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 $\sum a_i{g}_i$ has topological total variation $\sum \mathrm{\mid }{a}_i\mathrm{\mid }$. A varilet transform for $f$ is a varilet basis for which $f=\sum {\alpha }_i{g}_i$. Filtered versions of $f$ result from altering the coefficients ${\alpha }_i$.