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Hall thrusters are susceptible to large-amplitude plasma oscillations that impact thruster performance and lifetime and are also difficult to model. High-speed cameras are a popular tool to study these dynamics due to their spatial resolution and are a popular, nonintrusive complement to in-situ probes. High-speed video of thruster oscillations can be isolated (decomposed) into coherent structures (modes) with algorithms that help us better understand the evolution and interactions of each. This work provides an introduction, comparison, and step-by-step tutorial on established Fourier and newer Proper Orthogonal Decomposition (POD) algorithms as applied to high-speed video of the unshielded H6 6-kW laboratory model Hall thruster. From this dataset, both sets of algorithms identify and characterize $m=0$ and $m>0$ modes in the discharge channel and cathode regions of the thruster plume, as well as mode hopping between the $m=3$ and $m=4$ rotating spokes in the channel. The Fourier methods are ideal for characterizing linear modal structures and also provide intuitive dispersion relationships. By contrast, the POD method tailors a basis set using energy minimization techniques that better captures the nonlinear nature of these structures and with a simpler implementation. Together, the Fourier and POD methods provide a more complete toolkit for studying Hall thruster plasma instabilities and mode dynamics. Specifically, we recommend first applying POD first to quickly identify the nature and location of global dynamics and then using Fourier methods to isolate dispersion plots and other wave-based physics.