#include <haar.h>
Inheritance diagram for haar::
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... | |
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.
Definition at line 79 of file haar.h.
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One step in the forward wavelet transform, with normalization.
Reimplemented from liftbase. |
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One inverse wavelet transform step, with normalization.
Reimplemented from liftbase. |
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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
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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 }
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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 }
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1.2.8.1 written by Dimitri van Heesch,
© 1997-2001