lift::line Class Reference

Inheritance diagram for lift::line:

Inheritance graph
Collaboration diagram for lift::line:

Collaboration graph
List of all members.

Protected Member Functions

void predict (double[] vec, int N, int direction)
void update (double[] vec, int N, int direction)

Detailed Description

Line (with slope) wavelet

The wavelet Lifting Scheme "line" wavelet approximates the data set using a line with with slope (in contrast to the Haar wavelet where a line has zero slope is used to approximate the data).

The predict stage of the line wavelet "predicts" that an odd point will lie midway between its two neighboring even points. That is, that the odd point will lie on a line between the two adjacent even points. The difference between this "prediction" and the actual odd value replaces the odd element.

The update stage calculates the average of the odd and even element pairs, although the method is indirect, since the predict phase has over written the odd value.

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,, 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:

Ian Kaplan

Member Function Documentation

void lift::line::predict double[]  vec,
int  N,
int  direction
[inline, protected, virtual]

Predict phase of line Lifting Scheme wavelet

The predict step attempts to "predict" the value of an odd element from the even elements. The difference between the prediction and the actual element is stored as a wavelet coefficient.

The "predict" step takes place after the split step. The split step will move the odd elements (bj) to the second half of the array, leaving the even elements (ai) in the first half

    a0, a1, a1, a3, b0, b1, b2, b2, 

The predict step of the line wavelet "predicts" that the odd element will be on a line between two even elements.

    bj+1,i = bj,i - (aj,i + aj,i+1)/2

Note that when we get to the end of the data series the odd element is the last element in the data series (remember, wavelet algorithms work on data series with 2n elements). Here we "predict" that the odd element will be on a line that runs through the last two even elements. This can be calculated by assuming that the last two even elements are located at x-axis coordinates 0 and 1, respectively. The odd element will be at 2. The new_y() function is called to do this simple calculation.

Implements lift::liftbase.

00175   {
00176     int half = N >> 1;
00177     double predictVal;
00179     for (int i = 0; i < half; i++) {
00180       int j = i + half;
00181       if (i < half-1) {
00182         predictVal = (vec[i] + vec[i+1])/2;
00183       }
00184       else if (N == 2) {
00185         predictVal = vec[0];
00186       }
00187       else {
00188         // calculate the last "odd" prediction
00189         predictVal = new_y( vec[i-1], vec[i] );
00190       }
00192       if (direction == forward) {
00193         vec[j] = vec[j] - predictVal;
00194       }
00195       else if (direction == inverse) {
00196         vec[j] = vec[j] + predictVal;
00197       }
00198       else {
00199         System.out.println("predictline::predict: bad direction value");
00200       }
00201     }
00202   } // predict

void lift::line::update double[]  vec,
int  N,
int  direction
[inline, protected, virtual]

The predict phase works on the odd elements in the second half of the array. The update phase works on the even elements in the first half of the array. The update phase attempts to preserve the average. After the update phase is completed the average of the even elements should be approximately the same as the average of the input data set from the previous iteration. The result of the update phase becomes the input for the next iteration.

In a Haar wavelet the average that replaces the even element is calculated as the average of the even element and its associated odd element (e.g., its odd neighbor before the split). This is not possible in the line wavelet since the odd element has been replaced by the difference between the odd element and the mid-point of its two even neighbors. As a result, the odd element cannot be recovered.

The value that is added to the even element to preserve the average is calculated by the equation shown below. This equation is given in Wim Sweldens' journal articles and his tutorial (Building Your Own Wavelets at Home) and in Ripples in Mathematics. A somewhat more complete derivation of this equation is provided in Ripples in Mathematics by A. Jensen and A. la Cour-Harbo, Springer, 2001.

The equation used to calculate the average is shown below for a given iteratin i. Note that the predict phase has already completed, so the odd values belong to iteration i+1.

  eveni+1,j = eveni,j op (oddi+1,k-1 + oddi+1,k)/4

There is an edge problem here, when i = 0 and k = N/2 (e.g., there is no k-1 element). We assume that the oddi+1,k-1 is the same as oddk. So for the first element this becomes

      (2 * oddk)/4



Implements lift::liftbase.

00259   {
00260     int half = N >> 1;
00262     for (int i = 0; i < half; i++) {
00263       int j = i + half;
00264       double val;
00266       if (i == 0) {
00267         val = vec[j]/2.0;
00268       }
00269       else {
00270         val = (vec[j-1] + vec[j])/4.0;
00271       }
00272       if (direction == forward) {
00273         vec[i] = vec[i] + val;
00274       }
00275       else if (direction == inverse) {
00276         vec[i] = vec[i] - val;
00277       }
00278       else {
00279         System.out.println("update: bad direction value");
00280       }
00281     } // for    
00282   }

The documentation for this class was generated from the following file:
Generated on Sun Dec 11 20:01:10 2005 for LiftingScheme by  doxygen 1.4.5