Wavelet Analysis

One way of presenting reconstructions is in a multiresolution approximation plot. In this plot, reconstructions of the input data are shown by level. The reconstruction at any level uses only the detail and scaling coefficients that are defined below that level.

The following statement produces such a plot, starting at level 3. The results are shown in Figure 8.

call mraApprox(decomp, , 3, , "Quartz Spectrum");

Figure 8: Multiresolution Approximation

You can see that even at level 3, the basic form of the input signal has been captured. As noted earlier, the oscillation present in the absorbance data is captured in the detail coefficients higher than level 7. Thus, the reconstructions at level 7 and lower are largely free of oscillation because they do not use any of the higher detail coefficients. You can confirm this observation by plotting just this level in the multiresolution analysis as follows. The results are shown in Figure 9.

call mraApprox(decomp, , 7, 7, "Quartz Spectrum");

Figure 9: Level 7 of the Multiresolution Approximation

You can also plot the multiresolution approximations that are obtained by using thresholded detail coefficients. For example, the following statement plots the top-level reconstruction that is obtained by using the "SureShrink" threshold. The results are shown in Figure 10.

call mraApprox(decomp, &SureShrink, 10, 10, "Quartz Spectrum");

Figure 10: Top Level of Multiresolution Approximation with SureShrink Thresholding Applied

Notice that the high-frequency oscillation is still present in the reconstruction even with "SureShrink" thresholding applied.