## 2 Measuring System, Materials and Methods
### Methodology
#### 2.1.3 Features Extraction
Many attributes can be beneficial for exploring association rules (facilitating a more nuanced understanding of the relationships between waveform parameters and structural characteristics) but also cause problems with building identification models. As the number of attributes increases, the problem's dimensionality exponentially escalates the requisite size of the database (the number of records). This phenomenon is commonly referred to as the curse of dimensionality. Furthermore, in the NDT investigations, it is often observed that all shape attributes are highly correlated. A significant reduction in the attribute number is required for these two reasons. Usually, the number should be five at most. Techniques such as aggregation, filtering, wrappers, or more advanced tools like the rough set theory [71, 72] and Principal Component Analysis (PCA) may be employed to achieve this reduction. However, selecting relevant attributes is inherently time-consuming and highly subjective, requiring expertise and extensive empirical testing to identify appropriate tools effectively [7, 44]. Figure 5: Categorization and evolution of methods for determining waveform and signal features. The process of determining and selecting features of the waveform typically requires a specific compromise between reducing the number of attributes (reduction) and accurately capturing the shape of the waveform. The method of characteristic points can be used to minimize or entirely circumvent the need for reduction. Before determining the final attributes, it is usually helpful to investigate the association rules between the waveforms and the parameters of the studied structure. It can be achieved through mathematical analysis, statistical methods, graphical representations, and algorithms for discovering association rules, such as the Apriori algorithm (market basket analysis) [70]. Then, characteristic points are defined based on the results of the analyses. The tests can show which parameters of the waveform exhibit variability in response to changes in specific structure parameters. Also, using equal partition methods and reducing the number of attributes can uncover association rules and identify characteristic points. Particular emphasis is often placed on extreme values (both local and global) and inflection points, along with their coordinates. The number of attributes obtained through the method of characteristic points is typically small, rendering further reduction either straightforward or entirely unnecessary. However, it is essential to note that significant information regarding the shape of the waveform may be lost during this process (resulting in an imprecise representation). Consequently, this method is predominantly effective in standard, predetermined cases. For instance, the method of characteristic points has been applied in studies using the multi-frequency eddy current method (MMFM) [7,44]. The approximation method can be considered as an enhancement and extension of the characteristic points method. This approach sometimes allows describing (with high precision) a given curve (waveform or signal) using only a few parameters. Attributes are generated by fitting the mathematical function to the waveform with an approximating function and determining the parameters of that chosen function (the parameter values are considered as the attribute values). This method proves particularly effective when a theoretical description of the process is known or a specific function is fitted to the observed curve. Using universal solutions, such as polynomial functions, often leads to inadequate shape representation or representing the waveform with too many attributes, which subsequently leads to the need for further reduction. ACO (amplitude-correlation-offset) decomposition is a method (presented in [7]) that does not require prior knowledge of the studied process or the huge attribute processing and selection stage. The method uses ACO attributes created by comparing the parameters of the tested and the reference waveforms.