Motivated by recent developments in topological persistence for assessment of the importance of features in data sets, we study the ideas of persistence homology for one- and two-dimensional digital signals and its application in signal denoising. Recently, notions of persistence homology and persistence pairs were introduced in for measuring the topological complexity of point sets in 3-D. Persistence pairs and corresponding persistence diagrams are well suited to quantify the topological significance of data structures and to develop a formalism for topological simplification. These approaches are exploited for adaptive signal and image denoising.