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# haar Class Template Reference

Haar (flat line) wavelet. More...

`#include <haar.h>`

Inheritance diagram for haar::

List of all members.

## Protected Methods

void predict (T &vec, int N, transDirection direction)
Haar predict step. More...

void update (T &vec, int N, transDirection direction)
Update step of the Haar wavelet transform. More...

void normalize (T &vec, int N, transDirection direction)
The normalization step assures that each step of the wavelet transform has the constant "energy" where energy is defined as. More...

void inverseStep (T &vec, const int n)
One inverse wavelet transform step, with normalization. More...

void forwardStep (T &vec, const int n)
One step in the forward wavelet transform, with normalization. More...

## Detailed Description

### template<class T> class haar

Haar (flat line) wavelet.

As with all Lifting scheme wavelet transform functions, the first stage of a transform step is the split stage. The split step moves the even element to the first half of an N element region and the odd elements to the second half of the N element region.

The Lifting Scheme version of the Haar transform uses a wavelet function (predict stage) that "predicts" that an odd element will have the same value as it preceeding even element. Stated another way, the odd element is "predicted" to be on a flat (zero slope line) shared with the even point. The difference between this "prediction" and the actual odd value replaces the odd element.

The wavelet scaling function (a.k.a. smoothing function) used in the update stage calculates the average between an even and an odd element.

The merge stage at the end of the inverse transform interleaves odd and even elements from the two halves of the array (e.g., ordering them even0, odd0, even1, odd1, ...)

This is a template version of the Haar wavelet. The template must be instantiated with an array or an object that acts like an array. Objects that act like arrays define the left hand side and right hand side index operators: [].

See www.bearcave.com for more information on wavelets and the wavelet lifting scheme.

Author:
Ian Kaplan

Definition at line 79 of file haar.h.

## Member Function Documentation

 template void haar::forwardStep ( T & vec, const int n )` [inline, protected, virtual]`
 One step in the forward wavelet transform, with normalization. Reimplemented from liftbase. Definition at line 218 of file haar.h.```00219 { 00220 split( vec, n ); 00221 predict( vec, n, forward ); 00222 update( vec, n, forward ); 00223 normalize( vec, n, forward ); 00224 } // forwardStep ```

 template void haar::inverseStep ( T & vec, const int n )` [inline, protected, virtual]`
 One inverse wavelet transform step, with normalization. Reimplemented from liftbase. Definition at line 207 of file haar.h.```00208 { 00209 normalize( vec, n, inverse ); 00210 update( vec, n, inverse ); 00211 predict( vec, n, inverse ); 00212 merge( vec, n ); 00213 } // inverseStep ```

 template void haar::normalize ( T & vec, int N, transDirection direction )` [inline, protected]`
 The normalization step assures that each step of the wavelet transform has the constant "energy" where energy is defined as. ``` double energy = 0.0; for (int n = 0; n < N; n++) { energy = energy + (a[i] * a[i]); } ``` See 5.2.1 of Ripples in Mathematics by Jensen and la Cour-Harbo The most common implementation of the Haar transform leaves out the normalization step, since it does not make much of a difference in many cases. However, in the case of the wavelet packet transform, many of the cost functions are squares, so normalization produces smaller wavelet values (although the scaling function values are larger). This may lead to a better wavelet packet result (e.g., a few large values and lots of small values). Normalization does have the disadvantage of destroying the averaging property of the Haar wavelet algorithm. That is, the final scaling factor is no longer the mean of the time series. Definition at line 182 of file haar.h. Referenced by forwardStep(), and inverseStep(). ```00183 { 00184 const double sqrt2 = sqrt( 2.0 ); 00185 int half = N >> 1; 00186 00187 for (int i = 0; i < half; i++) { 00188 int j = i + half; 00189 00190 if (direction == forward) { 00191 vec[i] = sqrt2 * vec[i]; 00192 vec[j] = vec[j]/sqrt2; 00193 } 00194 else if (direction == inverse) { 00195 vec[i] = vec[i]/sqrt2; 00196 vec[j] = sqrt2 * vec[j]; 00197 } 00198 else { 00199 printf("normalize: bad direction value\n"); 00200 } 00201 } // for 00202 } // normalize ```

 template void haar::predict ( T & vec, int N, transDirection direction )` [inline, protected, virtual]`
 Haar predict step. Reimplemented from liftbase. Definition at line 85 of file haar.h.```00086 { 00087 int half = N >> 1; 00088 00089 for (int i = 0; i < half; i++) { 00090 double predictVal = vec[i]; 00091 int j = i + half; 00092 00093 if (direction == forward) { 00094 vec[j] = vec[j] - predictVal; 00095 } 00096 else if (direction == inverse) { 00097 vec[j] = vec[j] + predictVal; 00098 } 00099 else { 00100 printf("haar::predict: bad direction value\n"); 00101 } 00102 } 00103 } ```

 template void haar::update ( T & vec, int N, transDirection direction )` [inline, protected, virtual]`
 Update step of the Haar wavelet transform. The wavelet transform calculates a set of detail or difference coefficients in the predict step. These are stored in the upper half of the array. The update step calculates an average from the even-odd element pairs. The averages will replace the even elements in the lower half of the array. The Haar wavelet calculation used in the Lifting Scheme is ``` dj+1, i = oddj+1, i = oddj, i - evenj, i aj+1, i = evenj, i = (evenj, i + oddj, i)/2 ``` Note that the Lifting Scheme uses an in-place algorithm. The odd elements have been replaced by the detail coefficients in the predict step. With a little algebra we can substitute the coefficient calculation into the average calculation, which gives us ``` aj+1, i = evenj, i = evenj, i + (oddj, i/2) ``` Reimplemented from liftbase. Definition at line 134 of file haar.h.```00135 { 00136 int half = N >> 1; 00137 00138 for (int i = 0; i < half; i++) { 00139 int j = i + half; 00140 double updateVal = vec[j] / 2.0; 00141 00142 if (direction == forward) { 00143 vec[i] = vec[i] + updateVal; 00144 } 00145 else if (direction == inverse) { 00146 vec[i] = vec[i] - updateVal; 00147 } 00148 else { 00149 printf("update: bad direction value\n"); 00150 } 00151 } 00152 } ```

The documentation for this class was generated from the following file:
Generated at Sat Aug 10 13:23:36 2002 for Wavelet Packet Transform and Lossless Compression by 1.2.8.1 written by Dimitri van Heesch, © 1997-2001