text: 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 @math |i = 1...n , called varilets, such that every linear combination @math has topological total variation @math . A varilet transform for @math is a varilet basis for which @math . Filtered versions of @math result from altering the coefficients @math .
  @cite_14: 
  @cite_4: We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise, depending on its life-time or persistence within the filtration. We give fast algorithms for completing persistence and experimental evidence for their speed and utility.
  @cite_7: We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van
  @cite_8: 
  @cite_1: A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of these theories is Morse Theory. In this paper we present a combinatorial adaptation of Morse Theory, which we call discrete Morse Theory, that may be applied to any simplicial complex (or more general cell complex). The goal of this paper is to present an overview of the subject of discrete Morse Theory that is sucient both to understand the major applications of the theory to combinatorics, and to apply the the theory to new problems. We will not be presenting theorems in their most recent or most general form, and simple examples will often take the place of proofs. We hope to convey the fact that the theory is really very simple, and there is not much that one needs to know before one can become a "user".
  @cite_0: 
  @cite_27: In this thesis, I show how to use the topological information encoded manipulation of individual contour surfaces in an isosurface scene, using an interface called the flexible isosurface. Underpinning this interface are several improvements and extensions to existing work on the contour tree. The first, and most critical, extension, is the path seed: a new method of generating seeds from the contour tree for isosurface extraction. The second extension is to compute geometric information called local spatial measures for contours and store this information in the contour tree. The third extension is to use local spatial measures to simplify both the contour tree and isosurface displays. This simplification can also be used for noise removal. Lastly, this thesis extends work with contour trees from simplicial meshes to arbitrary meshes, interpolants, and tessellation cases.
  @cite_12: 
  @cite_17: Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.
summary: Varilets fit into the larger context of computational topology and data analysis @cite_17 @cite_14 . Although there may exist mathematical connections to persistent homology @cite_4 and discrete Morse theory @cite_1 , this paper focuses instead on the monotone-light factorization @cite_0 , which for piecewise monotone functions may be seen as a decorated version of the Reeb graph @cite_12 . Sometimes called the contour tree @cite_7 , and within persistence theory the merge tree @cite_17 , the Reeb graph has been often used for simplification of scalar fields @cite_27 @cite_8 and has been exploited throughout computational topology.