Application of Multiple-Scale Analysis with Less Than 2 Multiplicity Process Images

Abstract

The problem under study is of immediate interest since multiple-scale analysis of images is carried out using wavelets based on Gaussian function derivatives and wavelets constructed in frequency domain. This work describes possibility for applying algorithms of multiple-scale analysis ofone-dimensional and two-dimensional signals within the above wavelets. Alternatively to discrete wavelet transform (the Mallat algorithm), the authors suggest that multiple-scale frequency-domain analysis of images with less than 2 multiplicity should be made, i.e. coefficient of scale change should be less than 2. Although multiplicity of analysis is less than 2, signal may be presented in the form of successive approximations, as when applying discrete wavelet-transform. Reducing multiplicity allows to increase the depth of decomposition, and thereby improve accuracy of signals analysis and synthesis. Here, the number of decomposition levels is an order of magnitude more as compared with conventional multiple-scale analysis. This is accomplished by progressive scanning of image, i.e. image is not line-wise or column-wise processed; it is scanned progressively as a whole. Fast Fourier transformation decreases the time for transforming by four orders of magnitude compared to direct numerical integration. Thus, no increase in decomposition and reconstruction time is observed when compared to time spent on multiple-scale analysis using discrete wavelets. Application of multi-scale analysis with less than 2 multiplicity makes it possible to cleanse one-dimensional and two-dimensional signals of noise, compress them, determine mean-square deviation and mean size of objects on images obtained through electronic or optical microscopy, and space and aerial photographs.

Keywords: wavelet transform, decomposition, reconstruction, multiple-scale analysis, mean size, variance, satellite image, fractal.