## 1 Introduction
Additionally, the noise caused by the material grains and the measurement system may make cambered ends of the defect invisible. As shown in Figure 3, they can hardly be found. Therefore, the above method cannot be executed effectively. The height measurement accuracy is related to the arrival time measurements from the diffracted waves. However, the diffracted waves are likely to overlap and contain the noise, as exhibited in Figure 2a, which is very detrimental to the measurement. In addition, if large structures, such as pipelines and reactors, are inspected, the number of obtained data for interpretation will be considerable. Therefore, developing an effective and automatic data analysis method is imperative to enhancing measurement accuracy and efficiency. In this context, [PERSON] et al. first proved that when a crack appears in a structure, a group of parabolas with a principal axis is present in the ultrasonic diffraction image [7]. Thus, the crack in the structure can be located by examining these parabolas and determining the position of their summit points. [PERSON] used a mathematical morphology approach, the watershed technique, to extract relevant pixels corresponding to the presence of a defect, and subsequently adopted a skeletonization technique to characterize the defect [8]. Some works employed different means and then used the Hough transform (HT) to automatically detect parabolic shapes for locating crack tips in B-scan images [9; 10]. To save calculation and processing time, matched filtering was used to detect parabolic forms when the shapes in ultrasonic diffraction images are intact [11]. In terms of A-scan signal processing, various signal processing techniques have been employed for estimating the parameters of diffracted echoes, such as the methods based on cross-correlation (CC) and its enhanced forms [12; 13]. The CC method considers a white Gaussian noise presence in the acquired signal and needs a reference signal, which will affect the estimation accuracy. [PERSON] developed a model-based method [14; 15] to overcome the limitations and evaluate the parameters of ultrasonic signals by the Levenberg-Marquart or Gauss-Newton algorithms that can solve the non-linear optimization problem. However, these methods cannot ensure the convergence of an estimation problem due to dependence on the initial guesses. So, they can have trouble processing complex signals. [PERSON] et al. conquered this difficulty by applying an MP approach [16]. In the ap  
Figure 2: An example of ultrasonic diffraction data actually acquired. (**a**) is the A-scan signal at position 60 of the scanning axis; (**b**) is the ultrasonic diffraction image (B-scan) generated by all A-scan signals in the scanning path. proach, a signal is estimated through matching it to a minimum number of elementary functions selected from a large dictionary. Further, [PERSON] estimated the ultrasonic diffraction signals by the envelope extraction using Hilbert transform (HT) and subsequently the envelope estimation using a matching pursuit (MP) approach that is optimized by differential evolution to reach fast and accurate results [17]. [PERSON] et al. proposed a Faster R-CNN model for TOFD signal processing that can classify and localize various defect signals, providing a localization accuracy of 92.43 \(\pm\) 1.25% [18]. [PERSON] et al. proposed an ultrasound B-scan (S-scan) imaging technique that utilizes machine vision, which achieved a detection accuracy of 91.3% [19]. Although considerable successes have been achieved in ultrasonic diffraction data analysis by various techniques, they can only estimate the perfect ultrasonic data without noise or with simple noise (white Gaussian noise). The real data, however, contain multifarious complicated noises caused by grain or surface roughness of the structure to be detected. These methods can run into difficulty in complicated situations. In addition, systematic studies on both the length and height estimation of defects in the ultrasonic diffraction method have not been reported.