EM Algorithm Possibilities

updated: 21-Oct-04

Index

1. Two separate EM algorithms are run in the sliding windows chips (0-9). The results are used to fill outputs of different thresholds.
   • High Thresholds: An ATLAS-style algorithm is used (1x2 or 2x1)
   • Low Thresholds: Single TTs are used as EM objects (1x1)
2. Constructing the SW chip output
3. Numbering Convention for data in Sliding Windows Chips.
   Note: each cell in the grid is can also be referenced by its (eta,phi) coordinates within the chip.
   • eta = 0 - 8
   • phi = 0 - 8

ATLAS-Style Algorithm

The algorithm described below is a variant of the 2,1,1 algorithm. That is, no Local Maxima (LM) can exist in the 3x3 ring around a 2x2 region containing a LM. An illustration is available here.

Note: other options than those described in the table below are certainly possible. Some of them are mentioned at the end of this section.

Steps in the algorithm are described in the table below. In the table

• Horizontal refers to the branch of the algorithm that considers 2x1 ROIs in eta x phi
• Vertical refers to the branch of the algorithm that considers 1x2 ROIs in eta x phi
• Object Size refers to the eta x phi region used to make the object described
• SW Inputs refers to the eta x phi region in the SW chip that is required to perform this step for all objects in the chip.

		Horizontal		Vertical
Step	Description	Object Size	SW Inputs	Object Size	SW Inputs
1	Add ICR	1 TT	as needed	1 TT	as needed
2.a	ROI Sums	2x1	TTs: 0-8 x 0-7	1x2	TTs: 0-7 x 0-8
2.b	EM Rim Sums	4x3 ring	ROIs: 2-5 x 2-5	3x4 ring	ROIs: 2-5 x 2-5
2.c	HAD Sums	4x4	ROIs: 2-5 x 2-5	4x4	ROIs: 2-5 x 2-5
3	ROI Compares	2x1	ROIs: 0-8 x 0-7	1x2	ROIs: 0-7 x 0-8
4.a	EM Iso LM > 2a*Rim	2x1	ROIs: 2-5 x 2-5	1x2	ROIs: 2-5 x 2-5
4.b	EM Frac LM > 2b*HAD	2x1	ROIs: 2-5 x 2-5	1x2	ROIs: 2-5 x 2-5
?	Choose H or V	ROIs: 2-5 x 2-5
6	Compare to Et thresh's	4x4	Objs: 2-5 x 2-5	4x4	Objs: 2-5 x 2-5
7	Make Output Words	EM: 2-5 x 2-5

Rules for Choosing Horizontal or Vertical Objects

1. For a given 2x2 region choose the H or V object with the highest Et if both exist
2. If Et(H) = Et(V) then consider the region to be Horizontal (needs to be discussed).

1. Choose after step 2 based on rules 1 and 2 above. highest Et.
2. Choose the H or V LM remaining after step 3, or if both H and V LMs exist for a 2x2 region, use rules 1 and 2 above.
3. Choose the H or V EM object remaining after step 4, or if both H and V objects exist for a 2x2 region, use rules 1 and 2 above.

• Base Algorithm:
  • Use a 2,1,1-based algorithm
  • Use a 2,0,1-based algorithm
• Isolation Ring:
  • Use a 3x4 (H) or 4x3 (V) ring depending on the ROI chosen
  • Use a 4x4 ring for both H and V ROIs
  • Use a 3x3 ring for both H and V ROIs
    Presumably this does not include the TT used in the ROI !
• Hadronic Region:
  • Use a 4x4 region for both H and V ROIs
  • Use a 3x4 (H) or 4x3 (V) region depending on the ROI chosen

Single TT Algorithm

This algorithm would be included to reduce noise effects at low threshold. It is different than a sliding windows type algorithm in the following ways.

• LM Separation: No separation between LMs is possible. Only single TTs over threshold are used.
• Isolation Region: No EM isolation is included in the proposal below. Maybe it should be ?
• EM Fraction: In the current proposal, the single HAD TT behind the EM TT in question is used to define EM fraction. Making a bigger HAD region would require more resources.

Steps of the algorithm are shown in the table below, using the same notation as in the ATLAS table.

Step	Description	Object Size	SW Inputs
1	Add ICR	1 TT	as needed
2	EM Frac EM-TT > 2b*HAD-TT	1x1	TTs: 2-5 x 2-5
6	Compare to Et thresh's	4x4	Objs: 2-5 x 2-5
7	Make Output Words	EM: 2-5 x 2-5

Constructing the SW Chip Output

The SW chip outputs are described here. Briefly, a 3-bit word is used, for each of the 4x4 EM objects sent out from a SW chip, to indicate the highest threshold that was passed by that object. This has the following consequences:

• 7 thresholds are, in principle, available in the system - output = 1-7. Output=0 is used to indicate that no thresholds were passed.
• The construction of L1Cal And/Or terms in the GAB as well as the interpretation of Object data in the Cal-Track and L2 systems assumes that the events passing threshold i+1 are a subset of the events passing threshold i. Thus an output word of 4 will be interpreted as having fired thresholds 4,3,2,1 - even if one of the low thresholds did not actually pass.
• If the above is not the case, then triggers depending on threshold i also depend on whether threshold i+1 fired as well. This may turn out to be a small effect, but it should be kept in mind.

There are three possibilities for constructing the 3-bit output words.

1. One Algorithm: Use only one algorithm for all thresholds (this would probably be the ATLAS algorithm). Construction of the threshold output word would then be straightforward.
2. Inclusive Thresholds:
   
   • out[2..0] = 0    ==> no threshold passed
   • out[2..0] = 1,2 ==> highest thresh passed from Single-TT algo.
   • out[2..0] = 3-7 ==> highest thresh passed from ATLAS algo
   This allows multiple thresholds for the ATLAS and Single-TT algorithms and allows all 7 thresholds to be used, but has the problem of events passing threshold threshold 3 but failing threshold 2.
3. Exclusive Thresholds:
   
   • out[2..0] = 0       ==> no threshold passed
   • out[0]     = 1       ==> passed Single-TT algo thresh
   • out[1..0] = 1,2,3 ==> highest thresh passed from ATLAS algo
   This takes care of the subset problem, but allows only one Single-TT threshold and reduces the number of ATLAS algo thresholds to 3. So the system now has effectively only 4 thresholds.