Topological data analysis is a new area of study aimed at having applications in areas such as data mining and computer vision. Data mining is the process of Sorting through large amounts of data and picking out relevant information Computer vision is the science and technology of machines that see The main problems are (1) how one infers high-dimensional structure from low-dimensional representations; and (2) how one assembles discrete points into global structure.
The human brain can easily extract global structure from representations in a strictly lower dimension, i. e. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images. Continuity may refer to In mathematics: Continuous probability distribution or random variable in probability and statistics For E. g. dot-matrix printers and televisions communicate images via arrays of discrete points.
The main method used by topological data analysis is:
(1) replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments
(2) Analyse these topological complexes via algebraic topology — specifically, via the new theory of persistent homology. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic
(3) Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which will be called a barcode. In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing
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Point cloud data
Data is often represented as points in a Euclidean n-dimensional space En. The global shape of the data may provide information about the phenomena that the data represent.
One type of data set for which global features are certainly present is the so-called point cloud data coming from physical objects in 3D. E. g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object. Point cloud data refers to any collection of points in En or a (perhaps noisy) sample of points on a lower-dimensional subset.
For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of computer graphics and statistics. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections.
To convert a point cloud in a metric space into a global object use the point cloud as the vertices of a graph whose edges are determined by proximity, then turn the graph into a simplicial complex and use algebraic topology to study it. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments
Persistent homology
See also
- Dimensionality reduction
- Data mining
- Computer vision
- Computational topology
- Digital topology
- Digital Morse theory
- Shape analysis
- Structured data analysis (statistics)
References
- BARCODES: THE PERSISTENT TOPOLOGY OF DATA
- Topological Data Analysis: the algebraic topology of point data clouds?
- Sanjay Rana (2004). In Statistics, dimension reduction is the process of reducing the number of random variables under consideration and can be divided into Feature selection and Data mining is the process of Sorting through large amounts of data and picking out relevant information Computer vision is the science and technology of machines that see Algorithmic topology, or computational topology, is a subfield of Topology with an overlap with areas of Computer science, in particular Computational Digital topology deals with properties and features of Two-dimensional (2D or three-dimensional (3D Digital images that correspond to topological properties In Mathematics, digital Morse theory is a digital adaptation of continuum Morse theory for scalar volume data. This article describes shape analysis to analyze and process geometric shapes. Structured data analysis is the statistical data analysis of structured data Topological Data Structures for Surfaces. John Wiley and Sons. ISBN 978-0470851517.
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