1. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov1 Visualization of FTK & Tiny Triplet Finder Jinyuan Wu and Tiehui Liu Fermilab January 2010

2. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov2 Assumption Pixel and SCT barrels are considered. Total number of channels (From Fig. 2.1 of TP): –B0: 20K –B1: 16K –B2: 14K –SCT3, 4, 5, 6: 5K each. Total number of clusters (1/1.3 of above): –B0: 15K –B1: 12K –B2: 11K –SCT3, 4, 5, 6: 3.8K each.

3. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov3 The R-phi View of Pixels Pixel modules are tilted. For hits on B0 found in rows along z-axis, several “R-phi Corridors” can be defined. In each r-phi corridor, distances between pixel planes are relatively well defined.

4. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov4 The R-Phi Corridors Consider 1 mm impact parameter, the possible range of tracks passing through a 1 mm r-phi bin in B0 spread in large range. For Pt = +- 1 GeV, the range in B2 is about 14 mm. For a r-phi corridor passing a bin each in B0 and B1, 1 mm each, the range on B2 is about 4mm. More than 30 hit patterns for each bin in B0 can be counted above. There are about 30 patterns/bin x 300 bins ~ 9K patterns in r-phi view.

5. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov5 The Thin Road Bins The size of the bins in the Thin Road scheme is (1 mm x 60 mm) in (r-phi & z).

6. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov6 Random Hits in the Thin Road Bins 3.7 hits/bin 1.4 hits/bin 0.9 hits/bin Nearly all bins are on. With or without pixel will not help given this bin size.

7. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov7 Binning in Z Direction Let’s dream we can further divide in z-direction. Consider (1 mm x 1.9 mm) in r-phi & z for B1. In B2, the bin size above is (1.7 mm x 3 mm).

8. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov8 Hits with Finer Bins 0.045 hits/bin 0.084 hits/bin With binning in z-direction, things become not hopeless. In r-z view, there can be 60 roads (when eta = +-1) in each Pt corridor. (x20 than the Thin Road scheme) For each hit on B0, a z range of +- 90 mm on B2 must be checked in order to cover eta +- 1 range. With 3 mm binning in z: –180 mm / 3 mm = 60.

9. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov9 Roads in a Pt Corridor For each hit in B0, there can be many roads in r-z view, depending on eta range (eta +- 0.2 are shown above). The similar set of roads repeat along z.

10. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov10 The R-Z Region to Work With The 2 sigma collision point is +-100 mm and a corresponding cut can be made to reduce fake rate. The region to work with for eta = +- 1 is shown above. For each bin (3mm) at B0, a range of about 180 mm, or 60 bins on B2 is searched. For each bin in B0, there are more than 120 hit patterns. There are about 100 bins at B0. Total number of patterns: 120 x 100 = 12K.

11. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov11 Good, But Using the Thin Road scheme as a starting point, adding x20 z-binning provides finer constraints in r-z view and reducing fake tracks becomes possible. The number of required patterns in B0 alone is 12K x 9K = 108M, (hand estimate only without considering multiple scattering etc.). Increasing AM size? Maybe it is OK for 108M. But it is endless when bins become finer, when strip layers are included, etc. There exist clever approaches. Too many patterns? 1.Reuse 2.Reuse 3.Reuse Tiny Triplet Finder is a scheme of reusing hit patterns.

12. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov12 Classification of Pattern Recognition SoftwareHardware Typical Hardware Resource Saving Approaches Doublet Finding O(n 2 ) for(){ for(){…} } O(n)*O(N) Comparator Array Hash Sorter O(n)*O(N): in RAM Triplet Finding O(n 3 ) for(){ for(){…} } O(n)*O(N 2 ) CAM, Hugh Trans. Tiny Triplet Finder O(n)*O(N*logN) O(n 4 ) for(){ for(){ for() {…} }}}

13. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov13 The Tiny Triplet Finder Triplet: e.g., 3 hits in a straight line in a plane. A semi-formal definition: n items satisfying n-2 constraints. Tiny = Small silicon resource usage. Typical implementation of triplet finding with N bins: –Hough transform: O(N^2) histogram bins. –AM: O(N^2) roads. Tiny Triplet Finder: O(N*log(N)) shifting elements.

14. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov14 Tiny Triplet Finder for FTK, Step (1a) Choose a narrow row (0.5 mm) of pixels along z-axis on B0 as seeds. Divide B1 into bins in r-phi view. The width of B1 bin is chosen to be (0.5 mm)*R1/R0 = 1.1 mm. Define the r-phi corridors. An r-phi corridor contains the seed bin in B0, a bin in B1 and the projected range (3.7 mm) in B2. Note that ranges of different corridors in B2 overlap each other. Up to 7 corridors can be defined.

15. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov15 Tiny Triplet Finder for FTK, Step (1b) For each r-phi corridor, book hit patterns of B2 and B1 (not B0) to register arrays. Bin sizes (r-phi x z): –B1: (1 mm x 1.9 mm) no overlap. –B2: (3.7 mm x 3 mm) with overlap in r-phi but not z. 7 sets of the register arrays are needed corresponding to 7 r-phi corridors.

16. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov16 Tiny Triplet Finder for FTK, Step (2) For a hit in B0, align the two arrays to search for a valid road. For another hit in B0, shift the two arrays to reuse the road patterns. Only one set of the road patterns is needed. Other sets of patterns becomes available by shifting. However, shifting is not free. See next two pages.

17. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov17 Actual Hardware Implementation *R1/R3 *R2/R3 Triplet Map Output To Encoder Bit Array Logarithmic Shifter Bit Array Bit-wise Coincident Logic 1.Fill the B1 and B2 bit arrays. (n1 clock cycles) 2.Loop over B0 hits, shift bit arrays and check for coincidence. (n0 clock cycles) Logarithmic Shifter

18. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov18 The Logarithmic Shifter S1 S2 S4 # of bits: N Shift distance: L # of stages: log 2 L Total LE usage: N*log 2 L A shift of X bit of the bit pattern is done in one clock cycle rather than X cycles. Logarithmic shifter is also known as “barrel shifter”, but the term “logarithmic” reflects nature of implementation, resource usage and propagation delay better. If number of bins x2, just add one layer, rather than x4 of total silicon resource.

19. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov19 Irregular Geometry The modules are tilted and the r at different phi in a module are different. However, relative distances in each Pt corridor are well defined. The bin widths (in both r-phi and z) are adjusted according to the relative distance when the register array is booked. Bins in these corridors are slightly narrower. Bins in these corridors are slightly wider.

20. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov20 Adding Strip Layers The occupancy of the pixel layers are high. Additional constraints may be needed. It may be necessary of do coincidence for all 7 layers. (Perhaps, 6-out-of-7?).

21. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov21 Using Outer Strip Layer Hits as Seeds It is also possible to use strip layer hits as seeds.

22. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov22 Using Middle Strip Layer Hits as Seeds It is possible to choose seeding layer to for better performance.

23. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov23 Tiny Triplet Finder for More Then 3 Layers The triplet finder can also search coincidence in more than 3 layers. A coincidence for 5 layers is shown above. Seed Layer

24. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov24 Booking Bins in z for Strip Layers Use stereo and axial strips to book bins in both r-phi and z- directions. Book a bit pattern for the stereo layer. For any hit strip in axial layer, copy selected bit pattern for z-bins. Fake hits may exist but they can be eliminated after track finding.

25. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov25 How Tiny Triplet Finder Fits into FTK Project From DF (or maybe DO), output an additional copy of HITS information via S-links. Another set of crate carrying Tiny Triplet Finder or other R&D scheme can be tested. The found patterns + hits can be fed into existing track fitter for further cleanup. FTK Pattern Recognition R&D Platform FTK Pattern Recognition R&D Platform FTK Pattern Recognition R&D Platform

26. Jan. 2009Jinyuan Wu & Tiehui Liu, jywu168@fnal.gov26 Summary Using z information in pixel helps. However, number of patterns increases a lot. Tiny Triplet Finder reuse patterns so that binning in z becomes doable. The Tiny Triplet Finder or other pattern recognition schemes can be tested in parallel with the baseline AM approach. We wish to attract colleagues to work on simulation with z- binning to understand efficiency and fake triplet rate.