Morphological processing of univariate Gaussian distribution-valued images based on Poincaré upper-half plane representation. (English) Zbl 1345.62018
Nielsen, Frank (ed.), Geometric theory of information. Selected and revised contributions of the first conference on the geometric sciences of information, GSI, Paris, France, August 2013. Cham: Springer (ISBN 978-3-319-05316-5/hbk; 978-3-319-05317-2/ebook). Signals and Communication Technology, 331-366 (2014).
This work presents the Chapter 12 of the book by F. Nielsen (ed.) [Zbl 1291.94002].
Section 1 introduces the mathematical image model considered in the paper, a function \(f:\Omega\to N\), where \(\Omega\) is the support space of image \((\Omega\subset Z^2)\) and \(N\) denotes the family of univariate Gaussian probability distribution functions. In order to analyze an image the authors adopt an information geometry approach, which is based on considering that the univariate Gaussian densities are points in the hyperbolic space \(H^2=\{(x_1,x_2)\in \mathbb{R}^2/x_2>0\}\) with the Riemannian metric \(ds^2=((dx_1^2+dx_2^2)/(x_2^2))\). The aim of the paper is to endow \(H^2\) with partial orderings which lead to useful invariance properties in order to formulate morphological operators for images.
Section 2 summarizes the basic concepts and properties of \(H^2\): Riemannian metric, angle, distance, geodesics, hyperbolic polar coordinates, invariance and isometric symmetry, and hyperbolic circles and balls. Section 3 briefly discusses the link between the \(H^2\) model and geometry information Fisher of Gaussian distributions. Several partial orderings on \(H^2\) are studied in Section 4. Section 5 defines several morphological operators to images considered as functions \(f:\Omega\to H^2\) Then presents some applications to morphological processing univariate Gaussian distribution-valued images. Section 6 concludes the paper indicating possible future work on the subject.
The paper is well written, the proofs are rigorously established. Parts may be hard to read for non-mathematicians.
For the entire collection see [Zbl 1291.94002].
Section 1 introduces the mathematical image model considered in the paper, a function \(f:\Omega\to N\), where \(\Omega\) is the support space of image \((\Omega\subset Z^2)\) and \(N\) denotes the family of univariate Gaussian probability distribution functions. In order to analyze an image the authors adopt an information geometry approach, which is based on considering that the univariate Gaussian densities are points in the hyperbolic space \(H^2=\{(x_1,x_2)\in \mathbb{R}^2/x_2>0\}\) with the Riemannian metric \(ds^2=((dx_1^2+dx_2^2)/(x_2^2))\). The aim of the paper is to endow \(H^2\) with partial orderings which lead to useful invariance properties in order to formulate morphological operators for images.
Section 2 summarizes the basic concepts and properties of \(H^2\): Riemannian metric, angle, distance, geodesics, hyperbolic polar coordinates, invariance and isometric symmetry, and hyperbolic circles and balls. Section 3 briefly discusses the link between the \(H^2\) model and geometry information Fisher of Gaussian distributions. Several partial orderings on \(H^2\) are studied in Section 4. Section 5 defines several morphological operators to images considered as functions \(f:\Omega\to H^2\) Then presents some applications to morphological processing univariate Gaussian distribution-valued images. Section 6 concludes the paper indicating possible future work on the subject.
The paper is well written, the proofs are rigorously established. Parts may be hard to read for non-mathematicians.
For the entire collection see [Zbl 1291.94002].
Reviewer: Oscar Bustos (Córdoba)
MSC:
| 62B10 | Statistical aspects of information-theoretic topics |
| 92C55 | Biomedical imaging and signal processing |
| 62M30 | Inference from spatial processes |
| 53B21 | Methods of local Riemannian geometry |
