forward_sgwt | R Documentation |

`forward_sgwt`

computes the forward Spectral Graph Wavelet Transform (SGWT) for a given graph signal `f`

.

```
forward_sgwt(
f,
evalues,
evectors,
b = 2,
filter_func = zetav,
filter_params = list()
)
```

`f` |
Numeric vector representing the graph signal to analyze. |

`evalues` |
Numeric vector of eigenvalues of the Laplacian matrix. |

`evectors` |
Matrix of eigenvectors of the Laplacian matrix. |

`b` |
Numeric scalar that control the number of scales in the SGWT. It must be greater than 1. |

`filter_func` |
Function used to compute the filter values. By default, it uses the |

`filter_params` |
List of additional parameters required by |

The transform is constructed based on the frame defined by the `tight_frame`

function, without the need for its explicit calculation. Other filters can be passed as parameters. The SGWT provides a multi-scale analysis of graph signals.

Given a graph signal `f`

of length `N`

, `forward_sgwt`

computes the wavelet coefficients using SGWT.

The eigenvalues and eigenvectors of the graph Laplacian, are denoted as `\Lambda`

and `U`

respectively. The parameter `b`

controls the number of scales, and `\lambda_{\text{max}}`

is the largest eigenvalue.

For each scale `j = 0, 1, \ldots, J`

, where

`J = \left\lfloor \frac{\log(\lambda_{\text{max}})}{\log(b)} \right\rfloor + 2`

the wavelet coefficients are computed as:

```
\mathbf{w}_j = U \left( g_j \odot (U^T f) \right)
```

where

`g_j(\lambda) = \sqrt{\psi_j(\lambda)}`

and `\odot`

denotes element-wise multiplication.

The final result is a concatenated vector of these coefficients for all scales.

`wc`

A concatenated vector of wavelet coefficients.

`forward_sgwt`

can be adapted for other filters by passing a different filter function to the `filter_func`

parameter.

The computation of `k_{\text{max}}`

using `\lambda_{\text{max}}`

and `b`

applies primarily to the default `zetav`

filter. It can be overridden by providing it in the `filter_params`

list for other filters.

GĂ¶bel, F., Blanchard, G., von Luxburg, U. (2018). Construction of tight frames on graphs and application to denoising. In Handbook of Big Data Analytics (pp. 503-522). Springer, Cham.

Hammond, D. K., Vandergheynst, P., & Gribonval, R. (2011). Wavelets on graphs via spectral graph theory. Applied and Computational Harmonic Analysis, 30(2), 129-150.

de Loynes, B., Navarro, F., Olivier, B. (2021). Data-driven thresholding in denoising with Spectral Graph Wavelet Transform. Journal of Computational and Applied Mathematics, Vol. 389.

`inverse_sgwt`

, `tight_frame`

```
## Not run:
# Extract the adjacency matrix from the grid1 and compute the Laplacian
L <- laplacian_mat(grid1$sA)
# Compute the spectral decomposition of L
decomp <- eigensort(L)
# Create a sample graph signal
f <- rnorm(nrow(L))
# Compute the forward Spectral Graph Wavelet Transform
wc <- forward_sgwt(f, decomp$evalues, decomp$evectors)
## End(Not run)
```

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