#include <line_int.h>
Inheritance diagram for line_int::
Public Methods | |
| line_int () | |
| the constructor does nothing. More... | |
| ~line_int () | |
| the destructor does nothing. More... | |
| line_int (const line_int &rhs) | |
| declare, but do not define the copy constructor. More... | |
Protected Methods | |
| void | predict (T &vec, int N, transDirection direction) |
| Predict phase of Lifting Scheme linear interpolation wavelet. More... | |
| void | update (T &vec, int N, transDirection direction) |
| Update step of the linear interpolation wavelet. More... | |
Private Methods | |
| int | new_n_plus1 (int y1, int y2) |
| Given y1 at x-coordinate 0 and y2 at x-coordinate 1, calculate y, at x-coordinate 2. More... | |
| int | new_n_minus1 (int y1, int y2) |
| Given a point y1 at x-coordinate 0 and y2 at x-coordinate 1, calculate y at x-coordinate -1. More... | |
The linear interpolation wavelet uses a predict phase that "predicts" that an odd element in the data set will line on a line between its two even neighbors.
This is an integer version of the linear interpolation wavelet. It is interesting to note that unlike the S transform (the integer version of the Haar wavelet) or the TS transform (an integer version of the CDF(3,1) transform) this algorithm does not preserve the mean. That is, when the transform is calculated, the first element of the result array will not be the mean.
Definition at line 60 of file line_int.h.
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the constructor does nothing.
Definition at line 64 of file line_int.h. 00064 {}
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the destructor does nothing.
Definition at line 66 of file line_int.h. 00066 {}
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declare, but do not define the copy constructor.
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Given a point y1 at x-coordinate 0 and y2 at x-coordinate 1, calculate y at x-coordinate -1.
Definition at line 85 of file line_int.h. Referenced by update().
00086 {
00087 int y = 2 * y1 - y2;
00088 return y;
00089 }
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Given y1 at x-coordinate 0 and y2 at x-coordinate 1, calculate y, at x-coordinate 2.
Definition at line 74 of file line_int.h. Referenced by predict().
00075 {
00076 int y = 2 * y2 - y1;
00077 return y;
00078 }
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Predict phase of Lifting Scheme linear interpolation 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 Note that in the case where (N == 2), the algorithm becomes the same as the Haar wavelet. We "predict" that the odd value vec[1] will be the same as the even value, vec[0]. Reimplemented from liftbase. Definition at line 132 of file line_int.h. 00133 {
00134 int half = N >> 1;
00135 int predictVal;
00136
00137 for (int i = 0; i < half; i++) {
00138 int j = i + half;
00139 if (i < half-1) {
00140 predictVal = (int)((((float)vec[i] + (float)vec[i+1])/2.0) + 0.5);
00141 }
00142 else if (N == 2) {
00143 predictVal = vec[0];
00144 }
00145 else {
00146 // i == half-1
00147 // Calculate the last "odd" prediction
00148 int n_plus1 = new_n_plus1( vec[i-1], vec[i] );
00149 predictVal = (int)((((float)vec[i] + (float)n_plus1)/2.0) + 0.5);
00150 }
00151
00152 if (direction == forward) {
00153 vec[j] = vec[j] - predictVal;
00154 }
00155 else if (direction == inverse) {
00156 vec[j] = vec[j] + predictVal;
00157 }
00158 else {
00159 printf("line::predict: bad direction value\n");
00160 }
00161 }
00162 } // predict
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Update step of the linear interpolation wavelet. 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 neighboring odd element (e.g., its odd neighbor before the split). In the lifting scheme version of the Haar wavelet the odd element has been overwritten by the difference between the odd element and its even neighbor. In calculating the average (to replace the even element) the value of the odd element can be recovered via a simple algebraic manipulation. In the line wavelet the odd element has been replaced by the difference between the odd element and the mid-point of its two even neighbors. Recovering the value of the odd element to calculate the average is not as simple in this case. 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
This version of the line wavelet code implements an integer version of linear interpolating wavelet. This versoin comes from the paper Wavelet Transforms that Map Integers to Integers by A.R. Calderbank, ingrid daubechies, wim weldens and Boon-Lock Yeo, August 1996 This is the central reference that was used to develop this code. Parts 1 and 2 of this paper are for the mathematicially sophisticated (which is to say, they are not light reading). However, for the implementer, part 3 and part 4 of this paper provide excellent coverage of perfectly invertable wavelet transforms that map integers to integers. In fact, part 3 of this paper is worth reading in general for its discussion of the wavelet Lifting Scheme. The value added (or subtracted) from the eveni,j (depending on whether the forward or inverse transform is being calculated) is calculated from oddi+1,k-1 and oddi+1,k from the predict step. This means that there is missing value at the start of the set of odd elements (e.g., i = 0, j == half). This missing value assumed to line on a line with the first two odd elements. Because interpolated values are used, the average is not perfectly maintained. Reimplemented from liftbase. Definition at line 234 of file line_int.h. 00235 {
00236 int half = N >> 1;
00237
00238 for (int i = 0; i < half; i++) {
00239 int j = i + half;
00240 int val;
00241
00242 if (i == 0 && N == 2) {
00243 val = (int)(((float)vec[j]/2.0) + 0.5);
00244 }
00245 else if (i == 0 && N > 2) {
00246 int v_n_minus_1 = new_n_minus1( vec[j], vec[j+1] );
00247 val = (int)((((float)v_n_minus_1 + (float)vec[j])/4.0) + 0.5);
00248 }
00249 else {
00250 val = (int)((((float)vec[j-1] + (float)vec[j])/4.0) + 0.5);
00251 }
00252 if (direction == forward) {
00253 vec[i] = vec[i] + val;
00254 }
00255 else if (direction == inverse) {
00256 vec[i] = vec[i] - val;
00257 }
00258 else {
00259 printf("update: bad direction value\n");
00260 }
00261 } // for
00262 } // update
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1.2.8.1 written by Dimitri van Heesch,
© 1997-2001