*3.8. The Bayesian Classifier*

One of the most effective methods of extracting the cracking patterns from a digital image of the cracked surface of a cement composite is the use of machine learning algorithms. One of the most popular algorithms used for this purpose is the Bayesian classifier, which is a simple probabilistic classifier. The classifier is based on the assumption of mutual independence of predictors (independent variables). They often have no relation to reality and therefore the classifier is often referred to as the Naive Bayesian classifier. In this case, the probability model can be derived using the Bayesian theorem, which binds the conditional probabilities of two mutually dependent events. Depending on the type of accuracy of the probability model, the Bayesian classifier can be effectively taught in a supervised learning mode. In many practical applications, the estimation of the parameter for the Bayesian classifier occurs using the maximum probability method "a posteriori", which is a kind of paradox because the classifier can be used without believing in the Bayesian theorem. A characteristic feature of the classifier is that the classification is valid as long as the correct class is more likely than others. Despite its naive design and many simplified assumptions, the Bayesian classifier is very effective in real situations [87].

Schmugge et al. [88] developed an automatic crack identification method for use in concrete and steel construction elements of nuclear power plants. In this case the Bayesian classifier was used to classify whether a pair of line segments belong to the same segment or not. These linear segments were a segment of the cracks. Then, certain geometrical features of the line were determined for each segment, e.g., distance, orientation and similarity. The tests carried out on real images using the developed method showed an improvement in the ability to identify cracks by 38% compared to the reference methods. Hutchinson and Chen [89] proposed an automatic system for the detection of various types of defects in concrete, not only cracks but also cavities. Their procedure is based on the Bayesian decision theory. The detection of a probable crack is based on the implementation of two algorithms, i.e., the Canny and the fast Haar transform. The Bayesian classifier, on the other hand, decided whether the identified area in the image, which may be a crack, is really a crack. The advantage of the developed method is a quite accurate localization of concrete surface damage, with a small calculation effort. Valenca et al. [90] went one step further and created a powerful tool called "SurfCrete" for analyzing concrete surfaces. With this method, it is possible to classify an area into different classes, e.g., crack, biological corrosion, exposed aggregate grain or repair mortar. In their work they used two types of classifiers, i.e., the Bayesian classifier and the Multi-Layer Perceptron. Each of the classifiers was taught on a fragment of the surface showing a given morphological change of the concrete surface. The effect of the method is the possibility to create a map of concrete surface damage. After validation of the method, the accuracy of approx. 94% was obtained on the real concrete surface images. The Bayesian classifier is also popular in cracks detection systems in road surfaces [91,92].
