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 (real-valued) 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.