Highly Nonlinear Approximations for Sparse Signal Representation


Numerical Simulation

We test the proposed approaches, first on the simulation of Example [*] and then extend that simulation to consider a more realistic level of uncertainty in the data. Let us remark that the signal is meant to represent an emission spectrum consisting of the superposition of spectral lines (modeled by B-spline functions of support 0.04) which are centered at the positions $ (n-1)\Delta,  n=0,\ldots,102$, with $ \Delta = 0.01$. Since the errors in the data in Example [*] are not significant, both OBMP and the procedure outlined in the previous section accurately recovers the spectrum from the background, with any positive value of the $ q$-parameter less than or equal to one. The result (coinciding with the theoretical one) is shown in the right hand top graph of Fig. 5.

Now we transform the example into a more realistic situation by adding larger errors to the data. In this case, the data set is perturbed by Gaussian errors of variance up to $ 1\%$ of each data point. Such a piece of data is plotted in the left middle graph of Fig. 3 and the spectrum extracted by the $ q-$norm like approach (for $ q=0.8$) is represented by the broken line in the right middle graph of Fig. 5. The corresponding OBMP approach is plotted in the first graph of Fig. 6 and is slightly superior.

Finally we increase the data's error up to $ 3\%$ of each data point (left bottom graph of Fig. 5) and, in spite of the perceived significant distortion of the signal, we could still recover a spectrum which, as shown by the broken line in the right bottom graph of Fig.5 is a fairly good approximation of the true one (continuous line). The OBMP approach is again superior, as can be observed in the second graph of Fig. 6.

Figure 5: Top left graph: signal plus background. Top right graph: Recovered signal Middle left graph: signal distorted up to to $ 1\%$. Middle right graph: $ q$-norm like approach approximation (broken line) Bottom graphs: Same description as in the previous graphs but the data distorted up to $ 3\%$.
\includegraphics[width=7cm]{exasigo.eps} \includegraphics[width=7cm]{exp.eps}
\includegraphics[width=7cm]{sigo1.eps} \includegraphics[width=7cm]{exp01.eps}
\includegraphics[width=7cm]{sigo3.eps} \includegraphics[width=7cm]{exp03.eps}
Experiments for different realization of the errors (with the same variance) have produced results essentially equivalent. The same is true for other realizations of the signal.
Figure 6: Same description as the right middle and bottom graphs of Fig. 5 but applying the BOMP method discussed in Section [*]
\includegraphics[width=7cm]{exp.eps} \includegraphics[width=7cm]{obmp_exa3_tut.eps}