## 3. Analysis of Test Persons’ Handwritten Passwords and Signatures and Features Space

Nowadays, different orthogonal basis functions [

9] are used to describe the dynamics of a signature; however, there is no comparative evaluation of the information content for all attributes derived from these functions.

For the purpose of computer processing, the functions of the pen position are presented in the form of readings (sampling) and contain information about the dynamics of the pen movement and a signature’s image. Further, we will consider the signature in a system of coordinates O(x,y,p), where x and y are coordinates of the signature on a x- and y-planes, and p is a value of the pen pressure on the tablet. It is necessary to define robust attributes that characterize both a graphic picture and the dynamics of the handwritten password and to select such attributes that provide a minimum intersection of users’ personal samples in a multidimensional space.

Handwritten password instances recorded at various times differ in range (by amplitude and duration). In order to calculate attribute statistical characteristics, all instances must be the same duration and normalized by power. To scale signals by duration, the functions passed through direct transform into Fourier series and inverse transform with the same duration, thus providing a resampling (equal to an average duration of the scaling signals). Here, the process of changing signatures to equal duration is described:

Discard the first and last values for all dots with zero pressure.

Perform one-dimensional Fourier transform for x(t), y(t), and p(t).

Perform the inverse transform of these functions, taking into account that the output dimension should correspond to the nearest minimum integer multiple of the 2nd power.

The next step is the calculation of biometric features for all signatures. All mentioned features can be divided into groups as is shown in

Table 1. The following is a detailed description of all groups of features. This is reflected in the scientific literature; similar approaches to feature obtaining have been used by various researchers in the design of subject identification and authentication systems via signatures.

#### 3.3. Daubechies Wavelet Transform Coefficients

It is obvious that the analysis of the frequency domain provides advantages in evaluating noisy signals [

10]. A modern mathematical tool for the analysis of spectral characteristics of nonstationary signals is wavelet analysis. Some well-known studies where wavelet transform was applied to calculate attributes using signatures have been noted [

11,

12,

13,

14,

15]. The present paper proposes a transition from the time-domain representation of the functions of the pen position change to the frequency-domain representation, their research, and a search of dynamical characteristics using a method of multiresolution analysis. This approach is based on a discrete wavelet transform and uses Mallat pyramidal algorithm to decompose initial signals into sequences of wavelet coefficients d

_{jk} that characterize a structure of the process to be analyzed in different scales (j). These studies considered different bases of Daubechies wavelets (from D4 to D10 [

16]). To transit to a new analysis scale, the signal length must be multiple of 2

^{n}, where n is a number of factorization levels. If this requirement is not met, the numerical series may be added with deficient values using one of the following ways:

Periodic addition, which means the beginning of the sequence is put at the end of the numerical series.

Mirroring data at the ends of the sequence.

The calculation of special scaling and wavelet functions that are applied to the beginning and the end of the sequence, presupposed by Gram–Schmidt orthogonalization.

A numerical series may be added with zeros as well, but this approach leads to significant mistakes as a rule. In this paper, the time sequence is periodically added with deficient values.

The analyzing signals were discretized at a frequency of 200 Hz; according to the sampling theorem, the high signal frequency is 100 Hz. Thus, for a signal consisting of 256 readings for example, the wavelet coefficients of the first level of decomposition occupy the frequency bandwidth of 50–100 Hz. Wavelet coefficients of the second level describe the harmonics of the spectrum for the bandwidth of 25–50 Hz. The procedure is repeated until there is one wavelet coefficient and one approximation reading at the 9th level. In total, there are 256 coefficients (1 + 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128). This means that the number of coefficients is equal to the number of readings in the initial signal. If the major power of the signal is concentrated around a frequency of 2 Hz, the wavelet coefficients of the 6th level are significant, and the wavelet coefficients of the lower level may be discarded.

The number of wavelet coefficients at any decomposition level depends on the signal duration. The number of wavelet transform coefficients becomes equal and the spread of values decreases when the procedure of scaling functions to the mean duration is performed (

Figure 1 and

Figure 2). This operation allows transform implementations of the signature of one subject to the same number of features and improves the stability of feature values of this group. Herewith, the distinction of feature values is saved for different subjects (

Figure 3).

The moment of the “occurring” of a harmonic in a signal is defined as a multiplication of the wavelet coefficient number and the timing resolution for the corresponding level within the accuracy of the timing resolution value. Thus, the physical significance of the wavelet transform coefficients resulting from multidimensional analysis may be treated as characteristics of signal harmonics that belong to a certain frequency bandwidth and occur in the signal at a certain moment in time. These characteristics may be considered as values of signature characteristics.

The quality of the algorithm work depends on the selected wavelet a lot. This research covers the evaluation of stability for signals obtained using Daubechies wavelets D4, D6, D8, and D10. The analysis of resolution levels start from 3 to 6, which corresponds to the spectral range of 1.5625–25 Hz. This spectral range characterizes the dynamics of reproducing handwritten passwords. The statistical data processing was done using the Mann–Whitney U-test, and the differences were considered reliable when p < 0.05. The minimum time of analysis and consequently the worst-quality result is obtained for D2. The high stability (robustness) was proven in practice for the wavelet spectrum when handwritten signatures were input in different time. This will allow for the evaluation of complicated dynamically changing signals that are formed when handwritten passwords are being represented.

The possibilities of use features for each subject from this group can be different and depend on the mean of signature length.