Multiscale surveillance of Riemannian manifolds

When surveying a surface, such as a spherical shell around a planet or a mine field, it is often desirable to survey a pre-chosen distribution while simultaneously respecting all its length scales. We review the Mix Norm developed by Mathew and Mezic and their application for a multiscale coverage algorithm on the unit square and torus.We then develop tools to do multiscale surveillance on arbitrary Riemannian manifolds (M, g). Next, we study the effects of applying the mix norm algorithm to more realistic scenarios where we have vehicles with different maximum speeds and frequency responses. We demonstrate this by applying the algorithm on the unit square to the case of a slow vehicle and a fast vehicle simultaneously exploring different spatial scales; we then test how well the uniform coverage algorithm allows a vehicle to estimate a function in a least squares sense. We also study the effects of varying the parameter s in the definition of mix norm for (M, g). In future work, we plan to apply a motions primitive based approach and explore ways of working with more general surfaces using triangular meshes.