Hartley transform

About: Hartley transform is a research topic. Over the lifetime, 2709 publications have been published within this topic receiving 79944 citations.

Reconstruction of a real function from its Hartley-transform intensity

Abstract: A method for reconstructing a real object function from the intensity of its Hartley transform is proposed. This method is established as a closed-form expression by making use of the mathematical properties of entire functions of the exponential type. The usefulness of the method is shown in computer-simulation studies of the reconstruction of the one-dimensional real object from its Hartley intensity. In addition it is shown that the theory developed in one dimension is straightforwardly extended to two dimensions.

An explanation for the logarithmic connection between linear and morphological systems

Abstract: Since the introduction of the slope transform by Dorst/van den Boomgaard and Maragos as the morphological equivalent of the Fourier transform, people have been surprised about the almost logarithmic relation between linear and morphological system theory. This article gives an explanation by revealing that morphology in essence is linear system theory in a specific algebra. While classical linear system theory uses the standard (+,×)-algebra, the morphological system theory is based on the idempotent (max, +)-algebra and the (min, +)- algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions *e and *d induced by these idempotent algebras. The slope transform in the (max, +)-algebra, however, corresponds to the logarithmic multivariate Laplace transform in the (+, ×)-algebra. We study relevant properties of this transform and its links to convex analysis. This leads to the definition of the so-called Cramer transform as the Legendre-Fenchel transform of the logarithmic Laplace transform. Originally known from the theory of large deviations in stochastics, the Cramer transform maps standard convolution to *e-convolution, and it maps Gaussians to quadratic functions. The article is a step towards the unification of linear and morphological system theories on the basis of a general linear system theory in an appropriate algebra.

Fractionalization of the linear cyclic transforms

Abstract: In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.

Method of flow graph simplification for the 16-point discrete Fourier transform

Abstract: An efficient method for the realization of the paired algorithm for calculation of the one-dimensional (1-D) discrete Fourier transform (DFT), by simplifying the signal-flow graph of the transform, is described. The signal-flow graph is modified by separating the calculation for real and imaginary parts of all inputs and outputs in the signal-flow graph and using properties of the transform. The examples for calculation of the eight- and 16-point DFTs are considered in detail. The calculation of the 16-point DFT of real data requires 12 real multiplications and 58 additions. Two multiplications and 20 additions are used for the eight-point DFT.