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 .
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.
@cite_14 @cite_4 @cite_7 @cite_8 @cite_1 @cite_0 @cite_27 @cite_12 @cite_17
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The goal of this thesis is to bring together two different theories about critical points of a scalar function and their relation to topology: Discrete Morse theory and Persistent homology. While the goals and fundamental techniques are different, there are certain themes appearing in both theories that closely resemble each other. In certain cases, the two threads can be joined, leading to new insights beyond the classical realm of one particular theory.Discrete Morse theory provides combinatorial equivalents of several core concepts of classical Morse theory, such as discrete Morse functions, discrete gradient vector fields, critical points, and a cancelation theorem for the elimination of critical points of a vector field. Because of its simplicity, it not only maintains the intuition of the classical theory but allows to surpass it in a certain sense by providing explicit and canonical constructions that would become quite complicated in the smooth setting.Persistent homology quantifies topological features of a function. It defines the birth and death of homology classes at critical points, identifies pairs of these (persistence pairs), and provides a quantitative notion of their stability (persistence).Whereas (discrete) Morse theory makes statements about the homotopy type of the sublevel sets of a function, persistence is concerned with their homology. While homology is an invariant of homotopy equivalences, the converse is not true: not every map inducing an isomorphism in homology is a homotopy equivalence. In this thesis we establish a connection between both theories and use this combination to solve problems that are not easily accessibly by any single theory alone. In particular, we solve the problem of minimizing the number of critical points of a function on a surface within a certain tolerance from a given input function. This paper describes an effort to create new visualizations by exploiting hierarchical scalar topology. First, we build a hierarchical topology through synchronously constructing and simplifying Contour Tree (CT) and Morse-Smale (MS) complex of scalar fields. We then introduce three algorithms based on the hierarchical topology: (1) topology-based multi-resolution contouring — an overview provided for a scalar field by extracting iso-values from the simplified CT and tracing approximate contours across the MS complex cells; (2) topology based spaghetti plots for uncertainty — a seeding scheme based on the hierarchical topology for visualizing uncertainty among ensemble scalar data; (3) virtual ribbons — a new scheme for visualizing multivariate data invented by overlapping visual ribbons which encode the scalar variation of a region covered by uniform contours. We compare the new approaches with current alternatives. The present paper shows a mathematical formalization of---as well as algorithms and software for computing---volume-optimal cycles. Volume-optimal cycles are useful for understanding geometric features appearing in a persistence diagram. Volume-optimal cycles provide concrete and optimal homologous structures, such as rings or cavities, on a given dataset. The key idea is the optimality on a @math -chain complex for a @math th homology generator. This optimality formalization is suitable for persistent homology. We can solve the optimization problem using linear programming. For an alpha filtration on @math , volume-optimal cycles on an @math st persistence diagram are more efficiently computable using a merge-tree algorithm. The merge-tree algorithm also provides a tree structure on the diagram containing richer information than volume-optimal cycles. The key mathematical idea used here is Alexander duality. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. In case of the latter, extra homological structures are evident in an control group over the autistic group.