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 Bspline functions of support 0.04) which are centered
at the positions
, with
.
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 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 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 norm like approach (for ) 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 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.

