Bridging persistence of discrete and continuous 1D maps: Total variation, permutation equivalence, and diagonal affine transformations

Fundamental to the field of topological signal processing (TSP) is the $\mathit{\text{persistence}}\mathit{\text{diagram}}$ (PD), or barcode, of a signal, which gives noise-resistant measures of topologically significant features as distinct from noise. Existing literature proves that the total persistence computed from these diagrams is equal to the total variation, for certain continuous signals on compact domains. We provide a closed formula for this shared quantity, and generalize to all continuous and discrete functions on bounded domains, and to certain functions on non-compact 1D manifolds. Extending the identification of continuous and discrete persistence, we investigate the similarities of functions with the same PDs, and prove that, for any $n$, the persistence diagram can be calculated as a function of the sequence of critical values; we compute these formulas for $n\le 6$. Finally, we examine which function transformations cause predictable changes in the PD of the transformed function, characterizing affine and linear transformations.