Varilets: Additive Decomposition, Topological Total Variation, and Filtering of Scalar Fields

Continuous interpolation of real-valued data is characterized by piecewise monotone functions on a compact metric space. Topological total variation of piecewise monotone function f:X->R is a homeomorphism-invariant generalization of 1D total variation. A varilet basis is a collection of piecewise monotone functions { $g_i$ |i = 1...n}, called varilets, such that every linear combination $\sum a_ig_i$ has topological total variation $\sum |a_i|$. A varilet transform for $f$ is a varilet basis for which $f =\sum \alpha_ig_i$. Filtered versions of $f$ result from altering the coefficients $\alpha_i$.