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摘要
En 中文
The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace-Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.
Keyword:
Laplace equations
Transforms
Kernel
Linear systems
Chebyshev approximation
Adaptation models
Splines (mathematics)
Chebyshev polynomials
data-driven Fuzzy-transform (F-transform)
data-driven membership functions
F-transform
Laplacian spectrum
期刊
IF:
11.9
论文数:
5.0K
被引数:
2.9W
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