Abstract—The paper describes algorithms of data processing using methods of wavelet analysis. The main integral transforms of 2-D data based on ideas of multiscale analysis and the Radon transform are examined. The latter belongs to the class of beamlet transforms. They are briefly characterized and their application is illustrated by simple geophysical examples.

PACS numbers: 91.25.-r+91.25.Qi DOI: 10.1134/S106935130703007X

 

INTRODUCTION

In the last two decades, an autonomous branch of applied mathematics known as wavelet analysis has been developed [Goupillaud et al., 1984-1985; Daubechies, 2000; Chui, 1992; Mallat, 1999; Astaf’eva, 1996; Yudin et al., 2000; D’yakonov, 2004; Smolentsev, 2005]. At present, multiscale data transform plays a fundamental role in the approximation theory.

Wavelets deal with two main parameters, scale and location. Two-dimensional wavelets have, as a rule, only a fixed number of directional elements, but real signals can contain signatures of other directions that need to be emphasized by data transform. In the last decade, numerous publications have been devoted to the class of transforms of multidimensional data including such an additional parameter as the orientation of linear segments. Along with algorithms of multiscale analysis of signals induced by wavelets, this class of integral transforms utilizes ideas involved in the Radon transform, i.e., the mathematical apparatus underlying computer tomography. In contrast to wavelets, the main parameters of the transforms considered here are the scale, location, and direction of linear segments of data.

The integral transforms of the class under consideration include the following main modifications: beam- let, ridgelet, and curvelet transforms. These types of transforms of 2-D data possess a high sensitivity and accuracy in relation to the detection, identification, and localization of objects. For briefness, if this does not lead to misunderstanding, all of the aforementioned and similar integral transforms will be called here beamlet transforms because they involve linear stacking. We should note that there exist other, no less interesting approaches to data analysis that are not considered in the present paper.

The adaptive filtering of data based on the thresholding of coefficients in the image domain and the subsequent recovery of the signal is the main constituent of data compression and/or an increase in the signal/noise ratio. The crucial point here is the choice of an optimal threshold for the discrimination between useful information and the most probable noise.

The treatment of geophysical information is an application domain of integral transforms that stimulated the appearance, development, and elaboration of wavelet analysis [Goupillaud et al., 1984-1985]. In essence, many algorithms used for processing and interpretation of geophysical data (primarily, seismic and seismological) are close to multiscale (multiwindow) wavelet-like analysis of large volumes of information. These heuristic algorithms are commonly limited by various types of window technologies and their mathematical substantiation on the basis of multiscale analysis of multidimensional data is advantageous for the extraction of useful signals from random and/or regular noises [Hoekstra, 1996; Dudova and Yudin, 2005; Yudin and Yudin, 2005].

Below, we describe theoretical aspects of the algorithms and focus on the illustration of their operation by simple test examples using real seismic and electric data.