\(f_{beat}=\left|f_1-f_2\right|\)        Eq. [1]
The images are then captured at a 4 Hz rate, equivalent to 16 samples per cycle considering the beat frequency of 0.25 Hz. Those images are then processed in three main steps in order to make it possible to identify phases and phase changes from the  data. The first step is to identify separately each light source (window or external light source in the city nightscape) in the images. In order to do that, a stack is created from the median of 20 consecutive images. From this stack, by thresholding, the pixels with brightness higher than the 90th percentile of the distribution of all pixels are considered light sources and contiguous pixels are grouped together to form a single window.
The brightness time series for each window can be then extracted and analyzed. As mentioned before, not all light sources have an oscillatory behavior at 120 Hz, LED and some modern fluorescent lights, for instance, do not show this type of behavior. Besides that, for some sources it may be the case that the oscillatory signal-to-noise ratio is not good enough for further analysis. Therefore, the second step consists of a selection of the sources that present a clear oscillatory behavior at 120 Hz. That selection is made by performing a principal component analysis (PCA) on the light curves (time series of the brightness of each light source) and choosing the top 10% that have their variance explained by the first two principal components. In this decomposition, consistently the first two principal components are two nearly-sinusoidal curves with the expected beat  frequency of about 0.25 Hz. Although those two typically account for only 0.05-0.3 of the variance light curves, they encode precisely the oscillatory behavior and the phase information that we want to analyze.
Finally, the third step is to fit a sine curve to each of the selected light curves so that their phases and phase changes in time can be analyzed. That analysis in a relative manner, which means that one light curve is arbitrarily selected and all phase changes are measured in relation to that light curve. The reason to analyze always a relative phase change is to not have to account for eventual equipment instability when collecting the images.

Scope

Cost Effectiveness Analysis

As hyper temporal imaging is expected to be an affordable way to monitor the grid health of the city, it is critical to estimate the implementation cost in relation with its penetration rate and compare with the PMU, monitoring energy consumption at the neighbor level. The original observation system included a PointGrey Flea3 5.0 MP Color GigE Vision camera and a liquid crystal shutter. The total cost for the system is $5000. Our goal was to find out the coverage rate of this system in the previous study. The process was composed of two steps: Determining the area captured by the camera and estimating the number of building windows that can be seen in the image. The images used by the previous study was the pilot study for our cost-effectiveness analysis.