Legendre Wavelet - Legendre Multiresolution Filters

Legendre Multiresolution Filters

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 (MRA). 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 =1,3 and 5. 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.