Wavelet transformation on a finite interval

Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

Local transformations with a singular wavelet

The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.

Research of Image Processing Based on Lifting Wavelet Transform

This paper Study problems which based on lifting wavelet transform image processing. Coding and decoding a complete digital image by using W97-2 wavelet basis wavelet transform, combined with the embedded zerotree wavelet coding and binary arithmetic coding, and complete a lossless compression combined with the international standard test images. Experimental results show that graphics, image processing will come into a higher level because of wavelet analysis combined with image processing.