**4.2 Second algorithm: Multiresolution deconvolution**

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 thresholded by wavelet shrinkage.

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

Multi-Scale Deconvolution of Mass Spectrometry Signals 145

( )

( )

Following the Miller approach, the constraints are quadratically combined. We then have

2

2

<sup>2</sup> ( )

*j d j d j d*

*b r*

( ) <sup>2</sup> ( )

<sup>2</sup> ( ) 2 2 () () () () () () ( ) 2

*J a J JJ JJ J a a J aa a a j d j j j jj j d d j dd d d*

*b y Hx Dx b r*

( )

*b y Hx Dx b r*

() () ()

*H H D Dx H y*

() () ()

*H H D Dx H y*

In practice, regularity coefficients (ra(J))2, (rd(j))2 and noise energies <sup>2</sup> ( )*<sup>J</sup>*

The two deconvolutions are the solutions of the normal equations:

<sup>2</sup> ( )

*J a J a J a*

*b r*

<sup>2</sup> <sup>1</sup> () () () () ()

*y HH H y Trace I H*

To solve eq. (24), we must calculate the reverse of the matrices:

*J J J TJ J N a a*

<sup>1</sup> ( ) <sup>2</sup> ( )

*N*

The quality of the solutions ( )*<sup>J</sup>*

  *J*

 

*<sup>a</sup> <sup>x</sup>* and ( )*<sup>j</sup>*

( ) <sup>2</sup> ( )

and

( )

with

estimations are:

 V( ( )*<sup>J</sup> <sup>a</sup>* ) =

and Hd+.

( )

<sup>2</sup> ( ) 2 2 () () ( ) () () () ( ) 2

() () () ()( ) () () () ()

*aa aa a*

*dd dd d*

( )

unknown. Fortunately, these parameters can be estimated using generalized crossvalidation (Thompson et al, 1991; Weyrich et al, 1998). The mathematical formalisms of these

> () () () ()( ) () () () () () () () () () ()

The operators Da(J) and Dd(j) are selected with important eigenvalues when singular values of H(J) and H(j) are rather weak. Indeed, the choice of the regularization operators is conducted

*J J J JT J T a aa a j j T T jj j d dd d*

*HHH D D HH H DD*

, V( ( )*<sup>j</sup> <sup>d</sup>* ) =

*J T T J J JT J J J J*

() () () () () () ( ) ( )

*j j TT T jj jj j j*

j= 1,…, J. (23)

(J))2, (rd(j))2 are regularities of approximation

j= 1,…, J. (25)

j= 1,…, J. (26)

<sup>2</sup> <sup>1</sup> ( ) () () () ( )

*y HH H y Trace I H*

*j j j j Tj N d d*

<sup>1</sup> ( ) <sup>2</sup> ( )

*N*

*<sup>d</sup> x* depends on the conditioning of the matrices Ha

*j*

j= 1,…, J. (28)

j= 1,…, J. (24)

*<sup>a</sup> <sup>b</sup>* , <sup>2</sup> ( )*<sup>j</sup>*

*<sup>d</sup> b* are

(27)

+

() () () 2

*JJ J aa a jj j dd d*

*Dx r*

where da(J) and Dd(j) are high-pass filters, and (ra

and detail solutions at resolutions 2-J and 2-j, respectively.

*Dx r*

() () () 2

regularized deconvolution), the regularization parameter is applied comprehensively to all signal bands, which results in treating low frequencies which contain the useful signal as high frequencies mainly consisting of noise. The result is then an oscillatory signal, because the regularization parameter is insufficient to compensate high frequencies. Therefore, our idea is to locally adapt the regularization parameter in different frequency bands. This allows us to deconvolute signals previously decomposed by projection onto a wavelet basis.

We have seen in § 3 that the multiscale representation of the signal, or wavelet decomposition allows its associating with an approximation signal at low frequencies (scale coefficients) and a detail signal at high frequencies (wavelet coefficients). Indeed, the approximation signal is very regular (smooth) whereas the detail signal is irregular (rough). This information may be exploited a priori in the deconvolution algorithm. A regular wavelet base will be privileged if one wants to control this regularity, in particular if successive decompositions are used.

It should be noted that the use of a wavelet base with limited support allows preserving a priori knowledge of the signal support in its multiresolution representation. The effectiveness of the constraint of limited support is preserved if the wavelet support is small with respect to that of the signal. In the case of a positive signal, the approximation signal will be positive only if all low-pass filter coefficients are positive. The detail signal always averages to zero; this information can be used like a new soft constraint.

Considering all these advantages, the regularized multiresolution deconvolution can then be performed so that limits of classical monoresolution deconvolution methods are overcome, such as, generating oscillations with negatives components, which limit the depth resolution.

In sharp contrast with the usual multiresolution scheme, it has been established in refs. (Burdeau et al, 2000; Weyrich et al, 1998) that the decimation process is without interest in deconvolution and, in addition, that it incorporates errors in data, if this is the case, then the output of the filters are not decimated.

After wavelet decomposition, the observed noisy data of approximation and details are written under the following mathematical formalism:

$$\begin{cases} y\_a^{(l)} = H^{(l)} \mathbf{x}\_a^{(l)} + b\_a^{(l)} \\ y\_d^{(j)} = H^{(j)} \mathbf{x}\_d^{(j)} + b\_d^{(j)} \end{cases} \text{ j} = \mathbf{1}, \dots, \text{ J.} \tag{21}$$

where ba (J) and bd(j) represent the approximation and details of the noise at the resolutions 2-J and 2-j, respectively.

We use the Tikhonov regularization method to solve the two parts of eq. (21) separately. The following soft constraints about the solutions ( )*<sup>J</sup> <sup>a</sup> <sup>x</sup>* and ( )*<sup>j</sup> <sup>d</sup> x* are used:

$$\begin{aligned} \left\| y\_a^{(l)} - H^{(l)} \tilde{\boldsymbol{x}}\_a^{(l)} \right\|^2 &\le \left\| b\_a^{(l)} \right\|^2\\ \left\| y\_d^{(j)} - H^{(j)} \tilde{\boldsymbol{x}}\_d^{(j)} \right\|^2 &\le \left\| b\_d^{(j)} \right\|^2 \end{aligned} \quad \text{j} = 1, \dots, \text{J}\_\prime \tag{22}$$

$$\begin{aligned} \left\| D\_{a}^{(l)} \widetilde{\mathbf{x}}\_{a}^{(l)} \right\| \leq \left( r\_{a}^{(l)} \right)^{2} \\ \left\| D\_{d}^{(j)} \widetilde{\mathbf{x}}\_{d}^{(j)} \right\| \leq \left( r\_{d}^{(j)} \right)^{2} \end{aligned} \text{ j} = 1, \ldots, \text{J.} \tag{23}$$

where da (J) and Dd(j) are high-pass filters, and (ra (J))2, (rd(j))2 are regularities of approximation and detail solutions at resolutions 2-J and 2-j, respectively.

Following the Miller approach, the constraints are quadratically combined. We then have

$$\left\| y\_a^{(I)} - H^{(I)} \tilde{\mathbf{x}}\_a^{(I)} \right\|^2 + \frac{\left\| b\_a^{(I)} \right\|^2}{(r\_d^{(I)})^2} \left\| D\_a^{(I)} \tilde{\mathbf{x}}\_a^{(I)} \right\| \le 2 \left\| b\_a^{(I)} \right\|^2 \tag{24}$$
 
$$\left\| y\_d^{(j)} - H^{(j)} \tilde{\mathbf{x}}\_d^{(j)} \right\|^2 + \frac{\left\| b\_d^{(j)} \right\|^2}{(r\_d^{(j)})^2} \left\| D\_d^{(j)} \tilde{\mathbf{x}}\_d^{(j)} \right\| \le 2 \left\| b\_d^{(j)} \right\|^2$$

The two deconvolutions are the solutions of the normal equations:

$$\begin{bmatrix} \left( H^{(l)} \right)^{T} H^{(l)} + \alpha\_{a}^{(l)} (D\_{a}^{(l)})^{(T)} D\_{a}^{(l)} \end{bmatrix} \check{\mathbf{x}}\_{a}^{(l)} = (H^{(l)})^{T} y\_{a}^{(l)} \qquad \text{(} j=1,...,\text{ J.} \tag{25}$$
  $\left[ \left( H^{(j)} \right)^{T} H^{(j)} + \alpha\_{d}^{(j)} (D\_{d}^{(j)})^{T} D\_{d}^{(j)} \right] \check{\mathbf{x}}\_{d}^{(j)} = (H^{(j)})^{T} y\_{d}^{(j)} \qquad \text{(} j=1,...,\text{ J.} \tag{25}$ 

with

144 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

regularized deconvolution), the regularization parameter is applied comprehensively to all signal bands, which results in treating low frequencies which contain the useful signal as high frequencies mainly consisting of noise. The result is then an oscillatory signal, because the regularization parameter is insufficient to compensate high frequencies. Therefore, our idea is to locally adapt the regularization parameter in different frequency bands. This allows us to deconvolute signals previously decomposed by projection onto a wavelet basis. We have seen in § 3 that the multiscale representation of the signal, or wavelet decomposition allows its associating with an approximation signal at low frequencies (scale coefficients) and a detail signal at high frequencies (wavelet coefficients). Indeed, the approximation signal is very regular (smooth) whereas the detail signal is irregular (rough). This information may be exploited a priori in the deconvolution algorithm. A regular wavelet base will be privileged if one wants to control this regularity, in particular if

It should be noted that the use of a wavelet base with limited support allows preserving a priori knowledge of the signal support in its multiresolution representation. The effectiveness of the constraint of limited support is preserved if the wavelet support is small with respect to that of the signal. In the case of a positive signal, the approximation signal will be positive only if all low-pass filter coefficients are positive. The detail signal always

Considering all these advantages, the regularized multiresolution deconvolution can then be performed so that limits of classical monoresolution deconvolution methods are overcome, such as, generating oscillations with negatives components, which limit the depth

In sharp contrast with the usual multiresolution scheme, it has been established in refs. (Burdeau et al, 2000; Weyrich et al, 1998) that the decimation process is without interest in deconvolution and, in addition, that it incorporates errors in data, if this is the case, then the

After wavelet decomposition, the observed noisy data of approximation and details are

where ba(J) and bd(j) represent the approximation and details of the noise at the resolutions 2-J

We use the Tikhonov regularization method to solve the two parts of eq. (21) separately. The

*<sup>a</sup> <sup>x</sup>* and ( )*<sup>j</sup>*

*<sup>d</sup> x* are used:

j= 1,…, J. (21)

j= 1,…, J, (22)

() () () () () () () ( ) *J JJ J a aa j jj j d dd*

2 2 () () () ()

*J JJ J a aa j jj j d dd*

*y Hx b*

*y Hx b*

2 2 () () () ( )

*y Hx b y Hx b* 

averages to zero; this information can be used like a new soft constraint.

successive decompositions are used.

output of the filters are not decimated.

written under the following mathematical formalism:

following soft constraints about the solutions ( )*<sup>J</sup>*

resolution.

and 2-j, respectively.

$$\alpha\_a^{(f)} = \frac{\left\| b\_a^{(f)} \right\|^2}{\left( r\_a^{(f)} \right)^2} \text{ and } \alpha\_d^{(f)} = \frac{\left\| b\_d^{(f)} \right\|^2}{\left( r\_d^{(f)} \right)^2} \text{ j} = 1, \dots, \text{J.} \tag{26}$$

In practice, regularity coefficients (ra(J))2, (rd(j))2 and noise energies <sup>2</sup> ( )*<sup>J</sup> <sup>a</sup> <sup>b</sup>* , <sup>2</sup> ( )*<sup>j</sup> <sup>d</sup> b* are unknown. Fortunately, these parameters can be estimated using generalized crossvalidation (Thompson et al, 1991; Weyrich et al, 1998). The mathematical formalisms of these estimations are:

$$\mathbf{V}(\boldsymbol{\alpha}\_{d}^{\{\boldsymbol{l}\}}) = \frac{\frac{1}{N} \left\| \boldsymbol{y}\_{d}^{\{\boldsymbol{l}\}} - \boldsymbol{H}^{\{\boldsymbol{l}\}} \boldsymbol{H}^{+ \{\boldsymbol{l}\}} \boldsymbol{H}^{T(\boldsymbol{l})} \boldsymbol{y}\_{d}^{\{\boldsymbol{l}\}} \right\|^{2}}{\left[ \frac{1}{N} \text{Trace}\{\boldsymbol{I} - \boldsymbol{H}^{+ \{\boldsymbol{l}\}}\} \right]^{2}},\\\mathbf{V}(\boldsymbol{\alpha}\_{d}^{\{\boldsymbol{l}\}}) = \frac{\frac{1}{N} \left\| \boldsymbol{y}\_{d}^{\{\boldsymbol{l}\}} - \boldsymbol{H}^{\{\boldsymbol{l}\}} \boldsymbol{H}^{+ \{\boldsymbol{l}\}} \boldsymbol{H}^{T(\boldsymbol{l})} \boldsymbol{y}\_{d}^{\{\boldsymbol{l}\}} \right\|^{2}}{\left[ \frac{1}{N} \text{Trace}\{\boldsymbol{I} - \boldsymbol{H}^{+ \{\boldsymbol{l}\}}\} \right]^{2}} \tag{27}$$

To solve eq. (24), we must calculate the reverse of the matrices:

$$\begin{aligned} \mathbf{H}\_a^+ &= (\mathbf{H}^{(I)})^T \mathbf{H}^{(I)} + \mathbf{a}\_a^{(I)} (\mathbf{D}\_a^{(I)})^{(T)} \mathbf{D}\_a^{(I)} \\ \mathbf{H}\_d^+ &= (\mathbf{H}^{(j)})^T \mathbf{H}^{(j)} + \mathbf{a}\_d^{(j)} (\mathbf{D}\_d^{(j)})^T \mathbf{D}\_d^{(j)} \end{aligned} \quad \mathbf{j} = \mathbf{1}, \dots, \mathbf{J}. \tag{28}$$

The quality of the solutions ( )*<sup>J</sup> <sup>a</sup> <sup>x</sup>* and ( )*<sup>j</sup> <sup>d</sup> x* depends on the conditioning of the matrices Ha + and Hd+.

The operators Da (J) and Dd(j) are selected with important eigenvalues when singular values of H(J) and H(j) are rather weak. Indeed, the choice of the regularization operators is conducted

Multi-Scale Deconvolution of Mass Spectrometry Signals 147

Fig. 9. Results of multiresolution deconvolution of sample MD4 of boron in silicon matrix

The regularization parameter of approximation or detail takes different values according to the decomposition level. This enables it to be adapted in a local manner with the treated frequency bands, either low or high frequencies. This adaptation leads to compensate high frequencies contrary to a classical regularization parameter, which treats low frequencies

performed at 8.5 keV/O2+, 38.7°. (a) Linear scale plot. (b) Logarithmic scale plot. (c) Reconstruction of the measured profile from the deconvoluted profile and the DRF. The estimated threshold, obtained using soft universal shrinkage [eq. (17)], is λ = 55.7831 cps. The estimated level of noise, using eq. (18), is SNR = 40.9212 dB. The wavelet used

which contain useful signal as high frequencies mainly consisting of noise.

was *Sym4* with four vanishing moments.

based on the singular values of H(J) and H(j) but not by the considered frequency-band, because it is not useful to choose an operator for each frequency band. We construct Da(J) and Dd(j) from the same pulse response d(n); this operator is denoted as D(j) at resolution 2-j.

It is important to note that in a multiresolution scheme up to the resolution 2-J, the different filters responses of decomposition and reconstruction should be interpolated by 2j-1-1 zeros at the resolution 2-j in order to contract the filter bandwidth by a factor 2j-1-1. Each matrix should have a size in accordance with the size of the filtered vector that depends on the resolution level.

The different steps in the multiresolution deconvolution algorithm are as follows (Boulakroune, 2009).


By using multiresolution deconvolution, the results are quite satisfactory, suggesting that this approach is indeed self-consistent [Figs. 9(a) and 9(b)]. A significant improvement in contrast is observed; the delta layers are more separated. The shape of the results is symmetrical for all layers, indicating that the exponential features caused by SIMS analysis are removed.

The different regularization parameters obtained using the generalized cross validation [eq. (27)] at different levels necessary for a well regularized system are αa(1) = 3.34789×10-4, αa(2) = 6.7835×10-4, αa(3) = 0.0013, αa(4) = 0.0026, αa(5) = 0.0048, αd(1) = 1.1012, αd(2) = 2.3287, αd(3) = 4.0211, αd(4) = 9.1654, αd(5) = 16.0773. The classical regularization parameter is equal to 6.6552×10-5.

The approximation regularization parameter increases proportionally with the decomposition level. This behavior is explained by the decrease of the local regularity of the signal with the scale and inter-scale behavior of wavelet coefficients. The latter determines the visual appearance of the added details information (high frequency contents) in the reconstruction. Therefore, as the degree of accuracy is high, the signal regularity is better; hence, the regularization parameter decreases more.

The detail regularization parameter also decreases according to the decomposition level. This evolution is materialized by the degradation of the precision with the scale, which decreases the regularity from one level to another. As the noise is white and Gaussian and the decomposition is dyadic and regular, this parameter doubles in value from one scale to another.

based on the singular values of H(J) and H(j) but not by the considered frequency-band, because it is not useful to choose an operator for each frequency band. We construct Da

and Dd(j) from the same pulse response d(n); this operator is denoted as D(j) at resolution 2-j. It is important to note that in a multiresolution scheme up to the resolution 2-J, the different filters responses of decomposition and reconstruction should be interpolated by 2j-1-1 zeros at the resolution 2-j in order to contract the filter bandwidth by a factor 2j-1-1. Each matrix should have a size in accordance with the size of the filtered vector that depends on the

The different steps in the multiresolution deconvolution algorithm are as follows

4. Denoising of the wavelet-decomposed solution of the deconvolution problem by

5. Dyadic wavelet undecimated reconstruction of the restored signal up to the full

By using multiresolution deconvolution, the results are quite satisfactory, suggesting that this approach is indeed self-consistent [Figs. 9(a) and 9(b)]. A significant improvement in contrast is observed; the delta layers are more separated. The shape of the results is symmetrical for all layers, indicating that the exponential features caused by SIMS analysis

The different regularization parameters obtained using the generalized cross validation [eq. (27)] at different levels necessary for a well regularized system are αa(1) = 3.34789×10-4, αa(2) = 6.7835×10-4, αa(3) = 0.0013, αa(4) = 0.0026, αa(5) = 0.0048, αd(1) = 1.1012, αd(2) = 2.3287, αd(3) = 4.0211, αd(4) = 9.1654, αd(5) = 16.0773. The classical regularization parameter is equal to

The approximation regularization parameter increases proportionally with the decomposition level. This behavior is explained by the decrease of the local regularity of the signal with the scale and inter-scale behavior of wavelet coefficients. The latter determines the visual appearance of the added details information (high frequency contents) in the reconstruction. Therefore, as the degree of accuracy is high, the signal regularity is better;

The detail regularization parameter also decreases according to the decomposition level. This evolution is materialized by the degradation of the precision with the scale, which decreases the regularity from one level to another. As the noise is white and Gaussian and the decomposition is dyadic and regular, this parameter doubles in value from one scale to

1. Dyadic wavelet decomposition of the noisy signal up to the resolution 2-j (j = 1, 2, …). 2. Denoising of this signal by thresholding. One conserves only high-frequency components of details which are above the estimated threshold. One uses generalized cross-validation for threshold parameter evaluation without prior knowledge of the noise variance.(Weyrich, 1998) It can be noted that the wavelet should be orthogonal, therefore the noise in the approximation and detail remains white and Gaussian if it is,

3. Solving the two Tikhonov-Miller normal [eq. (22)] at each resolution level.

resolution level.

(Boulakroune, 2009).

thresholding.

resolution.

are removed.

6.6552×10-5.

another.

in the blurred signal, white and Gaussian.

hence, the regularization parameter decreases more.

(J)

Fig. 9. Results of multiresolution deconvolution of sample MD4 of boron in silicon matrix performed at 8.5 keV/O2 +, 38.7°. (a) Linear scale plot. (b) Logarithmic scale plot. (c) Reconstruction of the measured profile from the deconvoluted profile and the DRF. The estimated threshold, obtained using soft universal shrinkage [eq. (17)], is λ = 55.7831 cps. The estimated level of noise, using eq. (18), is SNR = 40.9212 dB. The wavelet used was *Sym4* with four vanishing moments.

The regularization parameter of approximation or detail takes different values according to the decomposition level. This enables it to be adapted in a local manner with the treated frequency bands, either low or high frequencies. This adaptation leads to compensate high frequencies contrary to a classical regularization parameter, which treats low frequencies which contain useful signal as high frequencies mainly consisting of noise.

Multi-Scale Deconvolution of Mass Spectrometry Signals 149

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

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

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

Allen, P. N., Dowsett, M. G. & Collins, R. (1993). SIMS profile quantification by maximum

Averbuch, A. & Zheludev, V. (2009). Spline-based deconvolution, *Elsevier, Signal Processing,*

Barakat, V., Guilpart, B., Goutte, R. & Prost, R. (1997). Model-based Tikhonov-Miller image

Boulakroune, M., Benatia, D. & Kezai, T. (2009). Improvement of depth resolution in

entropy deconvolution, *Surface and Interface Analysis*, Vol.20, (1993), pp. 696-702,

restoration, *IEEExplore, Proceedings International conference on Image processing (ICIP '97)*, pp. 310-31, ISBN: 0-8186-8183-7, Washington, DC, USA, October 26- 29, 1997 Berger, T., Stromberg, J. O. & Eltoft, T. (1999). Adaptive regularized constrained least

squares image restoration. *IEEE Transactions on Image Processing*, Vol.8, No9, (1999),

secondary ion mass spectrometry analysis using the multiresolution deconvolution.

also the optimum data deconvolution conditions for a specific experimental setup.

compared gains which show the influence of noise on the TMMS results.

Vol.89, (2009), pp. 1782–1797, ISSN 0165-1684

pp. 1191-1203, ISSN 1057-7149

become progressively less important.

ISSN 1096-9918

**6. References** 

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 closer to Gaussian than delta-layers.

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 classical regularization.

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 credibility to the deconvoluted profile.

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 the ultimate resolution
