# Spectral Projector-Based Graph Fourier Transforms

@article{Deri2017SpectralPG, title={Spectral Projector-Based Graph Fourier Transforms}, author={Joya A. Deri and Jos{\'e} M. F. Moura}, journal={IEEE Journal of Selected Topics in Signal Processing}, year={2017}, volume={11}, pages={785-795} }

This paper considers the definition of the graph Fourier transform (GFT) and of the spectral decomposition of graph signals. Current literature does not address the lack of unicity of the GFT. The GFT is the mapping from the signal set into its representation by a direct sum of irreducible shift invariant subspaces: 1) this decomposition may not be unique; and 2) there is freedom in the choice of basis for each component subspace. These issues are particularly relevant when the graph shift has…

## 48 Citations

### Graph Fourier transforms on directed product graphs

- Computer Science
- 2022

It is shown that the proposed GFTs could represent spatial-temporal data sets on directed networks with strong correlation eﬃciently, and in the undirected graph setting they are essentially the joint GFT in the literature.

### Agile Inexact Methods for Spectral Projector-Based Graph Fourier Transforms

- Computer ScienceArXiv
- 2017

The results show that identical highly expressed geolocations can be identified with the inexact method and the method based on eigenvector projections, while reducing computation time by a factor of 26,000 and reducing energy dispersal among the spectral components corresponding to the multiple zero eigenvalue.

### A Directed Graph Fourier Transform With Spread Frequency Components

- Computer ScienceIEEE Transactions on Signal Processing
- 2019

It is shown that the spectral-dispersion minimization problem can be cast as supermodular optimization over the set of candidate frequency components, whose orthonormality can be enforced via a matroid basis constraint.

### Graph Fourier transform based on singular value decomposition of directed Laplacian

- Computer Science
- 2022

This paper introduces a novel deﬁnition of GFT on directed graphs, and uses singular values of Laplacian to carry the notion of graph frequencies, and shows that frequencies and frequency components of the proposed GFT can be evaluated by solving some constrained minimization problems with low computational cost.

### A digraph fourier transform with spread frequency components

- Computer Science, Mathematics2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP)
- 2017

It is shown that orthonormality can be enforced via a matroid basis constraint, which motivates adopting a scalable greedy algorithm to obtain an approximate solution with provable performance guarantee.

### Digraph Fourier Transform via Spectral Dispersion Minimization

- Computer Science, Mathematics2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2018

This work addresses the problem of constructing a graph Fourier transform (GFT) for both undirected and directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network, and advocates a two-step design.

### Digraph Signal Processing With Generalized Boundary Conditions

- Computer ScienceIEEE Transactions on Signal Processing
- 2021

An algorithm is designed that adds a small number of edges to a given digraph to destroy nontrivial Jordan blocks and yields an approximate eigenbasis and Fourier transform for the original digraph, which can be viewed as generalized form of boundary conditions, a common practice in signal processing.

### Diagonalizable Shift and Filters for Directed Graphs Based on the Jordan-Chevalley Decomposition

- Computer ScienceICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2020

This paper proposes to replace a given adjacency shift A by a diagonalizable shift AD obtained via the Jordan-Chevalley decomposition, which means that AD generates the subalgebra of all diagonalizable filters and is itself a polynomial in A (i.e., a filter).

### Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms

- MathematicsArXiv
- 2017

The utility of two equivalence graph classes over which a spectral projector-based graph Fourier transform is equivalent are defined and methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed.

### Generalized Sampling on Graphs With Subspace and Smoothness Priors

- Computer ScienceIEEE Transactions on Signal Processing
- 2020

A framework for generalized sampling of graph signals that parallels sampling in shift invariant (SI) subspaces is proposed and the use of recovery techniques when the recovery filter can be optimized and under a setting in which a predefined filter must be used is suggested.

## References

SHOWING 1-10 OF 86 REFERENCES

### Agile Inexact Methods for Spectral Projector-Based Graph Fourier Transforms

- Computer ScienceArXiv
- 2017

The results show that identical highly expressed geolocations can be identified with the inexact method and the method based on eigenvector projections, while reducing computation time by a factor of 26,000 and reducing energy dispersal among the spectral components corresponding to the multiple zero eigenvalue.

### A Spectral Graph Uncertainty Principle

- Computer ScienceIEEE Transactions on Information Theory
- 2013

A spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed, which provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain.

### Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms

- MathematicsArXiv
- 2017

The utility of two equivalence graph classes over which a spectral projector-based graph Fourier transform is equivalent are defined and methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed.

### Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs

- MathematicsIEEE Transactions on Signal Processing
- 2013

This paper relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that are k-hop localized with compact spectral spread and still satisfy the perfect reconstruction conditions.

### On the Shift Operator, Graph Frequency, and Optimal Filtering in Graph Signal Processing

- Computer ScienceIEEE Transactions on Signal Processing
- 2017

A new shift operator based GSP framework enables the signal analysis along a correlation structure defined by a graph shift manifold as opposed to classical signal processing operating on the assumption of the correlation structure with a linear time shift manifold.

### Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

- EngineeringIEEE Transactions on Signal Processing
- 2017

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper and studies M-channel polynomial filter banks on graphs and shows that for M-block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks.

### Sampling of Graph Signals With Successive Local Aggregations

- Computer ScienceIEEE Transactions on Signal Processing
- 2016

A more general sampling scheme, under which, either the aggregation approach or the alternative approach of sampling a graph signal by observing the value of the signal at a subset of nodes can be both viewed as particular cases.

### Perfect Reconstruction Two-Channel Wavelet Filter Banks for Graph Structured Data

- Computer ScienceIEEE Transactions on Signal Processing
- 2012

This work proposes the construction of two-channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph, and proposes quadrature mirror filters for bipartite graph which cancel aliasing and lead to perfect reconstruction.

### The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms

- Computer Science, MathematicsSIAM J. Comput.
- 2003

This paper presents an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of Chebyshev polynomials.