm{y}):\mathrm{x}\in[1,\ \mathrm{M}],\mathrm{y}\in[1, \mathrm{N}],\ \mathrm{M}\ \mathrm{and}\ \mathrm{N}\in\mathrm{R}^{+}\right\}\). The processing of this image according to an operator-based algorithm can be described by the partial differential equation:  
\[\frac{\partial\Omega}{\partial t}\ =\ F[\Phi(\mathrm{x},\mathrm{y},t)] \tag{28}\]  
where an artificial parameter \(\mathrm{t}\) was introduced, \(\mathrm{F}\) being the operator that characterizes the desired processing algorithm (\(\mathrm{F}:\mathrm{R}^{2}\rightarrow\mathrm{R}\)). In general, the function \(\mathrm{F}\) depends on the initial image, its first and second spatial derivatives. The final image, obtained after processing, results as a solution of this partial differential equation. Applying the Dirichlet integral:  
\[\mathrm{E}(\Phi)\ =\ \int|\
abla\Phi|^{2}(\mathrm{x})d\mathrm{x} \tag{29}\]  
which is associated with the equation:  
\[\frac{\partial\Phi}{\partial t}(\mathrm{x},t)\ =\ \
abla\Phi(\mathrm{x}) \tag{30}\]  
Equation (27) becomes:  
\[\mathrm{E}(\mathrm{I},\mathrm{G})\ =\ \iint\|\
abla\mathrm{I}\|^{2}dxdy+ \lambda|\mathrm{G}|\ =\ \iint\limits_{\mathrm{x},\mathrm{y}\in\Omega}^{\mathrm{M},\mathrm{N}\in \mathrm{R}^{+}}\left[\left(\frac{\partial\mathrm{I}}{\partial\mathrm{x}} \right)^{2}+\left(\frac{\partial\mathrm{I}}{\partial\mathrm{y}}\right)^{2} \right]d\mathrm{x}d\mathrm{y}+\lambda|\mathrm{G}| \tag{31}\]  
The energy function \(\mathrm{E}\) has two terms: a term corresponding to the uniformity constraint and a term to penalize the discontinuities that, in general, correspond to the contours.