## 3 Method
### Offline Phase
Similarly, for the case where the feature is present (denoted through the superscript \((\cdot)^{fe}\)), it is calculated as follows:  
\[\mathbf{T}^{fe}(\omega_{i})=\frac{\mathbf{u}_{u}^{fe}(\omega_{i})}{u_{k}^{fe}(\omega_{ i})} \tag{18}\]  
For a feature of size 0, the frequency response at the feature is the same for the baseline and the case where the feature is present (\(u_{k}^{fe}(\omega_{i})=u_{k}^{bl}(\omega_{i})\)). Therefore, the baseline subtracted transmissibility function (denoted through the superscript \((\cdot)^{bs}\)) is found as follows:  
Figure 3:0  
This enables evaluating the response at each sensor location, if the excitation at the actuator location is different or if the distance between the actuator and the feature varies. In a similar way, the response corresponding to a line of sensors that are radially placed with respect to the actuator can be compressed and simplified. The only difference to a wave packet stemming from a feature lies in the application of the principle of FFRs rather than the principle of transmissibility functions. In this case, \(\mathbf{u}(\omega_{i})\) in Equation (5) reflects the response at the sensors and \(f(\omega_{i})\) the excitation at the actuator. At this point, it is possible to evaluate the response for any sensor along the radial line from the feature by performing the following steps:  
1. Evaluate the response due to the direct wave: 1.1  Evaluate the FRF for the distance \(d_{AS}\) from the actuator to the sensor;1.2 Multiply the obtained FRF with the excitation. 2. Evaluate the virtual response at the feature: 2.1 Evaluate the FRF for the distance \(d_{AF}\) from the actuator to the feature; 2.2 Multiply the obtained FRF with the excitation. 3. Evaluate the response for reflected and mode-converted wave packets: 3.1 Evaluate the transmissibility functions for all modes of interest for the distance \(d_{FS}\) from the feature to the sensor; 3.2 Multiply the obtained transmissibility functions with the response obtained in step 2. 4. Sum up the response obtained in step 1.2 and the response obtained in step 3.2. In what is described so far, each computed response corresponds to a sensor placed along a specific radial line. In order to more generally devise an approach to densely recover the response in the full field of an infinite, isotropic plate with a single feature, actuators along multiple radial lines can be employed covering a range of actuator-feature-sensor angles. The compression of each radial line of sensors into an amplitude (\(A\)) and a phase (\(\theta\)) proves advantageous in this respect. Using the same simulations that were previously used to infer the parameters (\(A\) and \(\theta\)) for a single line, an arbitrary number of lines can be compressed. Afterwards, the amplitude and the phase can be interpolated for each angle of interest. The limiting factors are the accuracy of the original simulation, the amount of data stored for the sensors, and the time needed for the optimization of the phase. These limitations do not hinder the evaluation and interpolation of a high number of parameters. Owing to the assumption of isotropic behavior, the evaluation for angles in the interval \([0,\pi]\) is sufficient. The offline process is summarized in Algorithm 1. ```
1 for each sensor locationdo
2 Remove direct wave
3 Evaluate transmissibility function as in Equation (11)
4 end for
5 for Each discrete frequencydo
6 Evaluate \(\omega_{dist}\)
7 for Each linedo
8 Evaluate the amplitude \(A\)
9 Evaluate the phase shift \(\theta\)
10 end for
11 Create Interpolation object of \(A\) for all angles
12 Create Interpolation object of \(\theta\) for all angles // ''Interpolation object''
13 describes a handle that can be passed to a function and can be evaluated for a respective angle
14 end
```  
**Algorithm 1**Offline phase