smile.wavelet.DaubechiesWavelet Maven / Gradle / Ivy
/*
* Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
*
* Smile is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Smile is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Smile. If not, see .
*/
package smile.wavelet;
/**
* Daubechies wavelets. The Daubechies wavelets are a family of orthogonal
* wavelets defining a discrete wavelet transform and characterized by a
* maximal number of vanishing moments for some given support.
*
* In general the Daubechies wavelets are chosen to have the highest number A
* of vanishing moments, (this does not imply the best smoothness) for give
* n support width N = 2A, and among the 2A?1 possible solutions the one is
* chosen whose scaling filter has extremal phase. The wavelet transform is
* also easy to put into practice using the fast wavelet transform. Daubechies
* wavelets are widely used in solving a broad range of problems,
* e.g. self-similarity properties of a signal or fractal problems, signal
* discontinuities, etc.
*
* The Daubechies wavelets are not defined in terms of the resulting scaling
* and wavelet functions; in fact, they are not possible to write down in
* closed form.
*
* Daubechies orthogonal wavelets D2-D20 (even index numbers only) are commonly
* used. The index number refers to the number N of coefficients. Each wavelet
* has a number of zero moments or vanishing moments equal to half the number
* of coefficients. For example, D2 (the Haar wavelet) has one vanishing
* moment, D4 has two, etc. A vanishing moment limits the wavelet's ability
* to represent polynomial behaviour or information in a signal. For example,
* D2, with one moment, easily encodes polynomials of one coefficient, or
* constant signal components. D4 encodes polynomials with two coefficients,
* i.e. constant and linear signal components; and D6 encodes 3-polynomials,
* i.e. constant, linear and quadratic signal components. This ability to
* encode signals is nonetheless subject to the phenomenon of scale leakage,
* and the lack of shift-invariance, which arise from the discrete shifting
* operation during application of the transform. Sub-sequences which represent
* linear, quadratic (for example) signal components are treated differently
* by the transform depending on whether the points align with even- or
* odd-numbered locations in the sequence. The lack of the important property
* of shift-invariance, has led to the development of several different versions
* of a shift-invariant (discrete) wavelet transform.
*
* @author Haifeng Li
*/
public class DaubechiesWavelet extends Wavelet {
/**
* D4 coefficients
*/
private static final double[] c4 = {
0.4829629131445341, 0.8365163037378079,
0.2241438680420134, -0.1294095225512604
};
/**
* D6 coefficients
*/
private static final double[] c6 = {
0.3326705529500825, 0.8068915093110924, 0.4598775021184914,
-0.1350110200102546, -0.0854412738820267, 0.0352262918857095
};
/**
* D8 coefficients
*/
private static final double[] c8 = {
0.2303778133088964, 0.7148465705529155, 0.6308807679398587,
-0.0279837694168599, -0.1870348117190931, 0.0308413818355607,
0.0328830116668852, -0.010597401785069
};
/**
* D10 coefficients
*/
private static final double[] c10 = {
0.1601023979741929, 0.6038292697971895, 0.7243085284377726,
0.1384281459013203, -0.2422948870663823, -0.0322448695846381,
0.0775714938400459, -0.0062414902127983, -0.012580751999082,
0.0033357252854738
};
/**
* D12 coefficients
*/
private static final double[] c12 = {
0.111540743350, 0.494623890398, 0.751133908021,
0.315250351709, -0.226264693965, -0.129766867567,
0.097501605587, 0.027522865530, -0.031582039318,
0.000553842201, 0.004777257511, -0.001077301085
};
/**
* D14 coefficients
*/
private static final double[] c14 = {
0.0778520540850037, 0.3965393194818912, 0.7291320908461957,
0.4697822874051889, -0.1439060039285212, -0.2240361849938412,
0.0713092192668272, 0.0806126091510774, -0.0380299369350104,
-0.0165745416306655, 0.0125509985560986, 4.295779729214E-4,
-0.0018016407040473, 3.537137999745E-4
};
/**
* D16 coefficients
*/
private static final double[] c16 = {
0.0544158422431072, 0.3128715909143166, 0.6756307362973195,
0.585354683654216, -0.0158291052563823, -0.2840155429615824,
4.724845739124E-4, 0.1287474266204893, -0.017369301001809,
-0.0440882539307971, 0.0139810279174001, 0.0087460940474065,
-0.004870352993452, -3.91740373377E-4, 6.754494064506E-4,
-1.174767841248E-4
};
/**
* D18 coefficients
*/
private static final double[] c18 = {
0.0380779473638778, 0.2438346746125858, 0.6048231236900955,
0.6572880780512736, 0.1331973858249883, -0.2932737832791663,
-0.0968407832229492, 0.1485407493381256, 0.0307256814793365,
-0.0676328290613279, 2.50947114834E-4, 0.0223616621236798,
-0.0047232047577518, -0.0042815036824635, 0.0018476468830563,
2.303857635232E-4, -2.519631889427E-4, 3.93473203163E-5
};
/**
* D20 coefficients
*/
private static final double[] c20 = {
0.026670057901, 0.188176800078, 0.527201188932,
0.688459039454, 0.281172343661, -0.249846424327,
-0.195946274377, 0.127369340336, 0.093057364604,
-0.071394147166, -0.029457536822, 0.033212674059,
0.003606553567, -0.010733175483, 0.001395351747,
0.001992405295, -0.000685856695, -0.000116466855,
0.000093588670, -0.000013264203
};
/**
* Constructor. Create a Daubechies wavelet with n coefficients.
* n = 4, 6, 8, 10, 12, 14, 16, 18, or 20 are supported. For n = 4,
* D4Wavelet can be used instead.
* @param n the number of wavelet coefficients.
*/
public DaubechiesWavelet(int n) {
super(n == 4 ? c4 :
n == 6 ? c6 :
n == 8 ? c8 :
n == 10 ? c10 :
n == 12 ? c12 :
n == 14 ? c14 :
n == 16 ? c16 :
n == 18 ? c18 :
n == 20 ? c20 : c4
);
if ( n < 4 || n > 20 || n % 2 != 0) {
throw new IllegalArgumentException(String.format("n = %d not yet implemented.", n));
}
}
}