#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 xcoordinate 0 and y2 at xcoordinate 1, calculate y, at xcoordinate 2. More...  
int  new_n_minus1 (int y1, int y2) 
Given a point y1 at xcoordinate 0 and y2 at xcoordinate 1, calculate y at xcoordinate 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.

the constructor does nothing.
Definition at line 64 of file line_int.h. 00064 {} 

the destructor does nothing.
Definition at line 66 of file line_int.h. 00066 {} 

declare, but do not define the copy constructor.


Given a point y1 at xcoordinate 0 and y2 at xcoordinate 1, calculate y at xcoordinate 1.
Definition at line 85 of file line_int.h. Referenced by update().
00086 { 00087 int y = 2 * y1  y2; 00088 return y; 00089 } 

Given y1 at xcoordinate 0 and y2 at xcoordinate 1, calculate y, at xcoordinate 2.
Definition at line 74 of file line_int.h. Referenced by predict().
00075 { 00076 int y = 2 * y2  y1; 00077 return y; 00078 } 

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 (b_{j}) to the second half of the array, leaving the even elements (a_{i}) in the first half
a_{0}, a_{1}, a_{1}, a_{3}, b_{0}, b_{1}, b_{2}, b_{2}, The predict step of the line wavelet "predicts" that the odd element will be on a line between two even elements.
b_{j+1,i} = b_{j,i}  (a_{j,i} + a_{j,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 2^{n} 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 xaxis 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 < half1) { 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 == half1 00147 // Calculate the last "odd" prediction 00148 int n_plus1 = new_n_plus1( vec[i1], 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 

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 midpoint 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 CourHarbo, 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.
even_{i+1,j} = even_{i,j} op (odd_{i+1,k1} + odd_{i+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 BoonLock 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 even_{i,j} (depending on whether the forward or inverse transform is being calculated) is calculated from odd_{i+1,k1} and odd_{i+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[j1] + (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 