class TSpectrum: public TNamed

        ONE-DIMENSIONAL BACKGROUND ESTIMATION FUNCTION - GENERAL FUNCTION

        This function calculates background spectrum from source spectrum.
        The result is placed in the vector pointed by spe1945ctrum pointer.

        Function parameters:
        spectrum-pointer to the vector of source spectrum
        ssize-length of the spectrum vector
        numberIterations-maximal width of clipping window,
        direction- direction of change of clipping window
               - possible values=kBackIncreasingWindow
                                 kBackDecreasingWindow
        filterOrder-order of clipping filter,
                  -possible values=kBackOrder2
                                   kBackOrder4
                                   kBackOrder6
                                   kBackOrder8
        smoothing- logical variable whether the smoothing operation
               in the estimation of background will be included
             - possible values=kFALSE
                               kTRUE
        smoothWindow-width of smoothing window,
                  -possible values=kBackSmoothing3
                                   kBackSmoothing5
                                   kBackSmoothing7
                                   kBackSmoothing9
                                   kBackSmoothing11
                                   kBackSmoothing13
                                   kBackSmoothing15
         compton- logical variable whether the estimation of Compton edge
                  will be included
             - possible values=kFALSE
                               kTRUE

Background estimation

 

Goal: Separation of useful information (peaks) from useless information (background)

•         method is based on Sensitive Nonlinear Iterative Peak (SNIP) clipping algorithm

•          new value in the channel “i” is calculated

 

 

 

 

 

where p = 1, 2, …, numberIterations. In fact it represents second order difference filter (-1,2,-1).

 

Function:

const char* Background(float *spectrum, int ssize, int numberIterations, int direction, int filterOrder,  bool smoothing,  int smoothingWindow, bool compton) 

 

This function calculates background spectrum from the source spectrum.  The result is placed in the vector pointed by spectrum pointer.  One can also change the direction of the change of the clipping window, the order of the clipping filter, to include smoothing, to set width of smoothing window and to include the estimation of Compton edges. On successful completion it returns 0. On error it returns pointer to the string describing error.

 

Parameters:

        spectrum-pointer to the vector of source spectrum                 

        ssize-length of the spectrum vector                                

        numberIterations-maximal width of clipping window,                                

        direction- direction of change of clipping window                 

               - possible values=kBackIncreasingWindow                     

                                            kBackDecreasingWindow                     

        filterOrder-order of clipping filter,                             

                  -possible values=kBackOrder2                             

                                              kBackOrder4                             

                                              kBackOrder6                             

                                              kBackOrder8                            

        smoothing- logical variable whether the smoothing operation in the estimation of

               background will be included             

             - possible values=kFALSE                       

                                          kTRUE                        

        smoothWindow-width of smoothing window,                           

                  -possible values=kBackSmoothing3                         

                                             kBackSmoothing5                         

                                             kBackSmoothing7                         

                                             kBackSmoothing9                         

                                             kBackSmoothing11                         

                                             kBackSmoothing13                        

                                             kBackSmoothing15                        

        compton- logical variable whether the estimation of Compton edge   will be included                                          

             - possible values=kFALSE                         

                                          kTRUE                          

 

References:

[1]  C. G Ryan et al.: SNIP, a statistics-sensitive background treatment for the quantitative analysis of PIXE spectra in geoscience applications. NIM, B34 (1988), 396-402.

[2]  M. Morháč, J. Kliman, V. Matoušek, M. Veselský, I. Turzo.: Background elimination methods for multidimensional gamma-ray spectra. NIM, A401 (1997) 113-132.

[3] D. D. Burgess, R. J. Tervo: Background estimation for gamma-ray spectroscopy. NIM 214 (1983), 431-434.

Example 1– script Background_incr.c :

Figure 1 Example of the estimation of background for number of iterations=6. Original spectrum is shown in black color, estimated background in red color.

 

Script:

// Example to illustrate the background estimator (class TSpectrum).

// To execute this example, do

// root > .x Background_incr.C

#include <TSpectrum>

void Background_incr() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   TH1F *back = new TH1F("back","",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   back=(TH1F*) f->Get("back1;1");

   TCanvas *Background = gROOT->GetListOfCanvases()->FindObject("Background");

   if (!Background) Background = new TCanvas("Background","Estimation of background with increasing window",10,10,1000,700);

   back->Draw("L");

   TSpectrum *s = new TSpectrum();

   for (i = 0; i < nbins; i++) source[i]=back->GetBinContent(i + 1);

   s->Background(source,nbins,6,kBackIncreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

   }

Example 2 – script Background_decr.c :

In Figure 1. one can notice that at the edges of the peaks the estimated background goes under the peaks. An alternative approach is to decrease the clipping window from a given value numberIterations to the value of one, which is presented in this example.

Figure 2 Example of the estimation of background for numberIterations=6 using decreasing clipping window algorithm. Original spectrum is shown in black color, estimated background in red color.

 

Script:

// Example to illustrate the background estimator (class TSpectrum).

// To execute this example, do

// root > .x Background_decr.C

#include <TSpectrum>  

void Background_decr() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   TH1F *back = new TH1F("back","",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   back=(TH1F*) f->Get("back1;1");

   TCanvas *Background = gROOT->GetListOfCanvases()->FindObject("Background");

   if (!Background) Background = new TCanvas("Background","Estimation of background with decreasing window",10,10,1000,700);

   back->Draw("L");

   TSpectrum *s = new TSpectrum();

   for (i = 0; i < nbins; i++) source[i]=back->GetBinContent(i + 1);

   s->Background(source,nbins,6,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

   }

Example 3 – script Background_width.c :

•         the question is how to choose the width of the clipping window, i.e.,  numberIterations   parameter. The influence of this parameter on the estimated background is illustrated in Figure 3.

Figure 3 Example of the influence of clipping window width on the estimated background for numberIterations=4 (red line), 6 (blue line) 8 (green line) using decreasing clipping window algorithm.

•        in general one should set this parameter so that the value 2*numberIterations+1 was greater than the widths of preserved objects (peaks).

 

Script:

// Example to illustrate the influence of the clipping window width on the estimated background

// To execute this example, do

// root > .x Background_width.C

#include <TSpectrum>

void Background_width() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   TH1F *h = new TH1F("h","",nbins,xmin,xmax);

   TH1F *d1 = new TH1F("d1","",nbins,xmin,xmax);     

   TH1F *d2 = new TH1F("d2","",nbins,xmin,xmax);        

   TH1F *d3 = new TH1F("d3","",nbins,xmin,xmax);        

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("back1;1");     

   TCanvas *background = gROOT->GetListOfCanvases()->FindObject("background");

   if (!background) background = new TCanvas("background","Influence of clipping window width on the estimated background",10,10,1000,700);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   s->Background(source,nbins,4,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);

   for (i = 0; i < nbins; i++) d1->SetBinContent(i + 1,source[i]);  

   d1->SetLineColor(kRed);

   d1->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,6,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d2->SetBinContent(i + 1,source[i]);  

   d2->SetLineColor(kBlue);

   d2->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,8,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d3->SetBinContent(i + 1,source[i]);  

   d3->SetLineColor(kGreen);

   d3->Draw("SAME L");        

}

Example 4 – script Background_width2.c :

•         another example for very complex spectrum is given in Figure 4.

Figure 4 Example of the influence of clipping window width on the estimated background for numberIterations=10 (red line), 20 (blue line), 30 (green line) and 40 (magenta line) using decreasing clipping window algorithm.

 

Script:

// Example to illustrate the influence of the clipping window width on the estimated background

// To execute this example, do

// root > .x Background_width2.C

#include <TSpectrum> 

void Background_width2() {

   Int_t i;

   Double_t nbins = 4096;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)4096;

   Float_t * source = new float[nbins];

   TH1F *h = new TH1F("h","",nbins,xmin,xmax);

   TH1F *d1 = new TH1F("d1","",nbins,xmin,xmax);     

   TH1F *d2 = new TH1F("d2","",nbins,xmin,xmax);        

   TH1F *d3 = new TH1F("d3","",nbins,xmin,xmax);        

   TH1F *d4 = new TH1F("d4","",nbins,xmin,xmax);           

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("back2;1");  

   TCanvas *background = gROOT->GetListOfCanvases()->FindObject("background");

   if (!background) background = new TCanvas("background","Influence of clipping window width on the estimated background",10,10,1000,700);

   h->SetAxisRange(0,1000);

   h->SetMaximum(20000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   s->Background(source,nbins,10,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);       

   for (i = 0; i < nbins; i++) d1->SetBinContent(i + 1,source[i]);  

   d1->SetLineColor(kRed);

   d1->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,20,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d2->SetBinContent(i + 1,source[i]);  

   d2->SetLineColor(kBlue);

   d2->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,30,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d3->SetBinContent(i + 1,source[i]);  

   d3->SetLineColor(kGreen);

   d3->Draw("SAME L");        

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,10,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d4->SetBinContent(i + 1,source[i]);  

   d4->SetLineColor(kMagenta);

   d4->Draw("SAME L");           

}

Example 5 – script Background_order.c :

•         second order difference filter removes linear (quasi-linear) background and preserves symmetrical peaks.

•         however if the shape of the background is more complex one can employ higher-order clipping filters (see example in Figure 5)

Figure 5 Example of the influence of clipping filter difference order on the estimated background for fNnumberIterations=40, 2-nd order red line, 4-th order blue line, 6-th order green line and 8-th order magenta line, and using decreasing clipping window algorithm.

 

Script:

// Example to illustrate the influence of the clipping filter difference order on the estimated background

// To execute this example, do

// root > .x Background_order.C

#include <TSpectrum>

 

void Background_order() {

   Int_t i;

   Double_t nbins = 4096;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)4096;

   Float_t * source = new float[nbins];

   TH1F *h = new TH1F("h","",nbins,xmin,xmax);

   TH1F *d1 = new TH1F("d1","",nbins,xmin,xmax);     

   TH1F *d2 = new TH1F("d2","",nbins,xmin,xmax);        

   TH1F *d3 = new TH1F("d3","",nbins,xmin,xmax);        

   TH1F *d4 = new TH1F("d4","",nbins,xmin,xmax);           

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("back2;1");

   TCanvas *background = gROOT->GetListOfCanvases()->FindObject("background");

   if (!background) background = new TCanvas("background","Influence of clipping filter difference order on the estimated background",10,10,1000,700);

   h->SetAxisRange(1220,1460);

   h->SetMaximum(11000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   s->Background(source,nbins,40,kBackDecreasingWindow,kBackOrder2,kFALSE,kBackSmoothing3,kFALSE);       

   for (i = 0; i < nbins; i++) d1->SetBinContent(i + 1,source[i]);  

   d1->SetLineColor(kRed);

   d1->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,40,kBackDecreasingWindow,kBackOrder4,kFALSE,kBackSmoothing3,kFALSE);  

   for (i = 0; i < nbins; i++) d2->SetBinContent(i + 1,source[i]);  

   d2->SetLineColor(kBlue);

   d2->Draw("SAME L");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,40,kBackDecreasingWindow,kBackOrder6,kFALSE,kBackSmoothing3,kFALSE);     

   for (i = 0; i < nbins; i++) d3->SetBinContent(i + 1,source[i]);  

   d3->SetLineColor(kGreen);

   d3->Draw("SAME L");         

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->Background(source,nbins,40,kBackDecreasingWindow,kBackOrder8,kFALSE,kBackSmoothing3,kFALSE);     

   for (i = 0; i < nbins; i++) d4->SetBinContent(i + 1,source[i]);  

   d4->SetLineColor(kMagenta);

   d4->Draw("SAME L");           

}

        ONE-DIMENSIONAL MARKOV SPECTRUM SMOOTHING FUNCTION

        This function calculates smoothed spectrum from source spectrum
        based on Markov chain method.
        The result is placed in the array pointed by source pointer.

        Function parameters:
        source-pointer to the array of source spectrum
        ssize-length of source array
        averWindow-width of averaging smoothing window

Smoothing

 

Goal: Suppression of statistical fluctuations

•         the algorithm is based on discrete Markov chain, which has very simple invariant distribution

 

                 

          being defined from the normalization condition

 

         n is the length of the smoothed spectrum and

 

 

 

is the probability of the change of the peak position from channel i to the channel i+1.  is the normalization constant so that  and m is a width of smoothing window.

 

Function:

const char* SmoothMarkov(float *spectrum, int ssize,  int averWindow) 

 

This function calculates smoothed spectrum from the source spectrum based on Markov chain method. The result is placed in the vector pointed by source pointer. On successful completion it returns 0. On error it returns pointer to the string describing error.

 

Parameters:

        spectrum-pointer to the vector of source spectrum                 

        ssize-length of the spectrum vector                                

        averWindow-width of averaging smoothing window

 

Reference:

[1] Z.K. Silagadze, A new algorithm for automatic photopeak searches. NIM A 376 (1996), 451. 

Example 14 – script Smoothing.c :

 

Fig. 23 Original noisy spectrum    Fig. 24 Smoothed spectrum m=3

Fig. 25 Smoothed spectrum m=7 Fig.26 Smoothed spectrum m=10

 

Script:

// Example to illustrate smoothing using Markov algorithm (class TSpectrum).

// To execute this example, do

// root > .x Smoothing.C

 

//#include <TSpectrum>

 

void Smoothing() {

   Int_t i;

   Double_t nbins = 1024;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   TH1F *h = new TH1F("h","Smoothed spectrum for m=3",nbins,xmin,xmax);

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("smooth1;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   TCanvas *Smooth1 = gROOT->GetListOfCanvases()->FindObject("Smooth1");

   if (!Smooth1) Smooth1 = new TCanvas("Smooth1","Smooth1",10,10,1000,700);

   TSpectrum *s = new TSpectrum();

   s->SmoothMarkov(source,1024,3);  //3, 7, 10

   for (i = 0; i < nbins; i++) h->SetBinContent(i + 1,source[i]);  

   h->SetAxisRange(330,880);

   h->Draw("L");

}

ONE-DIMENSIONAL DECONVOLUTION FUNCTION
This function calculates deconvolution from source spectrum
according to response spectrum using Gold algorithm
The result is placed in the vector pointed by source pointer.

Function parameters:
source:  pointer to the vector of source spectrum
response:     pointer to the vector of response spectrum
ssize:    length of source and response spectra
numberIterations, for details we refer to the reference given below
umberRepetitions, for repeated boosted deconvolutio
boost, boosting coefficient

M. Morhac, J. Kliman, V. Matousek, M. Veselský, I. Turzo.:
Efficient one- and two-dimensional Gold deconvolution and its
application to gamma-ray spectra decomposition.
NIM, A401 (1997) 385-408.

Deconvolution

 

Goal: Improvement of the resolution in spectra, decomposition of multiplets

 

Mathematical formulation of the convolution system is

 

 

 

 

 

where h(i) is the impulse response function, x, y are input and output vectors, respectively, N is the length of x and h vectors. In matrix form we have

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

•         let us assume that we know the response and the output vector (spectrum) of the above given system.

•         the deconvolution represents solution of the overdetermined system of linear equations, i.e.,  the calculation of the vector x.

•         from numerical stability point of view the operation of deconvolution is extremely critical (ill-posed  problem) as well as time consuming operation.

•         the Gold deconvolution algorithm proves to work very well, other methods (Fourier, VanCittert etc) oscillate.

•         it is suitable to process positive definite data (e.g. histograms).

 

Gold deconvolution algorithm

 

 

 

 

 

 

 

 

 

 

where L is given number of iterations (numberIterations parameter).

 

Boosted deconvolution

1.    Set the initial solution

2.    Set required number of repetitions R and iterations L

3.    Set r = 1.

4.    Using Gold deconvolution algorithm for k=1,2,...,L  find

5.    If  r = R stop calculation, else

a. apply boosting operation, i.e., set

    i=0,1,...N-1 and p is boosting coefficient >0.

b. r = r + 1

c. continue in 4.

 

Function:

const char* Deconvolution(float *source, const float *respMatrix, int ssize, int numberIterations, int numberRepetitions, double boost)

 

This function calculates deconvolution from source spectrum according to response spectrum using Gold deconvolution algorithm. The result is placed in the vector pointed by source pointer. On successful completion it returns 0. On error it returns pointer to the string describing error. If desired after every numberIterations one can apply boosting operation (exponential function with exponent given by boost coefficient) and repeat it numberRepetitions times.

 

Parameters:

        source-pointer to the vector of source spectrum                 

        respMatrix-pointer to the vector of response spectrum                 

        ssize-length of the spectrum vector                                

        numberIterations-number of iterations (parameter l in the Gold deconvolution 

        algorithm)

        numberRepetitions-number of repetitions for boosted deconvolution. It must be

        greater or equal to one.

        boost-boosting coefficient, applies only if numberRepetitions is greater than one. 

        Recommended range <1,2>.

 

References:

[1] Gold R., ANL-6984, Argonne National Laboratories, Argonne Ill, 1964.

[2] Coote G.E., Iterative smoothing and deconvolution of one- and two-dimensional elemental distribution data, NIM B 130 (1997) 118.

[3] M. Morháč, J. Kliman, V. Matoušek, M. Veselský, I. Turzo.: Efficient one- and two-dimensional Gold deconvolution and its application to gamma-ray spectra decomposition. NIM, A401 (1997) 385-408.

[4] Morháč M., Matoušek V., Kliman J., Efficient algorithm of multidimensional deconvolution and its application to nuclear data processing, Digital Signal Processing 13 (2003) 144.

 

Example 8 – script Deconvolution.c :

•         response function (usually peak) should be shifted left to the first non-zero channel (bin) (see Figure 9)

 

Figure 9 Response spectrum

Figure 10 Principle how the response matrix is composed inside of the Deconvolution function

Figure 11 Example of Gold deconvolution. The original source spectrum is drawn with black color, the spectrum after the deconvolution (10000 iterations) with red color

 

Script:

// Example to illustrate deconvolution function (class TSpectrum).

// To execute this example, do

// root > .x Deconvolution.C

 

#include <TSpectrum>

 

void Deconvolution() {

             Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * response = new float[nbins];  

   TH1F *h = new TH1F("h","Deconvolution",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon1;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   d=(TH1F*) fr->Get("decon_response;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   for (i = 0; i < nbins; i++) response[i]=d->GetBinContent(i + 1);  

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Decon1",10,10,1000,700);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   s->Deconvolution(source,response,256,1000,1,1);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

}

Examples of Gold deconvolution method

 

•         first let us study the influence of the number of iterations on the deconvolved spectrum  (Figure 12)

Figure 12 Study of Gold deconvolution algorithm. The original source spectrum is drawn with black color, spectrum after 100 iterations with red color, spectrum after 1000 iterations with blue color, spectrum after 10000 iterations with green color and  spectrum after 100000 iterations with magenta color.

 

•      for relatively narrow peaks in the above given example the Gold deconvolution method is able to decompose  overlapping peaks practically to delta - functions.

•         in the next example we have chosen a synthetic data (spectrum, 256 channels) consisting of 5 very closely positioned, relatively wide peaks (sigma =5), with added noise (Figure 13).

•         thin lines represent pure Gaussians (see Table 1); thick line is a resulting spectrum with additive noise (10% of the amplitude of small peaks).

Figure 13 Testing example of synthetic spectrum composed of 5 Gaussians with added noise

 

Peak #	Position	Height	Area
1	50	500	10159
2	70	3000	60957
3	80	1000	20319
4	100	5000	101596
5	110	500	10159

 

Table 1 Positions, heights and areas of peaks in the spectrum shown in Figure 13

 

•         in ideal case, we should obtain the result given in Figure 14. The areas of the Gaussian components of the spectrum are concentrated completely to delta –functions

•         when solving the overdetermined system of linear equations with data from Figure 13 in the sense of minimum least squares criterion without any regularization we obtain the result with large oscillations  (Figure 15).

•         from mathematical point of view, it is the optimal solution in the unconstrained space of independent variables. >From physical point of view we are interested only in a meaningful solution.

•         therefore, we have to employ regularization techniques (e.g. Gold deconvolution) and/or to confine the space of allowed solutions to subspace of positive solutions.

 

Figure 14 The same spectrum like in Figure 13, outlined bars show the contents of present components (peaks)

Figure 15 Least squares solution of the system of linear equations without regularization

 

Example 9 – script Deconvolution_wide.c :

•         when we employ Gold deconvolution algorithm we obtain the result given in Fig. 16. One can observe that the resulting spectrum is smooth. On the other hand the method is not able to decompose completely the peaks in the spectrum.

 

Figure 16 Example of Gold deconvolution for closely positioned wide peaks. The original source spectrum is drawn with black color, the spectrum after the deconvolution (10000 iterations) with red color

 

 

Script:

// Example to illustrate deconvolution function (class TSpectrum).

// To execute this example, do

// root > .x Deconvolution_wide.C

 

#include <TSpectrum>

 

void Deconvolution_wide() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * response = new float[nbins];  

   TH1F *h = new TH1F("h","Deconvolution",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon3;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   d=(TH1F*) fr->Get("decon_response_wide;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   for (i = 0; i < nbins; i++) response[i]=d->GetBinContent(i + 1);  

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Deconvolution of closely positioned overlapping peaks using Gold deconvolution method",10,10,1000,700);

   h->SetMaximum(30000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   s->Deconvolution(source,response,256,10000,1,1);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

}

 

Example 10 – script Deconvolution_wide_boost.c :

 

•         further let us employ boosting operation into deconvolution (Fig. 17)

 

Figure 17 The original source spectrum is drawn with black color, the spectrum after the deconvolution with red color. Number of iterations = 200, number of repetitions = 50 and boosting coefficient = 1.2.

 

Peak #	Original/Estimated (max) position	Original/Estimated area
1	50/49	10159/10419
2	70/70	60957/58933
3	80/79	20319/19935
4	100/100	101596/105413
5	110/117	10159/6676

 

Table 2 Results of the estimation of peaks in spectrum shown in Figure 17

 

•                     one can observe that peaks are decomposed practically to delta functions. Number of peaks is correct, positions of big peaks as well as their areas are relatively well estimated. However there is a considerable error in the estimation of the position of small right hand peak.

 

Script:

// Example to illustrate deconvolution function (class TSpectrum).

// To execute this example, do

// root > .x Deconvolution_wide_boost.C

 

#include <TSpectrum>

 

void Deconvolution_wide_boost() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * response = new float[nbins];  

   TH1F *h = new TH1F("h","Deconvolution",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon3;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   d=(TH1F*) fr->Get("decon_response_wide;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   for (i = 0; i < nbins; i++) response[i]=d->GetBinContent(i + 1);  

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Deconvolution of closely positioned overlapping peaks using boosted Gold deconvolution method",10,10,1000,700);

   h->SetMaximum(110000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   s->Deconvolution(source,response,256,200,50,1.2);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

}

ONE-DIMENSIONAL DECONVOLUTION FUNCTION
This function calculates deconvolution from source spectrum
according to response spectrum using Richardson-Lucy algorithm
The result is placed in the vector pointed by source pointer.

Function parameters:
source:  pointer to the vector of source spectrum
response:     pointer to the vector of response spectrum
ssize:    length of source and response spectra
numberIterations, for details we refer to the reference given above
umberRepetitions, for repeated boosted deconvolutio
boost, boosting coefficient

Richardson-Lucy deconvolution algorithm

·              for discrete systems it has the form

                             

 

·               for positive input data and response matrix this iterative method forces the deconvoluted spectra to be non-negative.

·              the Richardson-Lucy iteration converges to the maximum likelihood solution for Poisson statistics in the data.

 

Function:

const char* DeconvolutionRL(float *source, const float *respMatrix, int ssize, int numberIterations, int numberRepetitions, double boost)

 

This function calculates deconvolution from source spectrum according to response spectrum using Richardson-Lucy deconvolution algorithm. The result is placed in the vector pointed by source pointer. On successful completion it returns 0. On error it returns pointer to the string describing error. If desired after every numberIterations one can apply boosting operation (exponential function with exponent given by boost coefficient) and repeat it numberRepetitions times (see Gold deconvolution).

 

Parameters:

        source-pointer to the vector of source spectrum                 

        respMatrix-pointer to the vector of response spectrum                 

        ssize-length of the spectrum vector                                

        numberIterations-number of iterations (parameter l in the Gold deconvolution 

        algorithm)

        numberRepetitions-number of repetitions for boosted deconvolution. It must be

        greater or equal to one.

        boost-boosting coefficient, applies only if numberRepetitions is greater than one. 

        Recommended range <1,2>.

 

References:

[1] Abreu M.C. et al., A four-dimensional deconvolution method to correct NA38 experimental data, NIM A 405 (1998) 139.

[2] Lucy L.B., A.J. 79 (1974) 745.

[3] Richardson W.H., J. Opt. Soc. Am. 62 (1972) 55.

Examples of Richardson-Lucy deconvolution method

 

Example 11 – script DeconvolutionRL_wide.c :

•         when we employ Richardson-Lucy deconvolution algorithm to our data from Fig. 13 we obtain the result given in Fig. 18. One can observe improvements as compared to the result achieved by Gold deconvolution.

•         neverthless it is unable to decompose the multiplet.

 

Figure 18 Example of Richardson-Lucy deconvolution for closely positioned wide peaks. The original source spectrum is drawn with black color, the spectrum after the deconvolution (10000 iterations) with red color

 

 

Script:

 

// Example to illustrate deconvolution function (class TSpectrum).

// To execute this example, do

// root > .x DeconvolutionRL_wide.C

 

#include <TSpectrum>

 

void DeconvolutionRL_wide() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * response = new float[nbins];  

   TH1F *h = new TH1F("h","Deconvolution",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon3;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   d=(TH1F*) fr->Get("decon_response_wide;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   for (i = 0; i < nbins; i++) response[i]=d->GetBinContent(i + 1);  

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Deconvolution of closely positioned overlapping peaks using Richardson-Lucy deconvolution method",10,10,1000,700);

   h->SetMaximum(30000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   s->DeconvolutionRL(source,response,256,10000,1,1);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

}

 

Example 12 – script DeconvolutionRL_wide_boost.c :

 

•         further let us employ boosting operation into deconvolution (Fig. 19)

Figure 19 The original source spectrum is drawn with black color, the spectrum after the deconvolution with red color. Number of iterations = 200, number of repetitions = 50 and boosting coefficient = 1.2.

 

Peak #	Original/Estimated (max) position	Original/Estimated area
1	50/51	10159/11426
2	70/71	60957/65003
3	80/81	20319/12813
4	100/100	101596/101851
5	110/111	10159/8920

Table 3 Results of the estimation of peaks in spectrum shown in Figure 19

 

•                     one can observe improvements in the estimation of peak positions as compared to the results achieved by Gold deconvolution

 

Script:

 

// Example to illustrate deconvolution function (class TSpectrum).

// To execute this example, do

// root > .x DeconvolutionRL_wide_boost.C

#include <TSpectrum>

 

void DeconvolutionRL_wide_boost() {

   Int_t i;

   Double_t nbins = 256;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * response = new float[nbins];  

   TH1F *h = new TH1F("h","Deconvolution",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon3;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   d=(TH1F*) fr->Get("decon_response_wide;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   for (i = 0; i < nbins; i++) response[i]=d->GetBinContent(i + 1);  

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Deconvolution of closely positioned overlapping peaks using boosted Richardson-Lucy deconvolution method",10,10,1000,700);

   h->SetMaximum(110000);

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   s->DeconvolutionRL(source,response,256,200,50,1.2);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,source[i]);  

   d->SetLineColor(kRed);

   d->Draw("SAME L");  

}

        ONE-DIMENSIONAL UNFOLDING FUNCTION
        This function unfolds source spectrum
        according to response matrix columns.
        The result is placed in the vector pointed by source pointer.

        Function parameters:
        source-pointer to the vector of source spectrum
        respMatrix-pointer to the matrix of response spectra
        ssizex-length of source spectrum and # of columns of response matrix
        ssizey-length of destination spectrum and # of rows of
              response matrix
        numberIterations, for details we refer to manual
        Note!!! ssizex must be >= ssizey

Unfolding

 

Goal: Decomposition of spectrum to a given set of component spectra

 

Mathematical formulation of the discrete linear system is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Function:

const char* Unfolding(float *source, const float **respMatrix, int ssizex, int ssizey, int numberIterations, int numberRepetitions, double boost)

 

This function unfolds source spectrum according to response matrix columns. The result is placed in the vector pointed by source pointer.  The coefficients of the resulting vector represent contents of the columns (weights) in the input vector. On successful completion it returns 0. On error it returns pointer to the string describing error. If desired after every numberIterations one can apply boosting operation (exponential function with exponent given by boost coefficient) and repeat it numberRepetitions times. For details we refer to [1].

 

Parameters:

        source-pointer to the vector of source spectrum                 

        respMatrix-pointer to the matrix of response spectra                 

        ssizex-length of source spectrum and # of columns of the response matrix

        ssizey-length of destination spectrum and # of rows of the response matrix                                

        numberIterations-number of iterations

        numberRepetitions-number of repetitions for boosted deconvolution. It must be

        greater or equal to one.

        boost-boosting coefficient, applies only if numberRepetitions is greater than one. 

        Recommended range <1,2>.

 

Note!!! sizex must be >= sizey After decomposition the resulting channels are written back to the first sizey channels of the source spectrum.

 

Reference:

[1] Jandel M., Morháč M., Kliman J., Krupa L., Matoušek V., Hamilton J. H., Ramaya A. V.: Decomposition of continuum gamma-ray spectra using synthetized response matrix. NIM A 516 (2004), 172-183.

 

Example of unfolding

 

Example 13 – script Unfolding.c :

Fig. 20  Response matrix composed of neutron spectra of pure chemical elements

Fig. 21 Source neutron spectrum to be decomposed

Fig. 22 Spectrum after decomposition, contains 10 coefficients, which correspond to contents of chemical components (dominant 8-th and 10-th components, i.e. O, Si)

 

Script:

 

// Example to illustrate unfolding function (class TSpectrum).

// To execute this example, do

// root > .x Unfolding.C

 

#include <TSpectrum>

void Unfolding() {

   Int_t i, j;

   Int_t nbinsx = 2048;

   Int_t nbinsy = 10;  

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbinsx;

   Double_t ymin  = 0;

   Double_t ymax  = (Double_t)nbinsy;  

   Float_t * source = new float[nbinsx];

   Float_t ** response = new float *[nbinsy];  

   for (i=0;i<nbinsy;i++)

                                    response[i]=new float[nbinsx];  

   TH1F *h = new TH1F("h","",nbinsx,xmin,xmax);

   TH1F *d = new TH1F("d","Decomposition - unfolding",nbinsx,xmin,xmax);  

   TH2F *decon_unf_resp = new TH2F("decon_unf_resp","Root File",nbinsy,ymin,ymax,nbinsx,xmin,xmax);

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("decon_unf_in;1");

   TFile *fr = new TFile("spectra\\TSpectrum.root");

   decon_unf_resp = (TH2F*) fr->Get("decon_unf_resp;1");    

   for (i = 0; i < nbinsx; i++) source[i] = h->GetBinContent(i + 1);

   for (i = 0; i < nbinsy; i++){

      for (j = 0; j< nbinsx; j++){

             response[i][j] = decon_unf_resp->GetBinContent(i + 1, j + 1);

      }

   }    

   TCanvas *Decon1 = gROOT->GetListOfCanvases()->FindObject("Decon1");

   if (!Decon1) Decon1 = new TCanvas("Decon1","Decon1",10,10,1000,700);

   h->Draw("L");  

   TSpectrum *s = new TSpectrum();

   s->Unfolding(source,response,nbinsx,nbinsy,1000,1,1);

   for (i = 0; i < nbinsy; i++) d->SetBinContent(i + 1,source[i]);

   d->SetLineColor(kRed);  

   d->SetAxisRange(0,nbinsy);    

   d->Draw("");

}

        ONE-DIMENSIONAL HIGH-RESOLUTION PEAK SEARCH FUNCTION
        This function searches for peaks in source spectrum
      It is based on deconvolution method. First the background is
      removed (if desired), then Markov spectrum is calculated
      (if desired), then the response function is generated
      according to given sigma and deconvolution is carried out.

        Function parameters:
        source-pointer to the vector of source spectrum
        destVector-pointer to the vector of resulting deconvolved spectrum     *
        ssize-length of source spectrum
        sigma-sigma of searched peaks, for details we refer to manual
        threshold-threshold value in % for selected peaks, peaks with
                amplitude less than threshold*highest_peak/100
                are ignored, see manual
      backgroundRemove-logical variable, set if the removal of
                background before deconvolution is desired
      deconIterations-number of iterations in deconvolution operation
      markov-logical variable, if it is true, first the source spectrum
             is replaced by new spectrum calculated using Markov
             chains method.
        averWindow-averanging window of searched peaks, for details
                  we refer to manual (applies only for Markov method)

Peaks searching

 

Goal: to identify automatically the peaks in spectrum with the presence of the continuous background and statistical fluctuations - noise.

 

The common problems connected with correct peak identification are

• non-sensitivity to noise, i.e., only statistically relevant peaks should be identified.
• non-sensitivity of the algorithm to continuous background
• ability to identify peaks close to the edges of the spectrum region. Usually peak finders fail to detect them
• resolution, decomposition of doublets and multiplets. The algorithm should be able to recognize close positioned peaks.
• ability to identify peaks with different sigma

 

Function:

Int_t SearchHighRes(float *source,float *destVector, int ssize, float sigma, double threshold, bool backgroundRemove,int deconIterations, bool markov, int averWindow)   

 

This function searches for peaks in source spectrum. It is based on deconvolution method. First the background is removed (if desired), then Markov smoothed spectrum is calculated (if desired), then the response function is generated according to given sigma and deconvolution is carried out. The order of peaks is arranged according to their heights in the spectrum after background elimination. The highest peak is the first in the list. On success it returns number of found peaks.

 

Parameters:

        source-pointer to the vector of source spectrum                 

        destVector-resulting spectrum after deconvolution

        ssize-length of the source and destination spectra               

        sigma-sigma of searched peaks

threshold- threshold value in % for selected peaks, peaks with amplitude less than threshold*highest_peak/100 are ignored

backgroundRemove- background_remove-logical variable, true if the removal of background before deconvolution is desired 

deconIterations-number of iterations in deconvolution operation

markov-logical variable, if it is true, first the source spectrum is replaced by new spectrum calculated using Markov chains method

averWindow-width of averaging smoothing window

 

Fig. 27 An example of one-dimensional synthetic spectrum with found peaks denoted by markers

 

References:

[1] M.A. Mariscotti: A method for identification of peaks in the presence of background and its application to spectrum analysis. NIM 50 (1967), 309-320.

[2]  M. Morháč, J. Kliman, V. Matoušek, M. Veselský, I. Turzo.:Identification of peaks in multidimensional coincidence gamma-ray spectra. NIM, A443 (2000) 108-125.

[3] Z.K. Silagadze, A new algorithm for automatic photopeak searches. NIM A 376 (1996), 451.

Examples of peak searching method

 

SearchHighRes function provides users with the possibility to vary the input parameters and with the access to the output deconvolved data in the destination spectrum. Based on the output data one can tune the parameters.

Example 15 – script SearchHR1.c:

Fig. 28 One-dimensional spectrum with found peaks denoted by markers, 3 iterations steps in the deconvolution

 

Fig. 29 One-dimensional spectrum with found peaks denoted by markers, 8 iterations steps in the deconvolution

 

Script:

// Example to illustrate high resolution peak searching function (class TSpectrum).

// To execute this example, do

// root > .x SearchHR1.C

 

#include <TSpectrum>

 

void SearchHR1() {

           

   Float_t fPositionX[100];

   Float_t fPositionY[100];  

   Int_t fNPeaks = 0;    

   Int_t i,nfound,bin;

   Double_t nbins = 1024,a;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * dest = new float[nbins];  

   TH1F *h = new TH1F("h","High resolution peak searching, number of iterations = 3",nbins,xmin,xmax);

   TH1F *d = new TH1F("d","",nbins,xmin,xmax);     

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("search2;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   TCanvas *Search = gROOT->GetListOfCanvases()->FindObject("Search");

   if (!Search) Search = new TCanvas("Search","Search",10,10,1000,700);

   h->SetMaximum(4000);     

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   nfound = s->SearchHighRes(source, dest, nbins, 8, 2, kTRUE, 3, kTRUE, 3);

   Float_t *xpeaks = s->GetPositionX();

   for (i = 0; i < nfound; i++) {

        a=xpeaks[i];

        bin = 1 + Int_t(a + 0.5);

        fPositionX[i] = h->GetBinCenter(bin);

        fPositionY[i] = h->GetBinContent(bin);

   }

   TPolyMarker * pm = (TPolyMarker*)h->GetListOfFunctions()->FindObject("TPolyMarker");

   if (pm) {

      h->GetListOfFunctions()->Remove(pm);

      delete pm;

   }

   pm = new TPolyMarker(nfound, fPositionX, fPositionY);

   h->GetListOfFunctions()->Add(pm);

   pm->SetMarkerStyle(23);

   pm->SetMarkerColor(kRed);

   pm->SetMarkerSize(1.3);

   for (i = 0; i < nbins; i++) d->SetBinContent(i + 1,dest[i]);

   d->SetLineColor(kRed);  

   d->Draw("SAME");

   printf("Found %d candidate peaks\n",nfound); 

   for(i=0;i<nfound;i++)

      printf("posx= %d, posy= %d\n",fPositionX[i], fPositionY[i]);       

   }

 

Example 16 – script SearchHR3.c:

Peak #	Position	Sigma
1	118	26
2	162	41
3	310	4
4	330	8
5	482	22
6	491	26
7	740	21
8	852	15
9	954	12
10	989	13

 

Table 4 Positions and sigma of peaks in the following examples

 

Fig. 30 Influence of number of iterations (3-red, 10-blue, 100- green, 1000-magenta), sigma=8, smoothing width=3

 

Fig. 31 Influence of sigma (3-red, 8-blue, 20- green, 43-magenta), num. iter.=10, sm. width=3

 

Fig. 32 Influence smoothing width (0-red, 3-blue, 7- green, 20-magenta), num. iter.=10, sigma=8

 

Script:

 

// Example to illustrate the influence of number of iterations in deconvolution in high resolution peak searching function (class TSpectrum).

// To execute this example, do

// root > .x SearchHR3.C

 

#include <TSpectrum>

 

void SearchHR3() {

   Float_t fPositionX[100];

   Float_t fPositionY[100];  

   Int_t fNPeaks = 0;    

   Int_t i,nfound,bin;

   Double_t nbins = 1024,a;

   Double_t xmin  = 0;

   Double_t xmax  = (Double_t)nbins;

   Float_t * source = new float[nbins];

   Float_t * dest = new float[nbins];  

   TH1F *h = new TH1F("h","Influence of # of iterations in deconvolution in peak searching",nbins,xmin,xmax);

   TH1F *d1 = new TH1F("d1","",nbins,xmin,xmax);     

   TH1F *d2 = new TH1F("d2","",nbins,xmin,xmax);     

   TH1F *d3 = new TH1F("d3","",nbins,xmin,xmax);     

   TH1F *d4 = new TH1F("d4","",nbins,xmin,xmax);              

   TFile *f = new TFile("spectra\\TSpectrum.root");

   h=(TH1F*) f->Get("search3;1");  

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);  

   TCanvas *Search = gROOT->GetListOfCanvases()->FindObject("Search");

   if (!Search) Search = new TCanvas("Search","Search",10,10,1000,700);

   h->SetMaximum(1300);        

   h->Draw("L");

   TSpectrum *s = new TSpectrum();

   nfound = s->SearchHighRes(source, dest, nbins, 8, 2, kTRUE, 3, kTRUE, 3);

   Float_t *xpeaks = s->GetPositionX();

   for (i = 0; i < nfound; i++) {

        a=xpeaks[i];

        bin = 1 + Int_t(a + 0.5);

        fPositionX[i] = h->GetBinCenter(bin);

        fPositionY[i] = h->GetBinContent(bin);

   }  

   TPolyMarker * pm = (TPolyMarker*)h->GetListOfFunctions()->FindObject("TPolyMarker");

   if (pm) {

      h->GetListOfFunctions()->Remove(pm);

      delete pm;

   }

   pm = new TPolyMarker(nfound, fPositionX, fPositionY);

   h->GetListOfFunctions()->Add(pm);

   pm->SetMarkerStyle(23);

   pm->SetMarkerColor(kRed);

   pm->SetMarkerSize(1.3);

   for (i = 0; i < nbins; i++) d1->SetBinContent(i + 1,dest[i]);

   h->Draw("");

   d1->SetLineColor(kRed);     

   d1->Draw("SAME");

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->SearchHighRes(source, dest, nbins, 8, 2, kTRUE, 10, kTRUE, 3);     

   for (i = 0; i < nbins; i++) d2->SetBinContent(i + 1,dest[i]);

   d2->SetLineColor(kBlue);     

   d2->Draw("SAME");

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->SearchHighRes(source, dest, nbins, 8, 2, kTRUE, 100, kTRUE, 3);     

   for (i = 0; i < nbins; i++) d3->SetBinContent(i + 1,dest[i]);

   d3->SetLineColor(kGreen);     

   d3->Draw("SAME");      

   for (i = 0; i < nbins; i++) source[i]=h->GetBinContent(i + 1);

   s->SearchHighRes(source, dest, nbins, 8, 2, kTRUE, 1000, kTRUE, 3);     

   for (i = 0; i < nbins; i++) d4->SetBinContent(i + 1,dest[i]);

   d4->SetLineColor(kMagenta);     

   d4->Draw("SAME");  

   printf("Found %d candidate peaks\n",nfound); 

}