## 2 Materials and Methods
### Principal Component Background Suppression with Peak Threshold Detection
#### 2.2.2 Principal Component Background Suppression  
Once the pixels' frequency spectra were obtained, principal component background suppression (PCBS) was performed to reduce the amplitude of the frequencies generated by the overwhelming source. PCBS is an adaptable statistical technique that uses the principal components (PCs), or eigenvectors, of a data set to suppress the background or clutter in a scene. This background suppression is often used for applications such as jitter removal in surveillance videos [34], thermal background subtraction [35], and artifact suppression in non-destructive testing data [36]. For a generic data set, the data matrix rows are the measurable variables while the columns are the observations [37]. In this work, the measurable variables were the FFT frequency bins and the observations were the frequency amplitudes in each pixel. PCBS assumes that a data matrix, \(\mathbf{X}\), contains the sum of the background, \(\mathbf{B}\), and signal, \(\mathbf{S}\), such that,  
\[\mathbf{X}=\mathbf{B}+\mathbf{S}. \tag{1}\]  
If a \"good enough\" background model, \(\mathbf{\tilde{B}}\), can be created, then an approximated signal, \(\mathbf{\tilde{S}}\), can found by  
\[\mathbf{\tilde{S}}=\mathbf{X}-\mathbf{\tilde{B}}. \tag{2}\]  
This detection method estimates that the eigenvector matrix, \(\mathbf{V}\), can be decomposed as  
\[\mathbf{V}=\big{[}\mathbf{\hat{B}}\ \mathbf{\hat{S}}\big{]}, \tag{3}\]  
where the eigenvector matrix is found by solving the eigenvalue problem of the data covariance matrix, \(\mathbf{V}\) is sorted by corresponding eigenvalues in descending order, and \(\mathbf{\hat{B}}\) and \(\mathbf{\hat{S}}\) are separate background and signal subspaces. PCBS assumes some number of the largest eigenvectors are a \"good enough\" representation for the background model in the  
Figure 8: Flow chart depicting the process of transforming a signal to event data, and event data to a frequency spectra by summing ON events into time bins (5 ms for this example) and applying a fast Fourier transform (FFT) algorithm. eigenspace [37]. Since the overwhelming source is assumed to cover a large portion of the scene and be the most prominent feature, the leading PCs (\(\mathbf{\hat{B}}\)) should correspond to the frequency features generated by the overwhelming source. Additionally, this method can include sensor noise as an additional term in Equations (1) and (2) and as a third subspace in Equation (3) [38]. However, since the signal in this work is assumed to be sparse, the noise is included in the background. The signal could then be estimated by either subtracting the background projected data or by projecting the data onto the signal eigenspace, such that,  
\[\mathbf{\hat{S}}=\mathbf{X}-\mathbf{\hat{B}}\mathbf{\hat{B}}^{T}\mathbf{X}= \mathbf{\hat{S}}\mathbf{\hat{S}}^{T}\mathbf{X}. \tag{4}\]  
Different methods can be used, and were explored, to automatically determine the appropriate number of PCs: cumulative variance threshold [26], elbow point of the scree plot [39], maximum value frequency spectra difference derivative threshold, and maximum value frequency spectra normalized dot product threshold. These methods exhibited varying levels of success due to a large number of variables (sources' frequency and power, data modification method, FFT parameters, etc.) which can influence the appropriate number of PCs for a sufficiently accurate background model. In this work, the number of PCs used for each data set was determined by identifying the elbow in the difference curve of the PC subtracted maximum value frequency spectra. A maximum value frequency spectrum is the maximum amplitude in each frequency bin across all pixels, as seen in Figure 9. The elbow method provides an automatic determination of how many PCs to include in the background model by finding the point where subtracting more PCs does not significantly alter the maximum value frequency spectra.