## Wavelets on graphs via spectral graph theory.(English)Zbl 1213.42091

The classical continuous wavelet transform can be generated by the choice of a single “mother” wavelet $$\psi$$. Other wavelets (with different locations and spatial scales) are formed by transforming the mother wavelet, for example, $$\psi_{s,a}(x)=\frac1s\psi\left(\frac{x-a}{s}\right)$$. Then for any suitable function (i.e. signal), the wavelet coefficients at scale $$s$$ and location $$a$$ can be given with $$\psi_{s,a}$$. Conversely, the wavelet coefficients enable us to reconstruct the original signal.
Several practical applications require discrete underlying spaces. The authors work out a possible framework of discrete wavelet transformation on weighted graphs. The first problem is the following: how to define $$\psi(sx)$$ if $$x$$ is a vertex of a graph? There is no expressive meaning of $$sx$$, where $$s$$ is a real scalar.
The authors get around the problem as follows. They define the Laplacian $(\mathcal{L}f)(m)=\sum_{n\sim m}a_{m,n}(f(m)-f(n)),$ where $$a_{m,n}$$ is the adjacency matrix of the weighted graph and $$f$$ is a real valued function on the vertices.
Since the usual one-dimensional Fourier transform uses the eigenfunctions of the Laplacian (i.e. the functions $$e^{i\omega x}$$), one may define the Fourier transform according to the eigenvectors of the Laplacian above. The authors go further and work out the spectral graph wavelet transform (SGWT) and deduces a number of its properties. It is also shown how can one speed up the SGWT-computations and apply it in practical applications, like transportation networks.
An implemented algorithm can be found at wiki.epfl.ch/sgwt.

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 94A12 Signal theory (characterization, reconstruction, filtering, etc.)

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