The Cosine Number Transform: A Graph Signal Processing Approach In this paper, we provide a matrix that diagonalizes the Laplacian of a path graph over a finite field. By analogy with the fact that the discrete cosine transform matrix diagonalizes the Laplacian of (realvalued) path graphs, we identify the provided matrix as that of the cosine number transform (CNT). Together with the fact that the Fourier number transform matrix diagonalizes any circulant matrix (the Laplacian of a ring graph, for example), this motivates us to introduce a preliminary definition of what should be the graph Fourier number transform (GFNT). We revisit the application of the CNT in image encryption and give examples of the GFNT.

Approximate eigenvalue decompositions of orthonormal and symmetric transformations with a few Householder reflectors Abstract   The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The classic examples are the Fourier and wavelet transforms that enjoy numerically efficient implementations (FFT and FWT, respectively). Unfortunately, orthonormal transformations are in general unstructured, and therefore they do not enjoy low computational complexity properties. In this paper, based on Householder reflectors, we introduce a class of orthonormal matrices that are numerically efficient to manipulate: we control the complexity of matrixvector multiplications with these matrices using a given parameter. We provide numerical algorithms that approximate any orthonormal or symmetric transform with a new orthonormal or symmetric structure made up of products of a given number of Householder reflectors. We show analyses and numerical evidence to highlight the accuracy of the proposed approximations and provide an application to the case of learning fast Mahalanobis distance metric transformations.