Discrete Gabor transforms: The Gabor-Gram matrix approach
The author develops efficient algorithms for the computation of Gabor coefficients and dual Gabor windows. In contrast to the matrix of the frame operator \(\text{GG}^*\) considered by \textit{S. Qiu} [Optical Eng. 34, No. 10, 2872-2878 (1995)], he studies the matrix \(\text{GM}= \text{G}^*\text{G}\). He shows that GM consists of circulant blocks. Further, if \(n\) divides \(ab\), then these blocks are arranged in a circulant manner. Consequently, matrix-vector multiplication with GM can be performed in an efficient way by using fast Fourier transforms. Applying this fact, the author proposes an efficient iterative algorithm (conjugate gradient algorithm) to compute the Gabor coefficients and the dual Gabor window. Numerical examples are included.
scientific article; zbMATH DE number 1220784
scientific article