Decomposition of functions and separation of concerns are the key aspects of our algorithm.
The wavelet transform provides a mathematical tool for the hierarchical decomposition of functions to obtain a projective decomposition of the data into different scales and therefore extract local information.
Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian.
Consideration was given to identification of discrete processes, which is reducible to the functional decomposition of discrete functions, where by the decomposition is meant the representation of a function by a formula in the basis of binary operations.
Let us notice that assumption (2.1), is equivalent to the existence of the decomposition of the function as the difference of two twice continuously differentiable convex functions such that.
Figure 2 CWT based decomposition of block function at scale J = 10 (with ( Ψ ( t )) first derivative of Gaussian function) is shown with the dotted lines (where total signal size = 2 J ).
Considering (H = H^{2}(mathbb{R}^{N})) and (mathcal{K} = {u in H^{2}(mathbb{R}^{N}); u geq0}), if it is possible to prove that (mathcal{K}^ subset-mathcal{K}), then we have a decomposition of a function (u in H^{2}(mathbb{R}^{N})) in terms of a non-negative and a non-positive function such that many times we can substitute the trivial decomposition not available in our case.
This approach permits to compute the decomposition of a function into a lacunary wavelet basis, i.e., a basis constituted of a subset of all basis functions up to a certain scale, without modification.
All of them we can transform using new parameters to the presented form: Using the following equation for decomposition of rational function to partial fraction: (27).
In the present study, a multiresolution wavelet decomposition of the function f0 t) (Equation (18)) was performed using a wavelet from the coiflets family.
By performing multiresolution wavelet decomposition of the function f0 t) to the level m, we obtain its representation in the following form (see Equation 5): f 0 t = g − 1 t + g − 2 t + … + g − m t + f − m t = ∑ j = − 1 − m ∑ n d j, n Ψ j, n t + ∑ n c − m, n ϕ − m, n t. (18).
