The Haar–Wavelet Transform in Digital Image Processing: Its Status and Achievements

Image processing and analysis based on the continuous or discrete image transforms are classic techniques. The image transforms are widely used in image filtering, data description, etc. Nowadays the wavelet theorems make up very popular methods of image processing, denoising and compression. Considering that the Haar functions are the simplest wavelets, these forms are used in many methods of discrete image transforms and processing. The image transform theory is a well known area characterized by a precise mathematical background, but in many cases some transforms have particular properties which are not still investigated. This paper for the first time presents graphic dependences between parts of Haar and wavelets spectra. It also presents a method of image analysis by means of the wavelets–Haar spectrum. Some properties of the Haar and wavelets spectrum were investigated. The extraction of image features immediately from spectral coefficients distribution were shown. In this paper it is presented that two–dimensional both, the Haar and wavelets functions products man be treated as extractors of particular image features. Furthermore, it is also shown that some coefficients from both spectra are proportional, which simplify slightly computations and analyses.