**5. Conclusion**

This chapter proposes two robust algorithms for inverse problem to perform deconvolution and particularly restore signals from strongly noised blurred discrete data. These algorithms can be characterized as a regularized wavelet transform. There combine ideas from Tikhonov Miller regularization, wavelet analysis and deconvolution algorithms in order to benefit from the advantages of each. The first algorithm is Tikhonov-Miller deconvolution method, where a priori model of solution, is included. The latter is a denoisy and pre-deconvolved signal obtained firstly by the application of wavelet shrinkage algorithm and after, by the introduction of the obtained denoisy signal in an iterative deconvolution algorithm. The second algorithm is multiresolution deconvolution, based also on Tikhonov-Miller regularization and wavelet transformation. Both local applications of the regularization parameter and shrinking the wavelet coefficients of blurred and estimated solutions at each resolution level in multiresolution deconvolution provide to smoothed results without the risk of generating artifacts related to noise content in the profile. These algorithms were developed and applied to improve the depth resolution of secondary ion mass spectrometry profiles.

The multiscale deconvolution, in particular multiresolution deconvolution (2nd algorithm), shows how the denoising of wavelet coefficients plays an important role in the deconvolution procedure. The purpose of this new approach is to adapt the regularization parameter locally according to the treated frequency band. In particular, the proposed method appears to be very well adapted to the case where the signal-to-noise ratio is poor, because in this case the

The FWHM of the deconvoluted peaks is equal to 18.9 nm, which corresponds to an improvement in the depth resolution by a factor of 3.1587 [Figs. 9(a) and 9(b)]. The dynamic range is improved by a factor of 2.13 for all peaks. The width of the measured peaks indicates that the δ-layers are not real deltas – doping (are not very thin layers); they are

The main advantage of MD is the absence of oscillations which appear in TMMS algorithm results due to the noise effect. Actually, these oscillations appear in most of the classical regularization approaches. The question for the SIMS user is to know whether these small peaks (oscillations) are to be considered as physical features or as deconvolution artifacts. In our opinion, the origin of these oscillations is the presence of strong local concentrations of high frequencies of noise in the signal which cannot be correctly restored by a simple

Figure 9(c) represents the reconstructed depth profile, obtained by convolving the deconvoluted profile with the DRF along with the measured profile. It is in perfect agreement with the measured profile over the entire range of the profile depth. This is a figure of merit of the quality of the deconvolution, and it ensures that the deconvoluted profile is undoubtedly a signal which has produced the measured SIMS profile. A good reconstruction is one of the criteria that confirms the quality of the deconvolution and gives

Finally, by using the proposed MD, the SIMS profiles are recovered very satisfactorily. The artifacts, which appear in almost all monoresolution deconvolution schemes, have been corrected. Therefore, this new algorithm can push the limits of SIMS measurements towards

This chapter proposes two robust algorithms for inverse problem to perform deconvolution and particularly restore signals from strongly noised blurred discrete data. These algorithms can be characterized as a regularized wavelet transform. There combine ideas from Tikhonov Miller regularization, wavelet analysis and deconvolution algorithms in order to benefit from the advantages of each. The first algorithm is Tikhonov-Miller deconvolution method, where a priori model of solution, is included. The latter is a denoisy and pre-deconvolved signal obtained firstly by the application of wavelet shrinkage algorithm and after, by the introduction of the obtained denoisy signal in an iterative deconvolution algorithm. The second algorithm is multiresolution deconvolution, based also on Tikhonov-Miller regularization and wavelet transformation. Both local applications of the regularization parameter and shrinking the wavelet coefficients of blurred and estimated solutions at each resolution level in multiresolution deconvolution provide to smoothed results without the risk of generating artifacts related to noise content in the profile. These algorithms were developed and applied to improve the depth resolution of secondary ion mass spectrometry profiles.

The multiscale deconvolution, in particular multiresolution deconvolution (2nd algorithm), shows how the denoising of wavelet coefficients plays an important role in the deconvolution procedure. The purpose of this new approach is to adapt the regularization parameter locally according to the treated frequency band. In particular, the proposed method appears to be very well adapted to the case where the signal-to-noise ratio is poor, because in this case the

closer to Gaussian than delta-layers.

credibility to the deconvoluted profile.

classical regularization.

the ultimate resolution

**5. Conclusion** 

minimum in the variance of the wavelet coefficients comes out more clearly. Thus, this aspect may be very attractive because it is particularly important to optimize the choice of the regularization parameter, especially at high frequencies. Moreover, the possibility of introducing various *a priori* probabilities at several resolution levels by means of the wavelet analysis has been examined. Indeed, we showed that multiresolution deconvolution can be successfully used for the recovery of data, and hence, for the improvement of depth resolution in SIMS analysis. In particular, deconvolution of delta layers is the most important depth profiling data deconvolution, since it gives not only the shape of the resolution function, but also the optimum data deconvolution conditions for a specific experimental setup.

The comparison between the performance of the proposed algorithms and that of classical monoresolution deconvolution, which is Tikhonov-Miller regularization with model of solution (TMMS), shows that MD results are better than the results of the first proposed algorithm and TMMS algorithm. Because in the classical approaches of the regularization (including our first proposed algorithm), the regularization operator applies in a total way to all bands of the signal. This results in treating low frequencies which contain the useful signal like high frequencies mainly constituted by noise. The result is then an oscillating signal, because the regularization parameter is insufficient to compensate all high frequencies. However, the multiresolution deconvolution (2nd algorithm) helps to suppress the influence of instabilities in the measuring system and noise. Particularly this method works very well and does not deform the deconvolution result. It gives smoothed results without the risk of generating a comprehensive mathematical profile with no connection to the real profile, i.e., free- oscillation deconvoluted profiles are obtained. We can say unambiguously that the MD algorithm is more reliable with regards to the quality of the deconvoluted profiles and the compared gains which show the influence of noise on the TMMS results.

The MD can be used in two-dimension applications and generally in many problems in science and engineering involving the recovery of an object of interest from collected data. SIMS depth profiling is just one example thereof. Nevertheless, the major disadvantage of MD is the longer computing time compared to monoresolution deconvolution methods. However, due to the increase of computer power during recent years, this disadvantage has become progressively less important.
