Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates. The radial part of the solution varies from one potential to another, but the harmonics are always the same and are a consequence of spherical symmetry. Spherical harmonics are solutions of the Legendre -order differential equation, n integer: polynomials can be used to define the smoothing filter of a multiresolution analysis. Since the appropriate boundary conditions for an MRA are and, the smoothing filter of an MRA can be defined so that the magnitude of the low-pass can be associated to Legendre polynomials according to: Illustrative examples of filter transfer functions for a Legendre MRA are shown in figure 1, for A low-pass behaviour is exhibited for the filter H, as expected. The number of zeroes within is equal to the degree of the Legendre polynomial. Therefore, the roll-off of side-lobes with frequency is easily controlled by the parameter. The low-pass filter transfer function is given by The transfer function of the high-pass analysing filter is chosen according to Quadrature mirror filter condition, yielding: Indeed, and, as expected.
Legendre multiresolution filter coefficients
A suitable phase assignment is done so as to properly adjust the transfer function to the form The filter coefficients are given by: from which the symmetry: follows. There are just non-zero filter coefficients on, so that the Legendre wavelets have compact support for every odd integer.
MATLAB implementation of Legendre wavelets
Legendre wavelets can be easily loaded into the MATLAB wavelet toolbox—The m-files to allow the computation of Legendre wavelet transform, details and filter are available. The finite support width Legendre family is denoted by legd. Wavelets: 'legdN'. The parameter N in the legdN family is found according to . Legendre wavelets can be derived from the low-pass reconstruction filter by an iterative procedure. The wavelet has compact support and finite impulse response AMR filters are used. The first wavelet of the Legendre's family is exactly the well-known Haar wavelet. Figure 2 shows an emerging pattern that progressively looks like the wavelet's shape. The Legendre wavelet shape can be visualised using the wavemenu command of MATLAB. Figure 3 shows legd8 wavelet displayed using MATLABTM. Legendre Polynomials are also associated with windows families.
