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poly Class Template Reference

4-point polynomial interpolation wavelet. More...

#include <poly.h>

Inheritance diagram for poly::

List of all members.

Public Methods
	poly ()
	poly class constructor. More...
	~poly ()
	poly (const poly &rhs)
void	forwardStep (T &vec, const int n)
	One step in the forward wavelet transform. More...
virtual void	inverseStep (T &vec, const int n)
	One inverse wavelet transform step. More...
Protected Methods
void	update (T &vec, int N, transDirection direction)
	The update stage calculates the forward and inverse Haar scaling functions. More...
void	predict (T &vec, int N, transDirection direction)
	Predict an odd point from the even points, using 4-point polynomial interpolation. More...
Private Types
enum	bogus { numPts = 4 }
Private Methods
void	fill (T &vec, double d[], int N, int start)
	Copy four points or N (which ever is less) data points from vec into d These points are the "known" points used in the polynomial interpolation. More...
Private Attributes
polyinterp	fourPtInterp

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Detailed Description

template<class T> class poly

4-point polynomial interpolation wavelet.

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.

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

This code was originally implemented in Java and then ported to C++.

Definition at line 85 of file poly.h.

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Member Enumeration Documentation

template<class T> enum poly<T>::bogus [private]

Enumeration values: numPts   Definition at line 97 of file poly.h. { numPts = 4 } bogus;

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Constructor & Destructor Documentation

template<class T> poly<T>::poly<T> ( ) [inline]

poly class constructor. Definition at line 91 of file poly.h. {}

template<class T> poly<T>::~poly<T> ( ) [inline]

Definition at line 92 of file poly.h. {}

template<class T> poly<T>::poly<T> ( const poly<T> & rhs )

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Member Function Documentation

template<class T> void poly<T>::fill ( T & vec, double d[], int N, int start ) [inline, private]

Copy four points or N (which ever is less) data points from vec into d These points are the "known" points used in the polynomial interpolation. Parameters: vec   [] the input data set on which the wavelet is calculated d   [] an array into which N data points, starting at start are copied. N   the number of polynomial interpolation points start   the index in vec from which copying starts Definition at line 111 of file poly.h. Referenced by predict(). { int n = numPts; if (n > N) n = N; int end = start + n; int j = 0; for (int i = start; i < end; i++) { d[j] = vec[i]; j++; } } // fill

template<class T> void poly<T>::forwardStep ( T & vec, const int n ) [inline, virtual]

One step in the forward wavelet transform. Reimplemented from liftbase. Definition at line 248 of file poly.h. { split( vec, n ); update( vec, n, forward ); predict( vec, n, forward ); } // forwardStep

template<class T> void poly<T>::inverseStep ( T & vec, const int n ) [inline, virtual]

One inverse wavelet transform step. Reimplemented from liftbase. Definition at line 255 of file poly.h. { predict( vec, n, inverse ); update( vec, n, inverse ); merge( vec, n ); }

template<class T> void poly<T>::predict ( T & vec, int N, transDirection 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 Reimplemented from liftbase. Definition at line 200 of file poly.h. { int half = N >> 1; double d[4]; 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 = fourPtInterp.interpPoint( 0.5, half, d ); } } else if (i == 1) { predictVal = fourPtInterp.interpPoint( 1.5, half, d ); } else if (i == half-2) { predictVal = fourPtInterp.interpPoint( 2.5, half, d ); } else if (i == half-1) { predictVal = fourPtInterp.interpPoint( 3.5, half, d ); } else { fill( vec, d, N, i-1); predictVal = fourPtInterp.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 { printf("interp: bad direction value\n"); break; } } // for } // predict

template<class T> void poly<T>::update ( T & vec, int N, transDirection 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. Reimplemented from liftbase. Definition at line 144 of file poly.h. { 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 { printf("update: bad direction value\n"); } } } // update

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Member Data Documentation

template<class T> polyinterp poly<T>::fourPtInterp [private]

Definition at line 98 of file poly.h.

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

• poly.h

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