G-Lets: A New Signal Processing Algorithm

Based on the concept of group representation theory, new representations can be generated by direct product (or tensor product) of any two representations of a group. In such case, their irreducible representations are also the direct product. But the conditions under which these representations can be chosen and how to decompose them is silent. In this work, a clear and efficient method for generating and decomposing representations is presented. The study is restricted to geometric group D n of order 2n and its subgroups, where a new homomorphism called a transfer function based on the geometric group is constructed. Due to linearity of discrete-time signal, the generated transformations are used on signal space. Thus, a different approach to signal processing with the choice of a group of transformations is established.

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“…In this section, we review the work of Rajathilagam, [2]. A discrete signal is normally viewed as a function f(t), a set of points over time.…”

Section: Review Of Relevant Group Theoretical Conceptsmentioning

confidence: 99%

“…Any change to the signal may be seen as a change of every point (the domain) in the signal to another point (the range). The pair-wise correspondence of points in the domain and range may be pictured as a mapping [2]:…”

Section: Review Of Relevant Group Theoretical Conceptsmentioning

confidence: 99%

On Finite Group Presentations and Function Decomposition Based on Linearity of Discrete-Time Signal

Ngulde

Madu

Samaila

2018

ARJOM

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“…We consider only discrete signals. An illustration of using group theory for generating G-lets with simple examples of 3, 6 and 9 tuples are presented in the reference [5]. Using the theory of representations we can associate one matrix representation to each of the transformations in the group.…”

Section: Transformations For G-lets Basismentioning

confidence: 99%

“…Our basis is called G-lets for two reasons: one because we use group theory; the other because it is a generalized form of both Fourier and Wavelet analysis. A basic introduction to G-lets is found in the reference [5].…”

Section: Introductionmentioning

confidence: 99%

G-Lets: Signal Processing Using Transformation Groups

Rajathilagam,

Rangarajan,

Soman

2012

Preprint

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We present an algorithm using transformation groups and their irreducible representations to generate an orthogonal basis for a signal in the vector space of the signal. It is shown that multiresolution analysis can be done with amplitudes using a transformation group. G-lets is thus not a single transform, but a group of linear transformations related by group theory. The algorithm also specifies that a multiresolution and multiscale analysis for each resolution is possible in terms of frequencies. Separation of low and high frequency components of each amplitude resolution is facilitated by G-lets. Using conjugacy classes of the transformation group, more than one set of basis may be generated, giving a different perspective of the signal through each basis. Applications for this algorithm include edge detection, feature extraction, denoising, face recognition, compression, and more. We analyze this algorithm using dihedral groups as an example. We demonstrate the results with an ECG signal and the standard 'Lena' image.

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“…Signal Space[9]: If signal can be represented by n-tuple, then it can be treated in much the same way as n-dimensional vector space. Hence, the n-dimensional Euclidean space is called Signal space.…”

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confidence: 99%

Action of Finite Group Presentations on Signal Space

Samaila

Shu’aibu

Modu

2021

JAMCS

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The use of finite group presentations in signal processing has not been exploit in the current literature. Based on the existing signal processing algorithms (not necessarily group theoretic approach), various signal processing transforms have unique decomposition capabilities, that is, different types of signal has different transformation combination. This paper aimed at studying representation of finite groups via their actions on Signal space and to use more than one transformation to process a signal within the context of group theory. The objective is achieved by using group generators as actions on Signal space which produced output signal for every corresponding input signal. It is proved that the subgroup presentations act on signal space by conjugation. Hence, a different approach to signal processing using group of transformations and presentations is established.

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G-Lets: A New Signal Processing Algorithm

Cited by 6 publications

References 14 publications

On Finite Group Presentations and Function Decomposition Based on Linearity of Discrete-Time Signal

On Finite Group Presentations and Function Decomposition Based on Linearity of Discrete-Time Signal

G-Lets: Signal Processing Using Transformation Groups

Action of Finite Group Presentations on Signal Space

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