## 2 Methodology
### Scattered Decision-Level Fusion
#### 2.2.2 Estimation of Partial Densities  
As noted, our first modification to standard KDE is to carry out density estimation in a per-sensor manner. To this end, the computations in Step 2 are defined in the respective local system for each sensor. Consequently, the kernel function \(K_{S}\) can be defined as an axis-aligned product kernel to reduce the computational cost. The corresponding bandwidth parameters \(\mathbf{h}_{S}\) are chosen based on background knowledge about the nature of our data. Intuitively, the density estimator should always be able to smoothly interpolate between neighboring hit locations. For NDT data, the smallest possible distance between any two hits of the same sensor is given by the known spatial sampling intervals. For a measurement grid per sensor \(S\), the two spatial sampling distances are denoted by \(\Delta^{S}_{x},~{}\Delta^{S}_{y}\) in the sensor's coordinate system. To ensure that neighboring line scans crossing the same defect do not  
Figure 3: Coordinate transformation during the computation of the fused density. The two black coordinate systems in (**a**,**b**) are related through the coordinate transform \(T_{s}\). (**a**) Coordinate system in which the fused density will be evaluated at gridded points \(\mathbf{p}\). The coordinate systems of the individual sensors, where the hits \(\mathbf{d}\) are defined, are not axis-aligned with the global system. In particular, the kernels \(K_{h}\) would require non-diagonal bandwidth matrices; (**b**) Coordinate system of one of the sensors, given by its measurement grid. For single-sensor density estimation, kernels \(K_{h}\) are axis-aligned to the sensor’s system, thus facilitating product kernels. The single-sensor density is then evaluated at the transformed points \(T_{S}\left(\mathbf{p}\right)\). form disconnected density peaks, the kernel functions should at least stretch across one pixel in the sensor image. However, to avoid merging two unrelated indications, the kernels should not be made much larger. Therefore, we propose to use product kernels with minimum bandwidth parameters of \(\mathbf{h}_{s}=\left(h_{x},h_{y}\right)=\left(\Delta_{x}^{S},\Delta_{y}^{S}\right)\) for each sensor. It is natural to use product kernels for gridded individual measurements, because the bandwidths directly correspond to the physical pixel dimensions. The aforementioned kernel size is a minimal setting. In practice, the most significant factor contributing to the localization uncertainty may not be the spatial sampling, but inevitable registration errors. The kernel sizes should be set large enough to smooth out these unwanted variations and to associate poorly registered indications. As a general approach, we propose the following kernel size:  
\[\mathbf{h}_{s}=\left(h_{x},h_{y}\right)=\frac{\hat{u}}{min\left\{\Delta_{x}^{S}, \Delta_{y}^{S}\right\}}\left(\Delta_{x}^{S},\Delta_{y}^{S}\right) \tag{4}\]  
where \(\hat{u}\) denotes an estimate of the localization uncertainty, for instance the mean registration error. This formulation keeps the kernel size ratio \(h_{x}/h_{y}=\Delta_{x}^{S}/\Delta_{y}^{S}\) fixed, and scales the two-dimensional kernel size proportionally to \(\hat{u}\). Consequently, in the fine-sampled direction corresponding to \(min\left\{\Delta_{x}^{S},\Delta_{y}^{S}\right\}\), the kernel will be exactly \(\hat{u}\) wide. In the other direction, the kernel is larger to maintain the ratio. Note that with increasing kernel size, the advantage of having spatially accurate sensors may be lost. Also, closely situated defects become harder to separate. Therefore, fusion performance benefits from high-quality registration by facilitating narrow kernels. Setting the kernel size according to the localization uncertainty implies that sensors with fine spatial sampling produce more hits in the area of influence of a kernel than sensors with coarse sampling. To prevent finely-sampled sensors from having more influence on the fusion process by contributing larger densities, normalization is required.