Representation of images using Le Gall transform
It is now obvious that the traditional approach to the analysis of stationary signals based on the Fourier transform is ineffective for the representation of functions and signals with local singularities. A compromise approach for signal analysis is the wavelet transform of signals based on a multiscale analysis. In this paper, we propose a technique for approximating images using Le Gall transform. A multi-scale analysis defines a sequence of disjoint nested approximative spaces V, in L<sup>2</sup> (ft). The space W- is defined as the orthogonal complement to Vj in Vj<sub>+1</sub>, i.e. V<sub>J+1</sub> = Vj ®W- and Wj ± Vj, J e Z. As Vj = Vj_i ®Wj_i it seems possible to define the function f (x), written in terms of basis functions of space Vj, in terms of basis functions of spaces V,-_i and Wj_i a fast wavelet transformation having a complexity O(n) is written in the form: then the expressions for the decomposition of the signal will be written in the form: 1, 2<sup>V</sup><sup>c</sup>j,k = <sup>c</sup>j-1,2k +<sup>1</sup> (dj,k-1 + <sup>d</sup>j,k ) <sup>d</sup>j,k = <sup>c</sup>j-1,2k+1 <sup>-</sup>1 <sup>(c</sup>j-1,2k + <sup>c</sup>j-1,2k+2), 1 4 Reconstruction is carried out in the reverse order: <sup>c</sup>j-1,2k = <sup>c</sup>j,k <sup>-</sup>1 <sup>(d</sup>j,k-1 + <sup>d</sup> j,k )> <sup>c</sup>j-1,2k+1 = <sup>d</sup>j,k +1 <sup>(c</sup>j-1,2k + <sup>c</sup>j-1,2k+2 ) The application of the proposed image representation technique allows obtaining the required resolution, as well as controlled original image detalisation, which is especially important in image compression and pattern recognition problems in the multiscale representation of real-world objects in an image. Above the image represented as a set of wavelet transform coefficients, can be perform filtering operations, highlighting various characteristics, thresholding, and others. The proposed approach to computing is characterized by low computational complexity.