lift::poly Class Reference

Inheritance diagram for lift::poly:

[legend]Collaboration diagram for lift::poly:

[legend]List of all members.

Public Member Functions
	poly ()
void	forwardTrans (double[] vec)
void	inverseTrans (double[] vec)
Protected Member Functions
void	update (double[] vec, int N, int direction)
void	predict (double[] vec, int N, int direction)
Static Package Attributes
static final int	numPts = 4

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Detailed Description

Polynomial wavelets

This wavelet transform uses a polynomial interpolation wavelet (e.g., the function used to calculate the differences). A Haar scaling function (the calculation of the average for the even points) is used.

This wavelet transform uses a two stage version of the lifting scheme. In the "classic" two stage Lifting Scheme wavelet the predict stage preceeds the update stage. Also, the algorithm is absolutely symetric, with only the operators (usually addition and subtraction) interchanged.

The problem with the classic Lifting Scheme transform is that it can be difficult to determine how to calculate the smoothing (scaling) function in the update phase once the predict stage has altered the odd values. This version of the wavelet transform calculates the update stage first and then calculates the predict stage from the modified update values. In this case the predict stage uses 4-point polynomial interpolation using even values that result from the update stage.

In this version of the wavelet transform the update stage is no longer perfectly symetric, since the forward and inverse transform equations differ by more than an addition or subtraction operator. However, this version of the transform produces a better result than the Haar transform extended with a polynomial interpolation stage.

This algorithm was suggested to me from my reading of Wim Sweldens' tutorial Building Your Own Wavelets at Home.

  
  http://www.bearcave.com/misl/misl_tech/wavelets/lifting/index.html
  

Copyright and Use

You may use this source code without limitation and without fee as long as you include: <blockquote> This software was written and is copyrighted by Ian Kaplan, Bear Products International, www.bearcave.com, 2001. </blockquote>

This software is provided "as is", without any warrenty or claim as to its usefulness. Anyone who uses this source code uses it at their own risk. Nor is any support provided by Ian Kaplan and Bear Products International.

Please send any bug fixes or suggested source changes to:

     iank@bearcave.com

Author:

Ian Kaplan

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Constructor & Destructor Documentation

lift::poly::poly (   )  [inline]

poly class constructor { fourPt = new polyinterp(); }

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Member Function Documentation

void lift::poly::forwardTrans (  double[]  vec  )  [inline]

Polynomial wavelet lifting scheme transform. This version of the forwardTrans function overrides the function in the liftbase base class. This function introduces an extra polynomial interpolation stage at the end of the transform. Reimplemented from lift::liftbase. { final int N = vec.length; for (int n = N; n > 1; n = n >> 1) { split( vec, n ); update( vec, n, forward ); predict( vec, n, forward ); } // for } // forwardTrans

void lift::poly::inverseTrans (  double[]  vec  )  [inline]

Polynomial wavelet lifting Scheme inverse transform. This version of the inverseTrans function overrides the function in the liftbase base class. This function introduces an inverse polynomial interpolation stage at the start of the inverse transform. Reimplemented from lift::liftbase. { final int N = vec.length; for (int n = 2; n <= N; n = n << 1) { predict( vec, n, inverse ); update( vec, n, inverse ); merge( vec, n ); } } // inverseTrans

void lift::poly::predict (  double[]  vec, int  N, int  direction )  [inline, protected, virtual]

Predict an odd point from the even points, using 4-point polynomial interpolation. The four points used in the polynomial interpolation are the even points. We pretend that these four points are located at the x-coordinates 0,1,2,3. The first odd point interpolated will be located between the first and second even point, at 0.5. The next N-3 points are located at 1.5 (in the middle of the four points). The last two points are located at 2.5 and 3.5. For complete documentation see http://www.bearcave.com/misl/misl_tech/wavelets/lifting/index.html The difference between the predicted (interpolated) value and the actual odd value replaces the odd value in the forward transform. As the recursive steps proceed, N will eventually be 4 and then 2. When N = 4, linear interpolation is used. When N = 2, Haar interpolation is used (the prediction for the odd value is that it is equal to the even value). Parameters: vec  the input data on which the forward or inverse transform is calculated. N  the area of vec over which the transform is calculated direction  forward or inverse transform Implements lift::liftbase. { int half = N >> 1; double d[] = new double[ numPts ]; int k = 42; for (int i = 0; i < half; i++) { double predictVal; if (i == 0) { if (half == 1) { // e.g., N == 2, and we use Haar interpolation predictVal = vec[0]; } else { fill( vec, d, N, 0 ); predictVal = fourPt.interpPoint( 0.5, half, d ); } } else if (i == 1) { predictVal = fourPt.interpPoint( 1.5, half, d ); } else if (i == half-2) { predictVal = fourPt.interpPoint( 2.5, half, d ); } else if (i == half-1) { predictVal = fourPt.interpPoint( 3.5, half, d ); } else { fill( vec, d, N, i-1); predictVal = fourPt.interpPoint( 1.5, half, d ); } int j = i + half; if (direction == forward) { vec[j] = vec[j] - predictVal; } else if (direction == inverse) { vec[j] = vec[j] + predictVal; } else { System.out.println("poly::predict: bad direction value"); } } } // predict

void lift::poly::update (  double[]  vec, int  N, int  direction )  [inline, protected, virtual]

The update stage calculates the forward and inverse Haar scaling functions. The forward Haar scaling function is simply the average of the even and odd elements. The inverse function is found by simple algebraic manipulation, solving for the even element given the average and the odd element. In this version of the wavelet transform the update stage preceeds the predict stage in the forward transform. In the inverse transform the predict stage preceeds the update stage, reversing the calculation on the odd elements. Implements lift::liftbase. { int half = N >> 1; for (int i = 0; i < half; i++) { int j = i + half; double updateVal = vec[j] / 2.0; if (direction == forward) { vec[i] = (vec[i] + vec[j])/2; } else if (direction == inverse) { vec[i] = (2 * vec[i]) - vec[j]; } else { System.out.println("update: bad direction value"); } } }

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The documentation for this class was generated from the following file:

• lift/poly.java

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