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The Morphological Approach to Segmentation: The Watershed Transformation
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
Homotopy is a topological property of sets. Instead of defining homotopy in pure mathematical terms, let us simply give a practical definition: two sets X and Y are said to be homotopic if the first one can be superimposed onto the second one by means of continuous deformations. A transformation $ is said to be homotopic if it transforms any set X into an homotopic set Φ(X) (Figure 9). A simply connected set will be transformed into a simply connected set, a set with one hole into a set with one hole, and so on. A typical example of homotopic transform is given by the skeleton of a set [11,12].
Homotopy Algorithms for Engineering Analysis
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
Layne T. Watson, Manohar P. Kamat
Continuation is a well known and established procedure in numerical analysis. The idea is to continuously deform a simple (easy) problem into the given (hard) problem, while solving the family of deformed problems. The solutions to the deformed problems are related, and can be tracked as the deformation proceeds. The function describing the deformation is called a homotopy map. Homotopies are a traditional part of topology, and have found significant application in nonlinear functional analysis and differential geometry. Similar ideas, such as incremental loading, are also widely used in engineering.
A novel method of combined interval analysis and homotopy continuation in indoor building reconstruction
Published in Engineering Optimization, 2019
Ali Jamali, Francesc Antón Castro, Darka Mioc
A homotopy is a continuous deformation of geometric figures or paths or, more generally, functions: a function (or a path or a geometric figure) is continuously deformed into another one (Allgower and Georg 1990). A homotopy between two continuous functions f0 and f1 from a topological space X to a topological space Y is defined as a continuous map H: X × [0, 1] → Y from the Cartesian product of the topological space X with the unit interval [0, 1] to Y such that H(x, 0) = f0, and H(x, 1) = f1, where x ∈ X. The two functions f0 and f1 are called, respectively, the initial and terminal maps. The second parameter of H, λ, also called the homotopy parameter, allows for a continuous deformation of f0 to f1 for λ varying continuously from 0 to 1 (Allgower and Georg 1990). Two continuous functions f0 and f1 are said to be homotopic, denoted by f0 ≃ f1, if and only if there is a homotopy H taking f0 to f1. Being homotopic is an equivalence relation on the set C(X, Y) of all continuous functions from X to Y, where a linear homotopy can be defined by H(x, λ) = (1 − λ) f0(x) + λ f1(x), where λ ∈[0,1].