Statistics of natural images, including astronomical images, often have a very nice mathematical description--differentially Laplacian statistics. Other types of data--ocean temperatures, precipitation, digital elevation models, currency exchange rates, NYSE volumes, turbulent fluid velocities--also often follow this model. This statistical information is useful for inverse problems for this type of data, and frequently suggests the right way to regularize the problem and how to create an optimal algorithm. One such application is image denoising. This will be a self-contained lecture that will not assume a knowledge of statistics. The quite interesting mathematics underlying differentially Laplacian statistics will be explained, as well as the general process of how this statistical knowledge was used in designing the denoising algorithm. I will also explain the link between this type of statistics and PDE, especially the Helmholtz equation, and also with the world of random processes with long-term memory. These are not Markov random processes or Markov random fields, but nevertheless have a user-friendly mathematical description.
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