# Haar Wavelet-Based Classification Method for Visual Information Processing Systems
## 2 Materials and Methods  
The classification consists in assigning the presented objects to one class by comparing their parameters. It is based on the compactness hypothesis--the assumption that objects of the same class are similar in terms of parameter values. In the classification, we search for the minimum of the functional \(Q(x,c)\) over the vector of coefficients \(c=(c_{1},\ldots,c_{N})\). The probability of incorrect classification was estimated as the ratio of the number of incorrectly recognized objects to the total number of objects in the sample. These coefficients \(c=(c_{1},\ldots,c_{N})\) define the type of surface separating the classes in the parameter space. At the first stage (during training the classifier), the separating surface \(y=f(x)\) is constructed by training samples of known classes; at the second stage (in the \"working\" mode), the class of the object under study is determined. Depending on which dataset is provided for research (for pragmatic reasons), three possible approaches to classification in a system for visual information processing can be considered. 1. If the parameters of the general population are known, it is recommended to carry out a point estimation of the coefficient values separating the classes of surfaces;If random sample data with a known type of distribution law with unknown parameters are presented, it is recommended to carry out interval estimates of these coefficient values;
3. If the law of distribution is unknown, we suggest using the iterative method of intelligent data analysis--in particular, the method of classification with _WT_. This method enables us to select areas where the value of the coefficient is located, at which the necessary values of reliability and performance are achieved during the training of the classifier, as well as the adaptation of the system parameters using the Shannon entropy formula. When training the classifier, the class of separating surfaces \(y=f(x)\) is set after the formation of the training sample of parameters. After that the following functional is formulated--\(Q(x,c)=F\left(y-\hat{f}(x,c)\right)\), with \(\hat{f}(x,c)=\sum_{v=1}^{N}c_{v}\varphi_{v}(x)\)--the method of searching the extremum of this functional has to be selected. For this purpose, authors chose the _WT_-based method, employing the _WT_ property as equal to zero at the optimum point [32]. The _WT_ has this property if real wavelets are used as base ones in the form of odd symmetric functions that have compact or efficient support. At the same time, the _WT_ enables us to search efficiently for the extremum of objective functions for the \"ravine\" type, and it has high noise immunity (compared to the differentiation operation). The Haar wavelet function is also characterized by low computational complexity. The impulse response of the Haar wavelet function (WF) is shown in Figure 1a. The illustration of the estimation of the trend towards an extremum is given in Figure 1b. At the stage of classification with training, the coefficients of the ranges of the separating surfaces between classes are determined using Haar _WT_:  
\[c[n]=c[n-1]-\gamma[n]WT_{k}(Q(x[n],c[n-1])), \tag{1}\]  
where \(Q(x,c)\) is the functional that depends on the vector of coefficients for separating surfaces \(c=(c_{1},\ldots,c_{N})\) and measurement data \(x=(x_{1},\ldots,x_{M})\); \(\gamma[n]\) is the step; \(n\) is the iteration number (order); and \(k\) is the start number. In (1),  
\[WT_{k}(Q(x[n],c[n-1]))=\{G_{1k},G_{2k},\ldots,G_{Nk}\} \tag{2}\]  
determines the direction of movement to the extremum, where  
\[G_{jk}=\frac{1}{s_{k}}\sum_{ \begin{subarray}{c}i=-\frac{s_{k}}{2}\\ i\
eq 0\end{subarray} }^{\frac{s_{k}}{2}}Q\big{(}x[n],c_{j}+ia\big{)}\cdot \Psi_{k}(i)\big{)}. \tag{3}\]  
In (3), \(s_{k}\) is the carrier length of WF at \(k\)-th start; \(a\) is the sampling interval; \(\Psi_{k}(i)\) is that Haar WF at \(k\)-th start; \(j=1,\ldots N\) is the dimension of the parameter vector.