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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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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)
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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.
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
Thank you for attention!
Ososkov G. Wavelet analysis CBM Collaboration meeting
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”
Ososkov G. Wavelet analysis CBM Collaboration meeting
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
Ososkov G. Wavelet analysis CBM Collaboration meeting
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 .
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),(22
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5
2ba
xbG
a
AagbaWG
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5
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