Orthogonal wavelet

An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal.That is, the inverse wavelet transform is the adjoint of the wavelet transform.If this condition is weakened one may end up with biorthogonal wavelets. An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal.That is, the inverse wavelet transform is the adjoint of the wavelet transform.If this condition is weakened one may end up with biorthogonal wavelets. The scaling function is a refinable function.That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation): where the sequence ( a 0 , … , a N − 1 ) {displaystyle (a_{0},dots ,a_{N-1})} of real numbers is called a scaling sequence or scaling mask.The wavelet proper is obtained by a similar linear combination, where the sequence ( b 0 , … , b M − 1 ) {displaystyle (b_{0},dots ,b_{M-1})} of real numbers is called a wavelet sequence or wavelet mask. A necessary condition for the orthogonality of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients: where δ m , n {displaystyle delta _{m,n}} is the Kronecker delta. In this case there is the same number M=N of coefficients in the scaling as in the wavelet sequence, the wavelet sequence can be determined as b n = ( − 1 ) n a N − 1 − n {displaystyle b_{n}=(-1)^{n}a_{N-1-n}} . In some cases the opposite sign is chosen. A necessary condition for the existence of a solution to the refinement equation is that there exists a positive integer A such that (see Z-transform): The maximally possible power A is called polynomial approximation order (or pol. app. power) or number of vanishing moments. It describes the ability to represent polynomials up to degree A-1 with linear combinations of integer translates of the scaling function.