1. Igor Volobouev Texas Tech University i.volobouev@ttu.edu
Multiresolution Jet Reconstruction with FFTJet Calor2010, IHEP, Beijing, China May
Igor Volobouev
Texas Tech University

2. Iterative Cone and Mean Shift
The iterative cone algorithm we all know is called ”Mean Shift” clustering in the pattern recognition literature. The procedure was first described in a 1975 paper by K. Fukunaga and L.D. Hostetler.
In 1977, Sterman and Weinberg proposed jet definition based on cones
Important generalization of Mean Shift: Y. Cheng, “Mean Shift, Mode Seeking, and Clustering”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol 17, Aug 1995, p. 790.
Independently, S.D. Ellis invented the concept of “Snowmass potential” in 2002 (?)
Fast seedless cone algorithms: FSLC (2006), SISCone (2007)
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

3. Connecting Mean Shift and KDE
Cheng (and Ellis) have shown that, mathematically, the problem of searching for stable cones is equivalent to locating the peaks of the energy density built in the η-φ space using kernel density estimation (KDE) with the “Epanechnikov” kernel function. That is, one convolutes
with
and finds the peaks K
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

4. Problems with the Cone Algorithms and the Split-Merge Procedure
Inherent configuration ambiguity
The area of the jet is not fixed (this is a problem, e.g., for SISCone)
The energy of two nearby jets is miscalibrated because standard jet energy calibration procedures employ well-isolated jets, and the combined energy fraction leaked out-of-cone is different for nearby jets
Energy correlations are not handled
The goal of assigning each tower to just one particular jet is unphysical because of an irreducible spatial energy smearing during shower development in the calorimeter
Seeded cone algorithms (CDF, D0)
have additional problems
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

5. Jet Reconstruction as Spatial Filtering
Convolute the energy deposition picture (as a function of η and φ) with a low-pass filter. This can be done quickly by DFFT. Computational complexity is O(N log N).
Find the intensity peaks. These are the precluster locations.
Apply a threshold to the peak magnitudes
Assign a cluster membership function (MF) to each surviving precluster and to the pileup/noise
Distribute calorimeter towers among jets and pileup/noise with weights generated by the MFs (fuzzy clustering) or assign each tower to the jet with the highest MF value (crisp clustering)
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

6. Seedless Cone Algorithm as a Spatial Filter
Compare with Gaussian filtering
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

7. Multiresolution Jet Reconstruction with FFTJet
FFTJet is a recently developed framework for building global, efficient, collinear and infrared safe jet algorithms using the spatial filtering approach
The method is described in arXiv: v1.The code and the manual are at
FFTJet allows its users to take into account the detector properties (magnetic field, noise, etc) inside jet clustering and energy recombination procedures. Note that this goes against the “conventional wisdom” of keeping jet algorithms identical between theory and experiment. Magnetic field in the detectors and shower development inside the calorimeters are important effects!
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

8. Jets in the 3.8 T Magnetic Fieldη φ
pT = 10 GeV/c
pT = 20 GeV/c
Average angular energy profiles from the Pythia
jet gun (light quark)
Distributions are built 2 m away from the jet origin
A CMS dijet event
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

9. pT Dependence of the Jet Shape
CMS HCAL Barrel (simulated)
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

10. Reconstructing Jets: Better Jet Shape Model Wins
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

11. Multiresolution Jet Reconstruction
Jets can be reconstructed using many angular resolution scales. Technically, using the cone algorithm with many values of R is not the best way to do this (the split-merge stage is a problem). Gaussian filtering works much better.
FFTJet builds the “clustering tree”: preclusters formed at different resolution scales are related to each other by parent-daughter relationships. Using the parent and the “closest daughter”, precluster characteristics can be determined as a function of the resolution scale.
Studies of precluster behavior in the scale space result in
Advanced pattern recognition capabilities
Optimal determination of jet properties
Control over bifurcation points
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

12. Example Clustering Tree
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

13. Pattern Recognition in Scale Space
Multiresolution analysis opens new ways of thinking about jet reconstruction:
Choose an optimal resolution scale, best according to some criterion. Similar to “normal” analysis but with essentially continuous resolution choices.
Choose optimal resolution scale for each jet (perhaps, using scale space “blob detection” techniques)
Choose a configuration with a certain number of jets. Assess the stability of the configuration in the scale space.
Look for jet substructure expected in the signal (e.g., for boosted heavy particles)
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

14. Multiresolution Jet Reconstruction with FFTJet
Bifurcation Points
There are no “completely IR-safe” algorithms
The field of science which studies significant topological structure changes in response to small changes in parameters or initial conditions is known as “Bifurcation Theory”
We want to be as far away as possible from a bifurcation point when the jet 4-momentum is determined: jets near bifurcations are noisy!
The clustering tree provides a good handle for controlling bifurcations
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

15. Jet Energy Determination
The most precise jet energy reconstruction will take into account a variety of jet characteristics in addition to the “one-shot” 4-momentum:
Electromagnetic energy fraction
Charged energy fraction
Jet shape
Jet flavor
Detector noise and pileup
Separation from other jets
FFTJet “jet membership functions” is the mechanism by which such information can be included directly into the jet clustering and energy recombination process
A powerful jet shape model: detector-level fragmentation function
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet

16. Multiresolution Jet Reconstruction with FFTJet
Conclusions
Better pattern recognition improves jet reconstruction on many levels
Spatial filtering and multiresolution image analysis are powerful tools, already studied by mathematicians and computer vision scientists for many years. These tools should be incorporated into the next-generation jet reconstruction algorithms for Tevatron and LHC.
Multiresolution approaches shine when pattern recognition becomes nontrivial: for signal identification in multijet processes, or when jet-like objects are not really jets (boosted tops or Ws) and the jet substructure analysis becomes necessary
With these techniques, jet energy determination can be optimized on process-by-process and jet-by-jet basis
I am far from knowing where this road will lead us, just making a few steps out the door…
Igor Volobouev
Multiresolution Jet Reconstruction with FFTJet