## 3 Results
### Mapping Using LMA Approach  
The LMA was utilized to determine the magnetic source's location and parameters by minimizing the discrepancy between the theoretical and experimental magnetic field data. The algorithm iteratively adjusted the source's position and current until the residuals converged to a satisfactory level. Using nine PHE sensors in a \(3\times 3\) grid, the LMA estimated the centers of each segment of the coil with reasonable accuracy, as illustrated in Figure 10b. The real center of each segment of the coil is illustrated in Figure 10a.  
Based on the magnetic values produced by the LMA and the nine measurement points, we calculated the magnetic heat map. This heat map, as shown in Figure 11, shows some improvement over the heat map that is based on the measurement (Figure 8a).  
Figure 10: Coil locations: (**a**) Real, (**b**) Estimated with the LMA.  
Figure 9: A 2D interpolation of \(B_{IP}\) based on theoretical values assuming an array consisting of 50 (**a**), 500 (**b**), 1000 (**c**), and 10,000 (**d**) sensors.  
However, the sparsity of the sensor data led to significant deviations in the reconstructed field map, emphasizing the limitations of traditional optimization methods for sparse sensor arrays. The initial guess for the algorithm, set to (25, 25, 1), was chosen as the center of the 50 mm \(\times\) 50 mm plane, assuming that the source is approximately at the center due to its symmetric placement and for simplicity in calculations. Each coordinate took 15 ms to calculate with this algorithm. In general, for 15 points, the total run time of this algorithm is 45 ms. This approach aimed to reduce the likelihood of convergence issues by starting at a neutral point with respect to the array geometry. For setups with multiple magnetic sources, the algorithm's performance significantly degrades due to its reliance on single-source assumptions, making it challenging to resolve overlapping fields or distinguish between closely located sources. Compared to the ML method, LMA is constrained by its need for a good initial guess and struggles with complex, non-linear relationships in data. Moreover, the ML approach excels in handling multiple sources and extracting richer spatial information, offering superior flexibility and spatial resolution.  
The reconstruction of the magnetic field map using the LMA involved two key approaches illustrated in Figure 11. Here, the magnetic field map is derived based on the estimated source parameters obtained from the algorithm, ignoring direct sensor measurements. This approach highlights the limitations of LMA when interpolating sparse sensor data, as the reconstructed map often deviates from the expected spatial profile due to the algorithm's reliance on the estimated parameters alone. In contrast, the image in Figure 11 represents a 2D interpolation that combines the results of the LMA with the measured magnetic field values from the nine PHE sensors. By integrating these measurements, the map retains features influenced directly by the experimental data, offering a more accurate representation of the field. However, while this combined approach improves fidelity, the sparse sensor grid still limits the resolution, and the map remains unable to capture finer spatial details of the magnetic field. This reinforces the need for additional sensors or advanced methods, such as ML-based techniques, to enhance the spatial accuracy of the reconstruction.