Digital Image Compression Using Fourier Transform and Wavelet Technique

Abstract

Fourier analysis and wavelet analysis have often been used in time series analysis. Fourier analysis can be used to detect periodic components that have sinusoidal shape. However, it might be misleading when the periodic components are not sinusoidal. The resulting Fourier analysis is more difficult to interpret compared with classical Fourier analysis. Wavelet analysis is very useful in analyzing and describing time series with gradual frequency changes. Wavelet analysis also has a shortcoming by giving no exact meaning to the concept of frequency because wavelets are not periodic functions. In addition, the two analysis methods above require equally-spaced time series observations.
In this paper, by using a sequence of periodic step functions, a new analysis method, adaptive Fourier analysis, and its theory are developed. These can be applied to time series data where patterns may take general periodic shapes that include sinusoids as special cases. Most importantly, the resulting adaptive Fourier analysis does not require equally-spaced time series observations.