In this paper we study the problem of bivariate density estimation. The aim is to find a density function with the smallest number of local extreme values which is adequate with the given data. Adequacy is defined via Kuiper metrics. The concept of the taut-string algorithm which provides adequate data with a small number of local extrema is generalised for analysing high dimensional data, thereby using Delaunay triangulation and diffusion filtering. Our results are based on equivalence relations in space dimension one of the taut string algorithm with the total variation minimisation and the method of solving the discrete total variation flow equation. The generalisation and some modifications (for instance based on the Fisher information contents) are developed and the performance for density estimation is shown.