# Haar Wavelet-Based Classification Method for Visual Information Processing Systems
## 3 Case Study
The time (by timer) when determining the range for the coefficients \(c_{0}\) and \(c_{1}\) was, respectively, 0.08 s and 0.1 s.  
The step \(\gamma=0.3\) was selected for the further research. During calculation, the one range for the class \(\Delta_{1}c_{0}=[-4.0083,-4.00]\) and seven nested ranges for the class \(c_{1}\) are obtained. Figure 3 shows the result of separating these two classes with the help of \(\hat{f}(x,c)=-4.0+\Delta_{j}c_{1}x\) at \(\Delta_{1}c_{1}=[1.0,2.36]\) (a line 1 in Figure 3), \(\Delta_{3}c_{1}=[1.15,2.24]\) (a line 2 in Figure 3), and \(\Delta_{7}c_{1}=[1.43,1.68]\) (a line 3 in Figure 3). Range detection time for \(c_{0}\) was 0.23 s, and for \(c_{1}\) was 13.1 s.  
Based on results above, we may conclude firstly that reducing the parameter \(\gamma\) enables us to determine a greater number of ranges. At the same time, time costs can increase by more than two orders of magnitude. However, obtaining a set of ranges when debugging the classification method allows us to evaluate the relationship between the parameter \(\gamma\), the error \(\delta_{1}\), and the classification performance. Secondly, we may conclude that the time for determining the set of nested ranges for coefficients depends on the variance in the training sample at the training mode and the variance in the working mode of classification. Moreover, for the widest range, it can be less by several times than for subsequent ranges. To evaluate the classifier parameters in terms of pragmatic sufficiency, we propose to carry out the definition of the classification efficiency indicator based on the statistical  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{ \begin{tabular}{c} **Interval Width Changing for** \\ **Coefficient of the Separating** \\ **Surface, H** \\ \end{tabular}  } & \multicolumn{4}{c}{**Relative Value of the Average Risk, R**} \\ \cline{2-5}  & \(q_{p}\)=3\(q_{0}\) & \(q_{p}\)=4\(q_{0}\) & \(q_{p}\)=5\(q_{0}\) & \(q_{p}\)=6\(q_{0}\) \\ \cline{2-5}  & **1** & **2** & **3** & **4** \\ \hline
0.005 & 0.8 & 0.6 & 0.4 & 0.3 \\ \hline
0.007 & 1.4 & 0.9 & 0  
approach using the Shannon entropy formula for measuring the information content. For this purpose, the following designations were employed:  
\(P(\Omega)=\{P(\Omega_{1}),P(\Omega_{2}),\ldots,P(\Omega_{L})\}\)--the priori probability distribution for occurrence of a class of objects;  
\(P(D)=\{P(D_{1}),P(D_{2}),\ldots,P(D_{L})\}\)--the probability distribution of deciding whether objects belong to the appropriate class;  
\(P(\Omega)/D\)--the conditional probability distribution for occurrence of the object \(\Omega_{i}\) under the condition that the classifier makes a decision \(D_{j}\). Then, the information measure [33]:  
\[\mathrm{I}(\Omega,D)=\mathrm{I}(D,\Omega)=-\sum_{i=1}^{L}P(D_{i})\mathrm{log}_{ 2}\left(P(D_{i})\right)+\sum_{j=1}^{L}P(\Omega_{i})\sum_{i=1}^{L}P\big{(}D_{i} /\Omega_{j}\big{)}\mathrm{log}_{2}\big{(}P\big{(}D_{i}/\Omega_{j}\big{)}\big{)}. \tag{4}\]  
Usage of (4) is complicated by the difficulty of calculating the conditional probabilities \(P\big{(}\Omega_{i}/D_{j}\big{)}\). Therefore, the symmetry property of information measure is employed for calculations \(\mathrm{I}(\Omega,D)=\mathrm{I}(D,\Omega)\). The probability distributions can be found using the matrix \(P\) data:  
\[P(D_{i})=\sum_{j=1}^{L}P\big{(}\Omega_{j}\big{)}p_{ij},\ \forall i=\overline{1,L}, \tag{5}\]  
\[P\big{(}D_{j}/\Omega_{i}\big{)}=p_{ij},\ \forall i=\overline{1,L}. \tag{6}\]  
Since the priori probabilities are unknown, we assume \(P(\Omega_{i})=\frac{1}{L}\).