# Haar Wavelet-Based Classification Method for Visual Information Processing Systems
## 2 Materials and Methods
Figure 1: Features of the search for _WT_-based extremum: WF Haar impulse response (**a**)’ fragment of the objective function (**b**). According to (1), the iterative scheme is similar to the iterative scheme for finding a first-order optimum. In the last one, the direction of movement to an extremum is determined using a finite-difference estimate of the derivative (Figure 2a,b). To estimate the direction of search for the optimum in (2), the symmetric and nonstationary Haar WF were selected (see Figure 1). This enables us to use the integral character of \(WT\), as well as to identify the segment of the objective function where the global extremum is located. Changing the WF carrier length in the subsequent steps of the search allows us to reduce the error of extremum determination inside the segment of the objective function found in the previous steps (to narrow the interval). Determination of the variation range of the separating surfaces coefficients is based on Haar \(WT\) optimization with initial data: \(\delta_{1}\)--start error; \(\delta_{2}\)--error of coefficient value; \(\delta_{3}\)--error in determining the coefficient range. Step 1. Setting: \(c[0]\)--initial approximation to the optimum coordinate; \(\gamma[1]\)--step; \(a\)--WF sampling step; \(s_{1}\)--WF carrier length of the first start \(\Psi_{1}(i)\); \(s_{k}\)--the step of changing the length of \(\Psi_{1}(i)\) in determining the value ranges of coefficients (in this article \(s_{k}=1\)); start number \(k=1\); iteration number \(n=1\);  
Step 2. According to (2), the direction of search is estimated for the start \(k\). At \(k=1\), for this, a weighted sum with WF \(\Psi_{1}(i)\) is used. The carrier length \(s_{1}\) for WF \(\Psi_{1}(i)\) is determined by analyzing the objective. The integral character of such \(WT\) enables us to reduce the sensitivity to local extrema and allocate a segment of the objective function and determine the range of change for its coordinates with a low error;  
At this step, the sign of the estimate by (3) is checked. If the sign changes, then a number of nested ranges for extremum coordinate changes are determined. The maximum range is determined at the start--\(k=1\) c \(s=s_{1}\)--by the length of the WF carrier for the first start, \(\Psi_{1}(i)\), in the subsequent processing steps with the WF carrier length, which varies as \(s=s-s_{k}\);  
Step 3. The range of the coefficient value is searched using (1);  
Step 4. If the condition is satisfied at the iteration \(n\): \(|c[n]-c[n-1]|\leq\delta_{1}\), then the search at the current start ends, otherwise--\(n=n+1\) and go to step 2;  
Step 5. If \(k>1\) and the coefficient value found at the \(k\)-th step differs from the result of the \(k-1\) start by no more than \(\delta_{2}\), then the algorithm is ended. In the opposite case (or \(k<k_{max}\)), the start number is increased \(k=k+1\). In particular, having \(1<k<k_{max}\), the WF is represented by \(\Psi_{1}(i)\), and at \(k=k_{max}\), the search direction is evaluated by discrete differentiation (see Figure 1) and the jump to Step 2 is performed. When the sign changes, a number of nested ranges of extremum coordinate changes are defined. In classification, these are the ranges of coefficient values for the segments separating the classes in the feature space. The maximum range is defined with the  
Figure 2: Specific features of extremum search with gradient estimation: fragment of objective function (a); gradient estimation (b). maximum length of the WF carrier; the other ranges are defined with increasingly shorter lengths of the WF carrier.