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

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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 gᵢ |i = 1...n, called varilets, such that every linear combination ∑ aᵢgᵢ has topological total variation ∑ |aᵢ|. A varilet transform for f is a varilet basis for which f =∑ αᵢgᵢ. Filtered versions of f result from altering the coefficients αᵢ.