1. Sophisticated high-resolution deconvolution algorithms for spectroscopic data V. Matoušek, M. Morháč Institute of Physics, Slovak Academy of Sciences, Slovakia ACAT 2013, BeijingMay 16 – 21, 2013

2. One of the most delicate problems of any spectrometric method are those related to the extraction of the correct information out of the experimental spectra. The quality of the analysis of spectrometric data consists in correct identification of the existence of peaks and subsequently in good estimation of their positions and intensities (areas). The peaks as the main carrier of spectrometric information are very frequently positioned close to each other. The extraction of the correct information out of the spectra sections, where due to the limited resolution of the equipment, signals coming from various sources are overlapping, is a very complicated problem. The deconvolution methods represent an efficient tool to improve the resolution in the data. It is of great importance mainly in the tasks connected with decomposition of overlapped peaks (multiplets) and subsequently for the determination of positions and intensities of peaks in gamma-ray spectra. Introduction

3. where x(i) is the input into the system, h(i) is its impulse function (response), y(i) is the output from the system, n(i) is additive noise and “ * “ denotes the operation of the convolution. Stationary discrete system that satisfies the superposition principle can be described by convolution sum In matrix form it can be written where the marix H has dimensions N × M, the vectors y, n have length N and vector x has length M, while N ≥ M (overdetermined system). The problem of finding x, when H, y are known is a discrete ill-posed problem and requires regularization techniques to get adequate solution.

4. Three types of regularization methods are very often used: smoothing, constraints imposition (for example only non-negative data are accepted), choice of a prior information probability function - Bayesian statistical approach. Tikhonov-Miller regularization. The functional is minimized. The solution can be obtained by solving the equation Q, α are the regularization matrix and parameter, respectively, Zero-th order or Tikhonov regularization Riley Algorithm (commonly called iterated Tikhonov regularization). To obtain smoother solution one may use algorithm of Tikhonov-Miller regularization with iterative refinement However, because of its iterative nature, the Riley algorithm lends itself to another type of regularization, so called Projections On Convex Sets – POCS. There exist many regularization methods based on this kind of regularization. It means we set all negative elements to zero after each iteration.

5. Fig.:Solution obtained using Tikhonov method of regularization. Fig.:Illustration of Riley algorithm of deconvolution - thick line is original spectrum, thin line represents spectrum after deconvolution.

6. Fig. 64Riley deconvolution with POCS regularization - thick line is original spectrum, thin line represents spectrum after deconvolution. Van Cittert Algorithm. The basic form of Van Cittert algorithm for discrete convolution system is where: A is system Toeplitz matrix, n represents iteration number and μ is the relaxation factor.

7. Fig. 65Original spectrum (thick line), deconvolved spectrum using Van Cittert algorithm (without regularization) and deconvolved spectrum using Van Cittert algorithm and regularized via POCS method. Gold Algorithm. If we choose the local variable relaxation factor and we substitute it into Van Cittert formula we get the Gold deconvolution algorithm: Its solution is always positive when the input data are positive, which makes the algorithm suitable for the use for naturally positive definite data, i.e., spectroscopic data.

8. Fig.: Original spectrum (thick line) and deconvolved spectrum using Gold algorithm (thin line) after 10000 iterations. Channels are shown as small circles. Fig.: Original spectrum (thick line) and deconvolved spectrum using Gold algorithm (thin line) after 50000 iterations.

9. Boosted deconvolution algorithm Iterative positive definite deconvolutions (Gold, Richardson-Lucy, Muller and MAP) converge to stable states. It is useless to increase the number of iterations, the result obtained practically does not change. Instead of it we can stop iterations, apply a boosting operation and repeat this procedure. Boosting operation should decrease sigma of peaks. Then, the algorithm of boosted Gold or other iterative deconvolutions is as follows: 1. Set the initial solution 2. Set required number of repetitions R and iterations L 3. Set the number of repetitions r=1 4. According to either Gold, or other iterative algorithms for find solution x (L) 5. If r=R stop the calculation, else a. apply boosting operation, i.e., set p is boosting coefficient >0, b. r=r+1 c. continue in 4.

10. Fig.: Original spectrum (thick line) and deconvolved spectrum using boosted Gold algorithm (thin line) after 200 iterations and 50 repetitions (p=1.2). Fig.: Original spectrum (thick line) and deconvolved spectrum using boosted Richardson-Lucy algorithm (thin line) after 200 iterations and 50 repetitions. Fig.: Original spectrum (thick line) and deconvolved spectrum using boosted MAP algorithm (thin line) after 200 iterations and 50 repetitions.

11. Fig. Original gamma-ray spectrum (thick line) and deconvolved spectrum using classic Gold algorithm (thin line) after 10 000 iterations. Fig.: Gamma-ray spectrum (thick line) and deconvolved spectrum using boosted Gold algorithm (thin line) after 200 iterations and 50 repetitions (p=1.2)

12. Fig.: Experimental gamma-gamma-ray spectrum (after background elimination). Fig.: Spectrum after boosted Gold deconvolution (50 iterations repeated 20 times).

13. Blind Deconvolution. Sometimes we don’t know the response function, we only suppose its shape or, respectively, we know theoretically its analytical form, which can substantially differ from the reality. We can observe that the operation of the convolution is commutative, which gives us the possibility of implementation of iterative algorithm of blind deconvolution scheme: 1.Estimate an initial response function h (0) (i) from y(i). This can be accomplished in different ways. For example, we can choose the narrowest line lobe of y(i), and by Gaussian fitting of the selected part we can get an initial estimate of the response function. 2.Accomplish the deconvolution using the estimated response function h (0) (i) and given y(i) where ** denotes the operation of deconvolution. 3.Accomplish the deconvolution using the estimated spectral function for finding an improved estimation of the response function 4.Repeat steps 2, 3 for improving the estimation of x(i), h(i). One can use any of the presented deconvolution algorithms.

14. Fig.: Original spectrum (thick line) and deconvolved spectrum using blind Gold algorithm (thin line) after 1 000 iterations and 50 repetitions. Fig.: Deconvolved response function.

15. Linear systems with changing response In this case, the columns of the system matrix are not simply shifted as it was in the previous examples. In these systems, the shape of response changes with independent variable (time, in the case of nuclear spectra energy, etc.). Problem of identification of the system matrix. Fig.: Principle of interpolation of response function (EGSnrc simulation package). Fig.: Illustration of interpolation between two simulated responses For time dependent linear system analogously as in the case of invariant systems it holds

16. Fig.: Synthetized response matrix for continuum decomposition for experimental data from Gamma- sphere. Fig.: Detail of the response matrix.

17. Fig.: Detail of spectrum before deconvolution (dotted line), after classic (thin line) and boosted Gold deconvolution (bars). Fig.: Measured spectrum of Co56. Fig.: Spectrum of Co56 after Gold deconvolution using above displayed response matrix.

18. We have discussed and analyzed a series of deconvolution methods. There exists immense number of variations of deconvolution algorithms applied in various scientific fields, e.g. astronomy, image processing, geophysics, optics etc. We have introduced only basic classes of deconvolution algorithms that can be easily employed and implemented for the processing of spectrometric data. We have restricted our investigations only to positive definite methods. Though they improve substantially the resolution in the spectra they are not efficient enough to decompose closely positioned peaks. We have presented boosted deconvolution algorithms, wich are able to decompose the overlapped peaks practically to δ functions while concentrating the peak areas to one channel. Though the procedure is fully automatic, due to large variability of the data, some intervention of the user and tuning of some parameters are required. Several algorithms were also implemented in ROOT system in the form of TSpectrum, TSpectrum2 and TSpectrum3 classes, developed in collaboration with CERN. Conclusions

19. Some relevant publications: 1.M. Morháč, V. Matoušek: Multidimensional Experimental Data Processing in Nuclear Physics, In:Computer Physics, Eds.: S. Doherty and A. Molloy, Book Series, Nova Science Publishers, Inc., ISBN: 978-1-61324-790-7, 2012, pp. 1 - 236. 2.M. Morháč, V. Matoušek, High-resolution boosted deconvolution of spectroscopic data, Journal of Computational and Applied Mathematics, 235 (2011) 1629–1640. 3.M. Morháč, V. Matoušek: Complete positive deconvolution of spectrometric data. Digital Signal Processing, Volume 19, Issue 3, May 2009, Pages 372-392. 4.M. Morháč, V. Matoušek: Applied Spectroscopy 62, 91 (2008). 5.M. Morháč: Deconvolution methods and their applications in the analysis of gamma-ray spectra, Nucl. Instrum. Methods Phys. Res., Sect. A 559 (1) (2006), pp. 119-123.