Overlap–add method

Abstract: We introduce a new method of constructing kernels on sets whose elements are discrete structures like strings, trees and graphs. The method can be applied iteratively to build a kernel on a innnite set from kernels involving generators of the set. The family of kernels generated generalizes the family of radial basis kernels. It can also be used to deene kernels in the form of joint Gibbs probability distributions. Kernels can be built from hidden Markov random elds, generalized regular expressions, pair-HMMs, or ANOVA de-compositions. Uses of the method lead to open problems involving the theory of innnitely divisible positive deenite functions. Fundamentals of this theory and the theory of reproducing kernel Hilbert spaces are reviewed and applied in establishing the validity of the method.

Abstract: Two distinct methods for synthesizing a signal from its short-time Fourier transform have previously been proposed. We call these methods the filter-bank summation (FBS) method and the overlap add (OLA) method. Each of these synthesis techniques has unique advantages and disadvantages in various applications due to the way in which the signal is reconstructed. In this paper we unify the ideas behind the two synthesis techniques and discuss the similarities and differences between these methods. In particular, we explicitly show the effects of modifications made to the short-time transform (both fixed and time-varying modifications are considered) on the resulting signal and discuss applications where each of the techniques would be most useful The interesting case of nonlinear modifications (possibly signal dependent) to the short-time Fourier transform is also discussed. Finally it is shown that a formal duality exists between the two synthesis methods based on the properties of the window used for obtaining the short-time Fourier transform.

Abstract: A new technique is proposed for the mathematical process of reconstruction of a three-dimensional object from its transmission shadowgraphs; it uses convolutions with functions defined in the real space of the object, without using Fourier transforms. The object is rotated about an axis at right angles to the direction of a parallel beam of radiation, and sections of it normal to the axis are reconstructed from data obtained by scanning the corresponding linear strips in the shadowgraphs at different angular settings. Since the formulae in the convolution method involve only summations over one variable at a time, while a two-dimensional reconstruction with the Fourier transform technique requires double summations, the convolution method is much faster (typically by a factor of 30); the relative increase in speed is larger where greater resolution is required. Tests of the convolution method with computer-simulated shadowgraphs show that it is also more accurate than the Fourier transform method. It has good potentialities for application in electron microscopy and x-radiography. A new method of reconstructing helical structures by this technique is also suggested.