Continuous Diffusion Wavelet Transforms and Scale Space over Euclidean Spaces and Noncommutative Lie Groups

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.(a)Quotients of Euclidean spaces by isometric actions of finite groups.(b)$L_{2}$-Wasserstein spaces over Euclidean spaces.(c)Compact flat manifolds.(d)Compact flat orbifolds.(e)Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

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Observability and Symmetries of Linear Control Systems

Symmetry ◽

10.3390/sym12060953 ◽

2020 ◽

Vol 12 (6) ◽

pp. 953

Author(s):

Víctor Ayala ◽

Heriberto Román-Flores ◽

María Torreblanca Todco ◽

Erika Zapana

Keyword(s):

Euclidean Space ◽

Control Systems ◽

Lie Groups ◽

Lie Group ◽

Linear Control ◽

Linear Control Systems ◽

Euclidean Spaces ◽

General Setup ◽

Connected Lie Group

The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.

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Otolith shape contour analysis using affine transformation invariant wavelet transforms and curvature scale space representation

Marine and Freshwater Research ◽

10.1071/mf04162 ◽

2005 ◽

Vol 56 (5) ◽

pp. 795 ◽

Cited By ~ 61

Author(s):

V. Parisi-Baradad ◽

A. Lombarte ◽

E. Garcia-Ladona ◽

J. Cabestany ◽

J. Piera ◽

...

Keyword(s):

Wavelet Transforms ◽

Scale Space ◽

Space Representation ◽

Contour Analysis ◽

Precise Position ◽

Curvature Scale Space ◽

Otolith Morphology ◽

Affine Image ◽

Landmark Selection ◽

Image Transformations

Fish otolith morphology has been closely related to landmark selection in order to establish the most discriminating points that can help to differentiate or find common characteristics in sets of otolith images. Fourier analysis has traditionally been used to represent otolith images, since it can reconstruct a version of the contour that is close to the original by choosing a reduced set of harmonic terms. However, it is difficult to locate the contour’s singularities from this spectrum. As an alternative, wavelet transform and curvature scale space representation allow us to quantify the irregularities of the contour and determine its precise position. These properties make these techniques suitable for pattern recognition purposes, ageing, stock determination and species identification studies. In the present study both techniques are applied and used in an otolith classification system that shows robustness against affine image transformations, shears and the presence of noise. The results are interpreted and discussed in relation to traditional morphology studies.

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