Geometry-preserving topological landscapes

We propose a novel technique for building geometry-preserving topological landscapes. Our technique creates a direct correlation between a scalar function and its topological landscape. This correlation is accomplished by introducing the notion of geometric proximity into the topological landscapes, reflecting the distance of topological features within the function domain. Furthermore, this technique enables direct comparative analysis between scalar functions, as long as they are defined on the same domain. We describe a construction technique that consists of three stages: contour tree computation, contour tree layout, and landscape construction. We provide a detailed description for the latter two steps. For the contour tree layout stage, we discuss dimension reduction and edge routing techniques that produce a drawing of the contour tree on the plane that preserves the geometric proximity. For the landscape construction stage, we develop a contour construction algorithm that takes the contour tree layout as an input, adds contours at heights that correspond to saddles of the contour tree, and produces a contour map. After an additional triangulation step, this construction method results in the landscape that has the same contour tree as the original function.