Generalized Discrete Hartley Transforms

R.V. Hartley disclosed a real-valued transform closely related to the Fourier transform in 1942. Besides having interesting properties of its own, the transform introduced by Hartley allows an indirect computation of the Fourier power spectrum of a given function only using real arithmetic. In the last decade some new discrete real-valued orthogonal transforms have been proposed, which are Hartley-related to other known complex-valued ones. The present paper studies (1) the necessary conditions for the existence of a Hartley mate for any complex-valued orthogonal transform and (2) the relationship between the 2D-spectrum of a real-valued Matrix using the complex-valued and the corresponding Hartley transform. 2D transforms are used for picture processing and pattern analysis.