1-D and 2-D real-valued discrete Gabor transforms

By replacing the complex-valued Gabor basis functions of the complex-valued discrete Gabor transform (CDGT) with real-valued Gabor basis functions, we propose a real-valued discrete Gabor transform (RDGT) for finite discrete signal and image representations. The RDGT provides a simpler method than the CDGT to calculate the transform (or Gabor) coefficients from finite summations and to reconstruct the original signal or image exactly from the computed transform coefficients. The similarity between the RDGT and the discrete Hartley transform (DHT) enables the RDGT to utilize the fast DHT algorithms for fast computation. Moreover, the RDGT has a simple relationship with the CDGT such that the CDGT coefficients can be directly computed from the RDGT coefficients