85 research outputs found
Computing a discrete Morse gradient from a watershed decomposition
We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological characterization of the function and of the shape on which it is defined through a Morse decomposition. Unfortunately, discrete Morse theory and related algorithms assume that the input scalar function has no flat areas, whereas such areas are common in real data and are easily handled by watershed algorithms. We propose here a new approach for building a discrete Morse gradient on a triangulated 3D shape endowed by a scalar function starting from the decomposition of the shape induced by the watershed transform. This allows for treating flat areas without adding noise to the data. Experimental results show that our approach has significant advantages over existing ones, which eliminate noise through perturbation: it is faster and always precise in extracting the correct number of critical elements
Ascending and descending regions of a discrete Morse function
We present an algorithm which produces a decomposition of a regular cellular
complex with a discrete Morse function analogous to the Morse-Smale
decomposition of a smooth manifold with respect to a smooth Morse function. The
advantage of our algorithm compared to similar existing results is that it
works, at least theoretically, in any dimension. Practically, there are
dimensional restrictions due to the size of cellular complexes of higher
dimensions, though. We prove that the algorithm is correct in the sense that it
always produces a decomposition into descending and ascending regions of the
critical cells in a finite number of steps, and that, after a finite number of
subdivisions, all the regions are topological discs. The efficiency of the
algorithm is discussed and its performance on several examples is demonstrated.Comment: 23 pages, 12 figure
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
Morse-Smale decomposition of multivariate transfer function space for separably-sampled volume rendering
We present a topology-guided technique for improving performance of multifield volume rendering with peak finding and preintegration with 2D transfer functions. We apply Morse-Smale decomposition to segment the multidimensional transfer function domain. This segmentation helps to reduce the number of cases where sampling in transfer function space should be performed, effectively reducing the rendering cost for equivalent sampling quality. We show that the overall performance is increased depending on the topology of a transfer function
Effective homology of k-D digital objects (partially) calculated in parallel
In [18], a membrane parallel theoretical framework for computing (co)homology information of fore- ground or background of binary digital images is developed. Starting from this work, we progress here in two senses: (a) providing advanced topological information, such as (co)homology torsion and effi- ciently answering to any decision or classification problem for sum of k -xels related to be a (co)cycle or a (co)boundary; (b) optimizing the previous framework to be implemented in using GPGPU computing. Discrete Morse theory, Effective Homology Theory and parallel computing techniques are suitably com- bined for obtaining a homological encoding, called algebraic minimal model, of a Region-Of-Interest (seen as cubical complex) of a presegmented k -D digital image
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