## 2 Methodology
### Kernel Density Estimation (KDE)  
Before proposing our technique, we repeat fundamental concepts of KDE in this section. These concepts and the associated notation are adopted in the rest of the paper.  
KDE is a nonparametric statistical method to estimate a probability density function \(\hat{f}\) from a set of samples \(x_{i}\). The result is a continuous function, computed from a weighted sum of kernel functions \(K_{h}\) with an associated bandwidth \(h\), each centered over one of the samples:  
Figure 2: Flow chart of the fusion process. Gray boxes denote original sensor images, each containing an indication (crack symbol). Solid-edge boxes denote local (per sensor) coordinate systems; dashed boxes denote the global (registered) coordinate system. Cross markers denote hits. Two-dimensional spatial densities are indicated by contour plot symbols. Mathematical symbols below the boxes correspond to the notation used throughout this work. (**a**) Sensor images; (**b**) Hits; (**c**) Registered hits; (**d**) Spatial density; (**e**) Registered density; (**f**) Fused density.  
\(\hat{f}\left(y\right)=\left(\sum_{i}w_{i}\right)^{-1}\sum_{i}w_{i}K_{h}\left(y-x_{ i}\right)\), with \(K_{h}\left(x\right)=\frac{1}{h}K\left(\frac{x}{h}\right)\). Some functions qualifying as a kernel \(K\) are the uniform, triangle, Gaussian or Epanechnikov kernel functions. The bandwidth \(h\) controls the size of the neighborhood in which samples influence the density at a specific location. The choice of the bandwidth is critical for the overall performance of the algorithm. If the bandwidth is chosen too wide, KDE results in an overly smoothed density, thus losing important details of the distribution. On the other hand, if it is chosen too narrow, the estimate adapts too much to the specific realization of the sample set, thus missing the global features of the density. This problem has been well-studied, and several solutions have been proposed [14]. We will describe how to automatically compute a suitable bandwidth for our problem in Section 2.2.2.  
The general formulation given above for KDE includes the normalization constant \(\left(\sum_{i}w_{i}\right)^{-1}\), ensuring that the density integrates to one. Since we do not require a probabilistic interpretation, we proceed with a simpler unnormalized version of KDE by dropping the normalization constant. This also simplifies the notation. Furthermore, since the density estimate is a weighted sum with one term per data point, the data set can be partitioned to aggregate the total density function \(\hat{f}\) from the sub-densities \(\hat{f}_{J}\), each including only the samples \(x_{i}\) from partition \(j\):  
\[\hat{f}\left(y\right)=\sum_{j}\hat{f}_{J}\left(y\right)=\sum_{j}\sum_{i\in P_{ j}}w_{i}K_{h}\left(y-x_{i}\right) \tag{1}\]  
\(P_{j}\) denotes a subset of all points such that partitions do not overlap and the union of all partitions covers the complete data set. This re-arrangement is taken up in the following section to group hits by each sensor, as is illustrated in Figure 2d.  
KDE can be extended to vector-valued samples. To that end, let \(\mathbf{x}_{i}\) denote the ith vector-valued (bold face) sample, and let \(\mathbf{y}\) denote the vector-valued location where to evaluate the density. Multivariate KDE is computed from multivariate kernel functions and an associated bandwidth matrix \(H\), which describes the scale and the orientation of the kernels. A special kind of multivariate kernel is the product kernel, defined by \(K_{H}\left(\mathbf{x}\right)=\prod_{j}\frac{1}{h_{j}}K\left(\frac{\mathbf{x}_{j}}{h_{j }}\right)\), where one univariate kernel for each dimension \(j\) is evaluated. The \(h_{j}\) are the entries of the diagonal bandwidth matrix, _i.e._, product kernels are not arbitrarily oriented in the data space. This property reduces the computational demand, because the data dimensions are considered independently. We use product kernels in our approach, which is described next.