Point Representation Using Wavelets

The above graph represents all the points currently used at the start of the registration process. The points create an elliptical shape. To transform the points using wavelets, the x and y values were separated as shown in the following graphs.

Then Cubic B-Spline wavelets are created with j = 0,...,6, where j is the number of intervals. Scaling functions for j = 0 where also created. Then the scaling functions and wavelets were all put into a matrix by evaluating the function along the interval at 512 points for each wavelet or scaling function and storing those values along the rows. Then that matrix was multiplied by the actual function needing to be transformed, in this case the x or y values that made up 512 points, the result was a vector of coefficients that could be used to recreate the data by multiplying the scaling and wavelet matrix by the coefficient vector. The following graphs illustrate the results where the one on the left represents the x points and the one on the right represents the projection that resulted from the data being projected onto the wavelets.

The wavelet projection produces a set of points with a similar overall shape but the y axis is more then doubled in magnitude. This might be occurring because the wavelets need to be scaled.