Go With the Flow, on Jupiter and Snow. Coherence From Model-Free Video Data without Trajectories.
Viewing a data set such as the clouds of Jupiter, coherence is readilyapparent to human observers, especially the Great Red Spot, but also othergreat storms and persistent structures. There are now many differentdefinitions and perspectives mathematically describing coherent structures, butwe will take an image processing perspective here. We describe an imageprocessing perspective inference of coherent sets from a fluidic systemdirectly from image data, without attempting to first model underlying flowfields, related to a concept in image processing called motion tracking. Incontrast to standard spectral methods for image processing which are generallyrelated to a symmetric affinity matrix, leading to standard spectral graphtheory, we need a not symmetric affinity which arises naturally from theunderlying arrow of time. We develop an anisotropic, directed diffusionoperator corresponding to flow on a directed graph, from a directed affinitymatrix developed with coherence in mind, and corresponding spectral graphtheory from the graph Laplacian. Our methodology is not offered as moreaccurate than other traditional methods of finding coherent sets, but ratherour approach works with alternative kinds of data sets, in the absence ofvector field. Our examples will include partitioning the weather and cloudstructures of Jupiter, and a local to Potsdam, N.Y. lake-effect snow event onEarth, as well as the benchmark test double-gyre system.
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