In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, compactly supported
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s derived from
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications in which
spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
is appropriate.
[Colomer and Colomer] As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
associated to Legendre
multiresolution analysis
A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introd ...
is a
finite impulse response
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse r ...
(FIR) filter.
Wavelets associated to FIR filters are commonly preferred in most applications.
[ An extra appealing feature is that the Legendre filters are ''linear phase'' FIR (i.e. multiresolution analysis associated with ]linear phase
In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the sam ...
filters). These wavelets have been implemented on MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
(wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for ''N'' = 1).
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).][Mallat] 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
Roll-off is the steepness of a Transfer function, transfer function with frequency, particularly in network analysis (electrical circuits), electrical network analysis, and most especially in connection with filter (signal processing), filter cir ...
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 In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around \pi/2 of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror ...
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 .
:::''Table I - Smoothing Legendre FIR filter coefficients for ( is the wavelet order.)''
::: N.B. The minus signal can be suppressed.
MATLAB implementation of Legendre wavelets
Legendre wavelets can be easily loaded into the MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
wavelet toolbox—The m-files to allow the computation of Legendre wavelet transform, details and filter are (freeware) available. The finite support width Legendre family is denoted by legd (short name). Wavelets: 'legdN'. The parameter ''N'' in the legdN family is found according to (length of the MRA filters).
Legendre wavelets can be derived from the low-pass reconstruction filter by an iterative procedure (the cascade algorithm In the mathematical topic of wavelet theory, the cascade algorithm is a numerical method for calculating function values of the basic scaling and wavelet functions of a discrete wavelet transform
In numerical analysis and functional analysis, a d ...
). The wavelet has compact support and finite impulse response AMR filters (FIR) are used (table 1). The first wavelet of the Legendre's family is exactly the well-known Haar wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represe ...
. 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. Legendre Polynomials are also associated with windows families.[Jaskula]
Legendre wavelet packets
Wavelet packets Originally known as optimal subband tree structuring (SB-TS), also called wavelet packet decomposition (WPD)
(sometimes known as just wavelet packets or subband tree), is a wavelet transform where the discrete-time (sampled) signal is passed through ...
(WP) systems derived from Legendre wavelets can also be easily accomplished. Figure 5 illustrates the WP functions derived from legd2.
References
Bibliography
* M.M.S. Lira, H.M. de Oliveira, M.A. Carvalho Jr, R.M.C.Souza, Compactly Supported Wavelets Derived from Legendre Polynomials: Spherical Harmonic Wavelets, In: ''Computational Methods in Circuits and Systems Applications'', N.E. Mastorakis, I.A. Stahopulos, C. Manikopoulos, G.E. Antoniou, V.M. Mladenov, I.F. Gonos Eds., WSEAS press, pp. 211–215, 2003. {{isbn, 960-8052-88-2. Available a
ee.ufpe.br
* A. A. Colomer and A. A. Colomer, Adaptive ECG Data Compression Using Discrete Legendre Transform, ''Digital Signal Processing'', 7, 1997, pp. 222–228.
* A.G. Ramm, A.I. Zaslavsky, X-Ray Transform, the Legendre Transform, and Envelopes, ''J. of Math. Analysis and Appl''., 183, pp. 528–546, 1994.
* C. Herley, M. Vetterli, Orthogonalization of Compactly Supported Wavelet Bases, ''IEEE Digital Signal Process. Workshop'', 13-16 Sep., pp. 1.7.1-1.7.2, 1992.
* S. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation, ''IEEE Transactions on Pattern Analysis and Machine Intelligence'', 11, July pp. 674–693, 1989.
* M. Vetterli, C. Herly, Wavelets and Filter Banks: Theory and Design, ''IEEE Trans. on Acoustics, Speech, and Signal Processing'', 40, 9, p. 2207, 1992.
* M. Jaskula, New Windows Family Based on Modified Legendre Polynomials, ''IEEE Instrum. And Measurement Technol. Conf.'', Anchorage, AK, May, 2002, pp. 553–556.
Wavelets