Non-Archimedian Geometry and Applications

In hierarchical classification there is an application of p-adic geometry given by encoding data with p-adic numbers. Such an encoding is generated by embedding the dendrogram associated with the data into the Bruhat-Tits tree T_p. This enables a geometric description of time series of dendrograms (J. Classification, 25 (2008), 27-42) as well as p-adic classification algorithms (p-Adic Num. Ultram. Anal. Appl., 4 (2009), 271-285; Comp. J., 53 (2010), 393-404).
 
Applications in Computer Vision are given by a p-adic encoding of images. In this way RanSaC_p, the p-adic RANSAC of stereo vision, was developped by solving a p-adic version of the five-point algorithm and p-adically classifying the solution sets (p-Adic Num. Ultram. Anal. Appl., 2 (2010), 55-67). The p-adic diffusion equation

    ∂_t f + D f = 0

allows a processing of hierarchically encoded images. If images are generally viewed as functions on Riemann surfaces, then D is the Laplace-Beltrami operator which generalises the well known Laplace operator to curved coordinate systems. In the p-adic case, it is reasonable to use for D the Vladimirov operator. In this way, images become objects of p-adic string theory, a quite new and active field of mathematical physics (p-Adic Num. Ultram. Anal. Appl., 2 (2010) 293-304). More on diffusion and image processing can be found on Wikipedia.