## 4 Image Fusion Applications in NDT Inspection
### Learning-Based Approaches
#### 4.5.3 Bayesian and Dempster-Shafer Methods  
The Dempster-Shafer (D-S) method, known as evidence theory or belief functions, can be seen as a generalised form of the Bayes theorem, which allows revision of the predicted probabilities of an occurrence with additional knowledge. While the Bayes theorem uses probabilities for weighting, the D-S method accomplishes this by assigning masses to all possible subsets [111]. For instance, the authors of [112] proposed a Bayes-based fusion method for tomography inspection, calculating wave velocity probability density functions to improve the impact localisation accuracy. Two sensors that have a 90\({}^{\circ}\) angle difference were incorporated, and non-matched regions and measurement noise were minimised by the multiplication of related probabilities. [PERSON]'s evidence theory can meet poor conditions better than probability theory can, using the belief functions instead of probability. It is also possible to capture cognition notions such as uncertainty. It presents the concept of attributing beliefs and plausibilities to potential measurement hypotheses and the necessary combination rule to combine them [113]. Basically, in the power set of a universal set X, \(P(X)\) shows all possible subsets, including an empty (\(\phi\)) set with its mass function equal to zero:  
\[m(\phi)=0 \tag{3}\]  
The D-S theory employs the mass function (\(m\)), which reflects the level of support for the evidence, as well as the belief function and the plausibility function. The value of \(m(A)\) applies just to the set \(A\) and does not make extra claims about any subsets of \(A\), each of which has its mass by definition. For the two different sensors \(A_{1}\) and \(A_{2}\), the combination rule of the D-S theory is as follows [114]:  
\[m_{1,2}(A)=(\sum_{B\cap C=A\
eq\phi}m(A_{1})m(A_{2}))/(1-K) \tag{4}\]  
\[K=(\sum_{B\cap C=\phi}m(A_{1})m(A_{2})) \tag{5}\]  
\(K\) indicates the amount of conflict between the two mass functions from the sensors. The equation means that the D-S combination procedure computes the orthogonal sum of each possible proposition and then combines these two beliefs to update the joint belief mass function \(m_{1,2}\). In particular, the work in [115] used an image fusion strategy to investigate the bonding quality of images from ultrasonic and thermographic inspection. Three different types of artificial debonding defects formed, and the extracted features were fed to basic and statistical data fusion as the source. A multi-step preprocessing application that included noise reduction to eliminate error multiplication coordinate matching to remove the effects of different data collection environments, registration or interpolation to match values perfectly and normalisation was performed before the fusion of data. Here, the maximum amplitudes of the interface reflection and second principal component were saved as features indicating the defects for ultrasonic and induction thermography, respectively. In the fusion stage, three hypotheses--defected (positive), non-defected (negative) and unsure regions (doubt)--were identified, and their belief probabilities were calculated with global Gaussian curves of the local amplitude distributions. Figure 8 shows the calculation of the D-S belief percentage for a randomly chosen pixel and its surrounding area in the induction thermography feature matrix's no-defect zone and global distribution. The results emphasised that the D-S combination might be an effective technique, even in the case of basic mathematical fusion rules that have worse results than those based on individual sensors. Another study using the Bayes and D-S methods was recently introduced for multi-material image representation [116]. X-ray computed tomography was applied with various configurations to provide images for different materials of a particular object. This means that the D-S combination rule was implemented to produce a fusion image using three different energy images.