Legendre wavelet

[1] Legendre functions have widespread applications in which spherical coordinate system is appropriate.

The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter.

Wavelets associated to FIR filters are commonly preferred in most applications.

[3] An extra appealing feature is that the Legendre filters are linear phase FIR (i.e. multiresolution analysis associated with linear phase filters).

Although being compactly supported wavelet, legdN are not orthogonal (but for N = 1).

[5] 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.

[2] 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.

, the smoothing filter of an MRA can be defined so that the magnitude of the low-pass

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.

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,[6][7] yielding: Indeed,

A suitable phase assignment is done so as to properly adjust the transfer function

, so that the Legendre wavelets have compact support for every odd integer

The finite support width Legendre family is denoted by legd (short name).

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 cascade algorithm).

The wavelet has compact support and finite impulse response AMR filters (FIR) are used (table 1).

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 MATLAB.

Figure 5 illustrates the WP functions derived from legd2.

Figure 1 - Magnitude of the transfer function for Legendre multiresolution smoothing filters. Filter for orders 1, 3, and 5.
Figure 2 - Shape of Legendre Wavelets of degree (legd2) derived after 4 and 8 iteration of the cascade algorithm, respectively. Shape of Legendre Wavelets of degree (legd3) derived by the cascade algorithm after 4 and 8 iterations of the cascade algorithm, respectively.
Figure 3 - legd8 wavelet display over MATLAB using the wavemenu command.
Figure 5 - Legendre (legd2) Wavelet Packets W system functions: WP from 0 to 9.