ososkov g. wavelet analysis cbm collaboration meeting wavelet application for handling invariant...

Ososkov G. Wavelet analysis CBM Collaboration meeting

Wavelet application for handling invariant mass

spectraGennady Ososkov

LIT JINR, Dubna

Semeon LebedevGSI, Darmstadt and LIT JINR, Dubna

12th CBM Collaboration meeting Oct. 13-18 2008, JINR Dubna


Why do we need waveletsfor handling invariant mass spectra?

-we need them when S/B ratio is << 1

Smoothing after background subtractionwithout losing any essential information

resonance indicating even in presence of background

evaluating peak parameters


Estimating peak parameters in G2 wavelet domain

How it works:How it works:after background subtracting we have a noisy after background subtracting we have a noisy spectrumspectrum

It is transformed by GIt is transformed by G2 2 into wavelet domain, into wavelet domain,

where we look for the wavelet surface maximum where we look for the wavelet surface maximum

2

322

2

5 )ˆˆ(

ˆˆ

maxˆ

a

a

WA

(bmax amax) and then fit this surface by the analytical formula for WG2(a,b;x0,σ)g starting fit from x0=bmax and

Eventually, we should find the maximum of this fitted surface and use its coordinates as

estimations of peak parameters

5max

0

a

ax ˆ,ˆ0From them we can obtain halfwidth ,

amplitude

and even the integral 2AI

peak has bell-shape form

5

ˆˆ

a


Real problems 1. CBM spectra Λc invariant mass spectrum (by courtesy of Iou.Vassilev)

and its G2 spectrum

more andmore detailed

Wavelet method results:A=15.0σ =0.0116mean=2.2840Iw=0.435

PDG m=2.285

Igauss=0.365 (19% less)


Real problems 2. CBM spectra Low-mass dileptons (muon channel) ω. Gauss fit

of reco signalM=0.7785σ =0.0125A=1.8166Ig=0.0569

ω. WaveletsM=0.7700σ =0.0143A=1.8430Iw=0.0598

- ω– wavelet spectrum

ω.

ω-meson

φ-meson

Even φ- and mesons have been visiblein the wavelet space, so we could extract their parameters.

Thanks to Anna Kiseleva


Real problems 3. Resonance structure

Carbon

p, D

K.Abraamyan, V.Toneev et al., arXiv:0806/0806.2790

Resonance structure study in γγ invariant mass spectra

in dC-InteractionsA resonance structure in the invariant mass spectrum of two photons at Mγγ = 360 MeV was observed in the reaction dC → γ +γ+Xat momentum Pd=2.75 GeV/c per nucleon.

Schematic view of PHOTON-2 setup. S1, S2 are scintillation counters.

Dubna Nuclotron, Photon-2 setup

π0 decay from dC into γγ has bulky background. Trerefore models with channels η,ή, NN-bremsstrahlung, ω, Δ → Nγ, Σ →Λγ etc producing this background has been simulated, but the only generator which took into account the R resonance with small opening angle gave good correspondence with experimental data.


Gaussian wavelet application for resonance structure study

Invariant mass distributions of γγ pairs satisfying the above criteria without (upper panel) and with (bottom panel) the background subtraction.

Selection criteria to background subtraction: the number of photons in an event, Nγ =2; the energies of photons, Eγ ≥ 100 MeV; the summed energy ≤ 1.5 GeV


The invariant mass distribution of The invariant mass distribution of γγγγ pairs and the biparametri pairs and the biparametric c distribution of the GW of the 8-th order for distribution of the GW of the 8-th order for dC dC interactions. interactions. TThe he distribution is obtaineddistribution is obtained with an additional condition for photon with an additional condition for photon

energies Eγ1/Eγ2 > 0.8 and binning in 2energies Eγ1/Eγ2 > 0.8 and binning in 2MeV.MeV.

Therefore, the presented results of the continuous wavelet analysis with vanishing momenta confirm the finding of a peak at Mγγ (2 − 3)m∼ π in the γγ invariant mass distribution obtained within the standard method with the subtraction of the background from mixing events.

rresonant structure wasn’t observed in pC-interactions at 5.5 Gev/c


Continuous or discrete wavelets?

Continuous wavelets are remarkably resistant to noise (robust), but because of their non-orthogonality one obtains non-admissible signal distortions after inverse transform. Besides, real signals to be analysed by computer are always discrete.

So orthogonal discrete wavelets look preferable.

The discrete wavelet transform (DWT) was built by Mallat as multi-resolution analysis. It consists in representing a given data as a signal decomposition into basis functions φ and ψ, which must be compact.

Various types of discrete wavelets

One of Daubechie’s wavelets

Coiflet – most symmetric


DWT Peak Finder

Denoising by DWT shrinkingWavelet shrinkage means, certain wavelet coefficients are reduced to zero ,

so after the inverse transform one can eliminate from the spectrum part of high and low frequencies.

Our innovation is the adaptive shrinkage,i.e. λk= 3σk where k is decomposition level (k=scale1,...,scalen), σk is RMS of Wψ for this level (recall: sample size is 2n)

The discrete wavelets are a good tool for background eliminating and peak detecting.However the main problem of wavelet applications was the absence of corresponding C++ software in any of available frameworks. So we had to build it ourselves and

Simeon accomplished that successfully.

An example of Daub2 spectrum


CBM spectra. Result 3.Low-mass dileptons (muon channel)

Its fragment after adaptive shrinking with specially chosen multipliers M inλk= Mσk for each level k of DWT

Original inv. mass spectrum

Result of adaptive shrinking with λk= 3σk

inevitable Gibbs edge effect

no backround!

ω-meson

φ-meson


1.Wavelet approach looks quite applicable for handling CBM invariant mass spectra with small S/B ratio

2.Discrete wavelets could be considered as a new tool for detecting the peak existence.

3.Algorithms and programs have been developed for estimating resonance peak parameters on the basis of gaussian continuous wavelets

4.Algorithms and programs have been developed for resonance peak detecting on the basis of discrete wavelets

5.First attempts of the wavelet applications to CBM open charm and meson data are very promising

Summary and outlook


What to do

• Commit, eventually, wavelet software into the SVN repository

• Tuning of running software in close contacts with physicists interested in peak finding business

• Furnish this software by algorithms for automatic peak search and their parameters estimation


Thank you for attention!


Recall to wavelet introduction

•

One-dimensional wavelet transform (WT) of the signal f(x) has 2D form

where the function is the wavelet, b is a displacement (shift), and a is a scale. Condition Cψ < ∞ guarantees the existence of and the wavelet inverse

transform. Due to freedom in choice, many different wavelets were invented.

The family of continuous wavelets with vanishing momenta is presented here by Gaussian wavelets, which are generated by derivatives of Gaussian function

Two of them, we use, areand

Most knownwavelet G2 is named “the Mexican hat”


Recall to wavelet introduction (cont)

Applicatios for extracting special features of mixed and contaminated signal

G2 wavelet spectrum of this signal

Filtering results. Noise is removed and high frequency part perfectly localized

An example of the signal with a localized high frequency part and considerable contamination

then wavelet filtering is applied

Filtering works in the wavelet domain bythresholding of scales, to be eliminatedor extracted, and then by making the inverse transform


Peak parameter estimating by gaussian wavelets

When a signal is bell-shaped one, itcan be approximated by a gaussian

Thus, we can work directly in the wavelet domain instead of time/space domain and use this analytical formula for WG2(a,b;x0,σ)g surface in order to fit it to the surface, obtained for a real invariant mass spectrum. The most remarkable point is: since the fitting parameters x0 and σ, can be estimated directly in the G2 domain, we do not need the inverse transform!

Then it can be derived analytically that its wavelet transformation looksas the corresponding wavelet. For instance, for G2(x)

one has

Considering WG2 as a function of the dilation b we obtain its maximum

and then solving

the equation we obtain .

)()(

),(22

02

2

322

2

5

2ba

xbG

a

AagbaWG

2

322

2

5

2

)(

),(max

a

AabaWGb

0)(max

a

ab 5max a

ososkov g. wavelet analysis cbm collaboration meeting wavelet application for handling invariant...

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