**4. Multiscale deconvolution (MD): The proposed algorithms**

### **4.1 First algorithm: Tikhonov-Miller regularization with a denoisy and deconvoluted signal as model of solution**

We have seen in § 2.2 eq. 10 that Mancina (Mancina, 2000) proposed to reiterate the algorithm of Barakat (Barakat et al, 1997) and to use a pre-deconvoluted signal as model of the solution with sufficient regularization. The accuracy of the solution is referred to the accuracy of the model, which suggests a reasonable formulation. It is obvious that a significant lack of precision in the a priori model leads to an error restoration more important than the usual one without the model. Moreover, if the pre-deconvoluted signal is a noisy signal (which is the case for SIMS signals) or contains aberrations, the iterative process worsens these aberrations and the result is an oscillatory signal. For this reason, it is important to remove noise components from the signal (the model of solution). The idea is to introduce a denoisy and deconvoluted signal as model of solution in Barakat's approach, which constitutes our first contribution in this field (Boulakroune, 2008). The first proposed deconvolution scheme is constructed by the following steps:


The mathematical formulation, in Fourier space, of this algorithm is as follows:

$$\begin{cases} \mathbf{X}\_{\text{mod}\_0} = \tilde{\mathbf{F}} \mathbf{y}\_a^{(j)} + \tilde{\mathbf{G}} \hat{\mathbf{y}}\_d^{(j)} \\ \hat{\mathbf{X}}\_{n+1} = \frac{\mathbf{H}^\top Y + \alpha \left| D \right|^2 \mathbf{X}\_{\text{mod}\_n}}{\left| H \right|^2 + \alpha \left| D \right|^2} \\ \mathbf{X}\_{\text{mod}\_n} = \hat{\mathbf{X}}\_{n+1} \end{cases} \tag{20}$$

Multi-Scale Deconvolution of Mass Spectrometry Signals 141

the generation of artifacts. With an over estimated value of the regularization parameter, which leads to a more conditioned matrix (*H*+) (see Fig. 2), one can reduces the number and the amplitude of these oscillations. The solution can be stable and smooth, but this operation

By applying the positivity constraint, one reinforces the positivity of the final deconvolution profile. The solution stays in accordance with physical reality, as is illustrated in Fig. 7(b). However, positiviting the signal is an arbitrary operation; it is only to direct the solution so that it becomes positive, without making sure that it is exact. Furthermore, the measured dose (the number of ions counted) must be identical for all signals (original, measured, deconvoluted) except for the noise. This dose must be preserved in the resolution of convolution equation and must take into account the generated negative components. With the application of the positivity constraint, the dose of the deconvoluted and constrained signal is higher than the initial dose. A variation of a few percent cannot be tolerated in the quantification of SIMS profiles. It is important to note that in the case of SIMS analysis, physical coherence is of paramount importance. The deconvoluted profile must be physically acceptable. Thus, it is important to adopt a method whose result is acceptable; otherwise the result obtained may be mathematically correct but have no connection with

Fig. 7. Results of TMMS algorithm for sample MD4 containing four delta layers of boron in

To completely remove artifacts from the deconvoluted profiles, Gautier et al (Gautier at al, 1997) proposed the application of local confidence level deduced empirically from the reconstruction error in the deconvoluted profiles. The goal of this confidence level is to separate the parts of the signal belonging to the original profile from those generated artificially by the process of deconvolution. According to these authors (Gautier at al, 1997), when the signal falls to the noise level, at which point one cannot be confident in the deconvolution result, one must fix a limiting value of the deconvoluted signal below which one should not take into account the deconvolution result that likely belongs to the original

silicon (8.5 keV/O2+, 31.8°); αc = 5.6552×10-6, n= 150 iterations.

(a) Without application of positivity constraint. (b) With application of positivity constraint.

is arbitrary and not based on any physical or mathematical support.

physical reality.

It can be noted that denoising reduces the noise power in data; the regularization parameter should be evaluated by cross-validation in regards of the denoisy signal.

Since the noise is controlled by multiscale transforms, the regularization parameter does not have the same importance as in standard deconvolution methods. Clearly it will be lower than obtained without denoising.

In order to validate the robustness of the proposed algorithm, the results must be compared with those of the previous Tikhonov-Miller regularization algorithms. In particular, we have chosen to compare our results with those obtained by Mancina algorithm (Mancina, 2000).

The results of deconvolution by Mancina's approach (Mancina, 2000) are given in Figs. 7(a) and 7(b). It is obvious by using this algorithm, that the deconvolution has improved the slope and the regularity of the delta layers which are completely separated. Indeed, their shape is symmetrical for all peaks, indicating that the exponential features caused by the SIMS analysis are removed. The full width at half maximum (FWHM) of the deconvoluted delta-layers is equal to 19.5 nm. This can be considered a very good result if one takes into account that the FWHM of the measured profile is approximately 59.7 nm. This corresponds to an improvement in depth resolution by a factor of 3.06. The dynamic range is enhanced by a factor of 2.03 for all peaks.

At both sides of the deconvoluted peaks, oscillations with negative components [Fig. 7(a)] appear under the level of noise where the reliability of the deconvolution process cannot be guaranteed. These artifacts, which have been produced by the deconvolution algorithm, must not be taken for a real concentration distribution. The negative values of these artifacts are not physically accepted for concentration measurements in SIMS analysis. Although a compromise was made between the iteration number and the quality of the deconvoluted peaks, if one increases the iteration number with a relatively weak regularization parameter (obtained by cross validation, it equals 5.6552×10-6), the number and the level of these oscillations increase more which reinforces the limits of this algorithm. Indeed, these oscillations are directly related to the quantity of noise. Part of this information, in particular in high frequencies, is masked by the noise, and this lack of information is compensated by

2. Denoising of this signal by thresholding. One conserves only high-frequency

3. Reconstruction of the denoisy signal from the approximations and thersholded details

4. The obtained denoisy signal constitutes the model of solution in iterative Tikhonov-

( ) ( )

*j j a d*

*H D*

ˆ

mod

*n*

(20)

\* 2

*HY D X*

1 2 2

1. Dyadic wavelet decomposition of the noisy signal at the resolution 2-j.

components of details which are above the estimated threshold.

The mathematical formulation, in Fourier space, of this algorithm is as follows:

0

 

mod 1

*n*

*X X*

ˆ

*X Fy Gy*

*n*

It can be noted that denoising reduces the noise power in data; the regularization parameter

Since the noise is controlled by multiscale transforms, the regularization parameter does not have the same importance as in standard deconvolution methods. Clearly it will be lower

In order to validate the robustness of the proposed algorithm, the results must be compared with those of the previous Tikhonov-Miller regularization algorithms. In particular, we have chosen to compare our results with those obtained by Mancina algorithm (Mancina, 2000). The results of deconvolution by Mancina's approach (Mancina, 2000) are given in Figs. 7(a) and 7(b). It is obvious by using this algorithm, that the deconvolution has improved the slope and the regularity of the delta layers which are completely separated. Indeed, their shape is symmetrical for all peaks, indicating that the exponential features caused by the SIMS analysis are removed. The full width at half maximum (FWHM) of the deconvoluted delta-layers is equal to 19.5 nm. This can be considered a very good result if one takes into account that the FWHM of the measured profile is approximately 59.7 nm. This corresponds to an improvement in depth resolution by a factor of 3.06. The dynamic range is enhanced

At both sides of the deconvoluted peaks, oscillations with negative components [Fig. 7(a)] appear under the level of noise where the reliability of the deconvolution process cannot be guaranteed. These artifacts, which have been produced by the deconvolution algorithm, must not be taken for a real concentration distribution. The negative values of these artifacts are not physically accepted for concentration measurements in SIMS analysis. Although a compromise was made between the iteration number and the quality of the deconvoluted peaks, if one increases the iteration number with a relatively weak regularization parameter (obtained by cross validation, it equals 5.6552×10-6), the number and the level of these oscillations increase more which reinforces the limits of this algorithm. Indeed, these oscillations are directly related to the quantity of noise. Part of this information, in particular in high frequencies, is masked by the noise, and this lack of information is compensated by

mod

ˆ

 

*X*

*n*

should be evaluated by cross-validation in regards of the denoisy signal.

Miller regularization at the first iteration.

using eq.19.

than obtained without denoising.

by a factor of 2.03 for all peaks.

the generation of artifacts. With an over estimated value of the regularization parameter, which leads to a more conditioned matrix (*H*+) (see Fig. 2), one can reduces the number and the amplitude of these oscillations. The solution can be stable and smooth, but this operation is arbitrary and not based on any physical or mathematical support.

By applying the positivity constraint, one reinforces the positivity of the final deconvolution profile. The solution stays in accordance with physical reality, as is illustrated in Fig. 7(b). However, positiviting the signal is an arbitrary operation; it is only to direct the solution so that it becomes positive, without making sure that it is exact. Furthermore, the measured dose (the number of ions counted) must be identical for all signals (original, measured, deconvoluted) except for the noise. This dose must be preserved in the resolution of convolution equation and must take into account the generated negative components. With the application of the positivity constraint, the dose of the deconvoluted and constrained signal is higher than the initial dose. A variation of a few percent cannot be tolerated in the quantification of SIMS profiles. It is important to note that in the case of SIMS analysis, physical coherence is of paramount importance. The deconvoluted profile must be physically acceptable. Thus, it is important to adopt a method whose result is acceptable; otherwise the result obtained may be mathematically correct but have no connection with physical reality.

Fig. 7. Results of TMMS algorithm for sample MD4 containing four delta layers of boron in silicon (8.5 keV/O2+, 31.8°); αc = 5.6552×10-6, n= 150 iterations. (a) Without application of positivity constraint. (b) With application of positivity constraint.

To completely remove artifacts from the deconvoluted profiles, Gautier et al (Gautier at al, 1997) proposed the application of local confidence level deduced empirically from the reconstruction error in the deconvoluted profiles. The goal of this confidence level is to separate the parts of the signal belonging to the original profile from those generated artificially by the process of deconvolution. According to these authors (Gautier at al, 1997), when the signal falls to the noise level, at which point one cannot be confident in the deconvolution result, one must fix a limiting value of the deconvoluted signal below which one should not take into account the deconvolution result that likely belongs to the original

Multi-Scale Deconvolution of Mass Spectrometry Signals 143

application of positivity constraint any more, which validates this approach. The question for the SIMS user is to know whether these peaks are to be considered as physical features or as deconvolution artifacts. The origin of these *positive* oscillations lies in the strong local concentrations of the high frequencies of noise, and which cannot be correctly restored. It should be noted that these small peaks can be eliminated by the support constraint, but we consider that the application of any kind of constraints is a purely arbitrary operation.

In the classical approaches of the regularization (including our first 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. To overcome these limits, it is important to adopt a powerful deconvolution that leads to a smoothed and stable solution. In this context, multiresolution deconvolution, which is never used to recover SIMS profiles,

Because of the very abrupt concentration gradients in circuits produced by the microelectronics industry the original SIMS depth profiles are likely to contain some very high frequencies (Gautier et al, 1998). SIMS signals can extend over several decades in a very short range of depth. The intention of any SIMS analyst, as well as of any deconvolution user, is to recover completely all the frequencies lost by the measurement process. Unfortunately, considering again the fact that the resolution function is a low-pass filter, the recovery of high frequencies is always limited, and the recovery of the highest frequencies is definitely impossible, particularly when the profile to be recovered is noisy, which is always the case. It is possible to produce some very high frequencies during the deconvolution process, but there are many chances that these high frequencies are only produced by the high-frequency noise or are created during the inversion of eq. (3) from the very small components of H(υ). High frequencies in the results of a deconvolution must be regarded suspiciously, except if we are just trying to recover very sharp spikes with no interesting low frequencies. This is definitely not the case for SIMS signals, which contain an appreciable amount of low frequencies, too. Therefore, the purpose of this work is to solve this problem by separating high frequencies and low frequencies in the signal, and then further recovering correctly the high frequencies which are not attributable to noise and which contain useful information. Using multiresolution deconvolution, the final result of the deconvolution should be reasonably smooth. This arises from the observation that, even though the SIMS profiles are likely to contain very high frequencies, which can be

In classical regularization approaches, in order to limit the noise content, one must give a higher bound to the quantity of high frequencies that are likely to be present in the result of the deconvolution [eq. (5)], which might be invalid. However, by this process one limits the quantity of high frequencies, not the quantity of noise. The best solution is to recover correctly the frequencies in different bands of the signal and to find an objective criterion to separate the high frequencies which contain noise from those containing the useful information. Moreover, in these traditional regularized methods (monoresolution

may be the most appropriate technique.

thresholded by wavelet shrinkage.

**4.2 Second algorithm: Multiresolution deconvolution** 

signal. However, a confidence level that authorizes to take into account certain parts of the signal and eliminates the lower parts in which the signal should not be taken into account any more, does not bring any information about the quality of information. One of the advantages of SIMS analysis is the great dynamic range of the signal, and allowing the deconvoluted signal to be restricted to a dynamic range which does not exceed two decades and thus does not reflect the original signal. The parts filtered by the confidence level can provide precious information about the sample. In ref. (Mancina, 2000), Mancina showed that the artifacts are not always aberrations of the deconvolution; they can be structures with low concentrations. The interpretation of the artifacts must be measured, especially if their amount is not negligible, in which case, one cannot eliminate them from the deconvoluted profiles. Therefore, it is important to find another tool which leads to a solution lacking of any non physical features and without any arbitrary operations.

Fig. 8. Result of deconvolution by the first proposed algorithm of SIMS profile containing four delta-layers of boron in silicon (8,5 keV/O2+, 38,1°), α = 5,6552.10-6, n = 250 iterations. The level of estimated noise, by using (17) et (18), is of SNR = 40,92 dB. The threshold λ = 55.7831 counts/s. The used wavelet is Sym4.

By using the first proposed algorithm, the results are quite satisfactory suggesting that this approach is indeed self-consistent, see Fig. 8. A significant improvement in the contrast is observed; the delta layers are more separated. The shape of the results is symmetrical for all layers, indicating that the exponential features (in particular the distorted tail shape observed in the boron profile is due to a significantly larger ion mixing effect) caused by SIMS analysis are removed. The same gains that those obtained by Mancina approach of the depth resolution and maximum of picks (dynamic range) are obtained. It can be noted that the width of measured peaks indicates that the δ-layers are not truths deltas - doping, they are close to gaussian more than delta-layers. At the right side of the main deconvoluted peaks some other small peaks appear without any negative component and without

signal. However, a confidence level that authorizes to take into account certain parts of the signal and eliminates the lower parts in which the signal should not be taken into account any more, does not bring any information about the quality of information. One of the advantages of SIMS analysis is the great dynamic range of the signal, and allowing the deconvoluted signal to be restricted to a dynamic range which does not exceed two decades and thus does not reflect the original signal. The parts filtered by the confidence level can provide precious information about the sample. In ref. (Mancina, 2000), Mancina showed that the artifacts are not always aberrations of the deconvolution; they can be structures with low concentrations. The interpretation of the artifacts must be measured, especially if their amount is not negligible, in which case, one cannot eliminate them from the deconvoluted profiles. Therefore, it is important to find another tool which leads to a

solution lacking of any non physical features and without any arbitrary operations.

Fig. 8. Result of deconvolution by the first proposed algorithm of SIMS profile containing

By using the first proposed algorithm, the results are quite satisfactory suggesting that this approach is indeed self-consistent, see Fig. 8. A significant improvement in the contrast is observed; the delta layers are more separated. The shape of the results is symmetrical for all layers, indicating that the exponential features (in particular the distorted tail shape observed in the boron profile is due to a significantly larger ion mixing effect) caused by SIMS analysis are removed. The same gains that those obtained by Mancina approach of the depth resolution and maximum of picks (dynamic range) are obtained. It can be noted that the width of measured peaks indicates that the δ-layers are not truths deltas - doping, they are close to gaussian more than delta-layers. At the right side of the main deconvoluted peaks some other small peaks appear without any negative component and without

The level of estimated noise, by using (17) et (18), is of SNR = 40,92 dB. The threshold

+, 38,1°), α = 5,6552.10-6, n = 250 iterations.

four delta-layers of boron in silicon (8,5 keV/O2

λ = 55.7831 counts/s. The used wavelet is Sym4.

application of positivity constraint any more, which validates this approach. The question for the SIMS user is to know whether these peaks are to be considered as physical features or as deconvolution artifacts. The origin of these *positive* oscillations lies in the strong local concentrations of the high frequencies of noise, and which cannot be correctly restored. It should be noted that these small peaks can be eliminated by the support constraint, but we consider that the application of any kind of constraints is a purely arbitrary operation.

In the classical approaches of the regularization (including our first 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. To overcome these limits, it is important to adopt a powerful deconvolution that leads to a smoothed and stable solution. In this context, multiresolution deconvolution, which is never used to recover SIMS profiles, may be the most appropriate technique.
