System of 57 spectral kernels, as show below:
which were used in e.g. the following papers for estimating the energy content of fMRI graph signals on the normalized graph Laplacian spectrum:
Behjat, H. and Larsson, M., 2020. Spectral characterization of functional MRI data on voxel-resolution cortical graphs. In Proc. IEEE Int. Symp. Biomed. Imaging, pp. 558–562. paper
Behjat, H., et al., 2021. Characterization of spatial dynamics of fMRI data in white matter using diffusion-informed white matter harmonics. In Proc. IEEE Int. Symp. Biomed. Imaging, pp. 1586-1590. paper
The kernels are defined on the range 0 to 2, which are the lower and upper bounds for eigenvalues of any given symmetric normalized Laplacian matrix defined as L = I - D^{-1/2} A D^{-1/2}, where A and D denote the graph adjacency matrix and degree matrix, respectively, and I denotes the identity matrix. The kernels at the lower end of the spectrum are narrower since many graph signals (in particular, fMRI voxel-wise graph signals) have most of their energy content in that part of the spectrum, and thus, such narrow kernels enable more detailed estimation of signal energy profiles in this part of the spectrum.
The system of kernels are precomputed and made available as function handles; in particular, 57 function handles, which are saved in two files (due to the large file size) and can be found in folder mats: sosks57_lower.mat (the first 20 kernels) and sosks57_upper.mat (the remaining 37 kernels). The central (peak) value of each kernel is stored in file sosks57_cents.mat.
For large graphs, graph signals can not be represented in the spectral domain via direct computation of the Graph Fourier Transform (GFT) since it is impractical to fully diagonalize the graph Laplacian matrix. As such, it is not possible to obtain a spectral representation of a given graph signal on the graph at the resolution of eigenvalues, and, therefore, the distribution of a graph signal's energy cannot be directly computed using these kernels within the spectral domain in a similar way as Frequency filtering is implemented in conventional signal processing via a masking procedure. Nevertheless, one can approximate the distribution of a graph signal's energy in association to a given spectral kernel using a polynomial approximation scheme, e.g. using Chebyshev filters as used in:
Hammond, D.K., et al, 2011. Wavelets on graphs via spectral graph theory. Applied and Computational Harmonic Analysis, 30(2), pp. 129-150. paper (see Section 6)
To do this, one should use a suitable Chebyshev polynomial order to ensure that (i) each approximated polynomial matches the desired kernel well, (ii) all the kernels togetehr satisfy the Parseval frame condition to an acceptable degree. File mats/sosks57_chebyOrds.mat provides suitable polynomial orders for each of the 57 kernels such that the resulting set of approximated system of spectral kernels do not deviate from the Parseval frame condition by more than 0.01 at any point across the spectral range 0 to 2.
Script construct_multires_uniform_tight_frame.m shows how this 57 system of spectral kernels were generated, in part using the signal-adapted tight frame design theory proposed in:
Behjat, H., et al., 2016. Signal-adapted tight frames on graphs. IEEE Trans. Signal Process., 64(22), pp.6017-6029. paper