<html><head><title></title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style2, li.Style2 /*Heading 1*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 13.26315789473684pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style3, li.Style3 /*Heading 2*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: bold; font-style: italic; vertical-align: baseline; } p.Style4, li.Style4 /*Heading 3*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 12pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style5, li.Style5 /*Paragraph*/ { margin-left: 0pt; margin-right: 0pt; text-indent: 21pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style6, li.Style6 /*List*/ { margin-left: 14.25pt; margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-61" style="position: absolute; top: -0.75pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1" style="text-indent: 0pt; list-style-type: none;"><span style="font-size: 13.26315789473684pt; "><b>Probability of finding the correct hit using weighted chi- square hit selection.</b></span></p></div><div id="mc-region-484" style="position: absolute; top: 44.25pt; left: 6pt; width: 82.25pt; "><a name="" /><p class="Style1">31-Dec-2006</p><p class="Style1">H. Wieman</p><p class="Style1">V. Perevoztchikov</p></div><div id="mc-region-684" style="position: absolute; top: 92.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1"><b>Abstract</b></p><p class="Style1">This note provides a derivation of the probability for selecting the correct hit when using a minimum weighted chi-square to make the hit selection. A comparison between two hit selection methods is presented for the case where the track projection and detector resolution have different values in the x and y direction.</p></div><div id="mc-region-686" style="position: absolute; top: 170.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1"><b>Introduction</b></p><p class="Style1">In tracking from one detector layer to the next one has a track model that points to a detector layer. The detector layer has multiple hits, only one of which belongs with the modeled track. In the STAR tracking code the hit selection is done using a weighted minimum chi-square to pick the hit to associate with the track. This method is the preferred way to select the hit where the track model projection uncertainly is different in the x and y direction. It is of interest to quantify the probability for correct hit assignment using this method. In this document we derive a formula for this probability. The derivation is an extension of the <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2D.htm">derivation used for a simpler hit selection method</a></span></u></b>, namely just picking the closest hit. The approach presented here confirms the earlier derivation of Perevoztchikov.</p></div><div id="mc-region-97" style="position: absolute; top: 300.75pt; left: 12pt; width: 132.5pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>Items to be presented:</b></span></p></div><div id="mc-region-99" style="position: absolute; top: 332.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"> 1. Formula for probability of finding the correct hit with asymmetric pointing resolution and a finite rectangular search window. Loss of efficiency can result from either finding the wrong hit or from finding no hit within the search window.</p><p class="Style1"></p><p class="Style1">2. Formula for probability of picking the correct hit with an asymmetric pointing resolution and an infinite search window.</p><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;"></p><p class="Style1">3. Derivation of the formula</p><p class="Style1"></p><p class="Style1">4. An example result comparing the minimum weighted chi-squared hit selection method with the simple closest hit method.</p></div><div id="mc-region-129" style="position: absolute; top: 486.75pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>1. Probability for picking the correct hit with a finite search window using the chi-squared method</b></span></p></div><div id="mc-region-289" style="position: absolute; top: 560.25pt; left: 372pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 1</p></div><div id="mc-region-290" style="position: absolute; top: 534.75pt; left: 6pt; width: 265.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0002_136693484.PNG" id="generatedImage290" path=".\HitFinding2DXsq_images/IMG0002_136693484.PNG" style="width: 265.5pt;height: 57.75pt;"></img></div><div id="mc-region-292" style="position: absolute; top: 615pt; left: 18pt; width: 24.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0003_136693484.PNG" id="generatedImage292" path=".\HitFinding2DXsq_images/IMG0003_136693484.PNG" style="width: 24.75pt;height: 12.75pt;"></img></div><div id="mc-region-155" style="position: absolute; top: 644.25pt; left: 24pt; width: 7.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0004_136693484.PNG" id="generatedImage155" path=".\HitFinding2DXsq_images/IMG0004_136693484.PNG" style="width: 7.5pt;height: 13.5pt;"></img></div><div id="mc-region-137" style="position: absolute; top: 644.25pt; left: 66pt; width: 233.75pt; "><a name="" /><p class="Style1">is the density of hits per area on the detector surface</p></div><div id="mc-region-154" style="position: absolute; top: 674.25pt; left: 24pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0005_136693484.PNG" id="generatedImage154" path=".\HitFinding2DXsq_images/IMG0005_136693484.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-139" style="position: absolute; top: 674.25pt; left: 66pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the x direction ( x uncertainty)</p></div><div id="mc-region-153" style="position: absolute; top: 704.25pt; left: 24pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0006_136693500.PNG" id="generatedImage153" path=".\HitFinding2DXsq_images/IMG0006_136693500.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-141" style="position: absolute; top: 704.25pt; left: 66pt; width: 313.25pt; "><a name="" /><p class="Style1">is the Gaussian distribution parameter in the y direction ( y uncertainty)</p></div><div id="mc-region-152" style="position: absolute; top: 735pt; left: 24pt; width: 7.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0007_136693500.PNG" id="generatedImage152" path=".\HitFinding2DXsq_images/IMG0007_136693500.PNG" style="width: 7.5pt;height: 12.75pt;"></img></div><div id="mc-region-145" style="position: absolute; top: 734.25pt; left: 66pt; width: 292.25pt; "><a name="" /><p class="Style1">is the half width of the rectangular search window in the x direction</p></div><div id="mc-region-156" style="position: absolute; top: 777pt; left: 24pt; width: 6.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0008_136693500.PNG" id="generatedImage156" path=".\HitFinding2DXsq_images/IMG0008_136693500.PNG" style="width: 6.75pt;height: 12.75pt;"></img></div><div id="mc-region-150" style="position: absolute; top: 776.25pt; left: 66pt; width: 292.25pt; "><a name="" /><p class="Style1">is the half width of the rectangular search window in the y direction</p></div><div id="mc-region-293" style="position: absolute; top: 822.75pt; left: 12pt; width: 395.75pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>2. Probability for picking the correct hit with an infinite search window using the chi-squared method</b></span></p></div><div id="mc-region-294" style="position: absolute; top: 872.25pt; left: 24pt; width: 98.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0009_136693500.PNG" id="generatedImage294" path=".\HitFinding2DXsq_images/IMG0009_136693500.PNG" style="width: 98.25pt;height: 32.25pt;"></img></div><div id="mc-region-295" style="position: absolute; top: 878.25pt; left: 264pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 2 </p></div><div id="mc-region-303" style="position: absolute; top: 926.25pt; left: 18pt; width: 284.75pt; "><a name="" /><p class="Style1">Note, as expected eq. 1 reduces to the <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2D.htm">closest hit form</a></span></u></b> when </p></div><div id="mc-region-493" style="position: absolute; top: 926.25pt; left: 294pt; width: 33.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0010_136693500.PNG" id="generatedImage493" path=".\HitFinding2DXsq_images/IMG0010_136693500.PNG" style="width: 33.75pt;height: 17.25pt;"></img></div><div id="mc-region-158" style="position: absolute; top: 966.75pt; left: 6pt; width: 137pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; "><b>3. Derivation of formula</b></span></p></div><div id="mc-region-161" style="position: absolute; top: 992.25pt; left: 6pt; width: 437.75pt; "><a name="" /><p class="Style1">The projected track pointing resolution to the detector layer is a 2D Gaussian distribution, that is the probability density for where the correct hit will be found relative to the projected track point is</p></div><div id="mc-region-162" style="position: absolute; top: 1034.25pt; left: 6pt; width: 181.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0011_136693500.PNG" id="generatedImage162" path=".\HitFinding2DXsq_images/IMG0011_136693500.PNG" style="width: 181.5pt;height: 44.25pt;"></img></div><div id="mc-region-183" style="position: absolute; top: 1052.25pt; left: 252pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 3</p></div><div id="mc-region-176" style="position: absolute; top: 1094.25pt; left: 12pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0012_136693500.PNG" id="generatedImage176" path=".\HitFinding2DXsq_images/IMG0012_136693500.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-178" style="position: absolute; top: 1094.25pt; left: 60pt; width: 365pt; "><a name="" /><p class="Style1">is the projection resolution in the x direction plus hit resolution of the detector plane</p></div><div id="mc-region-175" style="position: absolute; top: 1124.25pt; left: 12pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0013_136693500.PNG" id="generatedImage175" path=".\HitFinding2DXsq_images/IMG0013_136693500.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-177" style="position: absolute; top: 1124.25pt; left: 60pt; width: 370.25pt; "><a name="" /><p class="Style1">is the projection resolution in the y direction plus hit resolution of the detector plane</p></div><div id="mc-region-185" style="position: absolute; top: 1160.25pt; left: 12pt; width: 432.5pt; "><a name="" /><p class="Style1">It is assumed that the detectors are correctly spatially mapped and that the distribution is centered on the true hit location</p></div><div id="mc-region-187" style="position: absolute; top: 1202.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">The probability for the weighted minimum chi-squareed selected hit being the correct hit is found by integrating the above distribution for the projected hit location over the plane weighted by the probability that no background hit has a smaller chi-square closer to the track projection point.</p></div><div id="mc-region-189" style="position: absolute; top: 1255.5pt; left: 6pt; width: 159pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0014_136693515.PNG" id="generatedImage189" path=".\HitFinding2DXsq_images/IMG0014_136693515.PNG" style="width: 159pt;height: 51.75pt;"></img></div><div id="mc-region-191" style="position: absolute; top: 1274.25pt; left: 204pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 4</p></div><div id="mc-region-192" style="position: absolute; top: 1274.25pt; left: 282pt; width: 134.75pt; "><a name="" /><p class="Style1">probability for min chi-squared selected hit being correct.</p></div><div id="mc-region-314" style="position: absolute; top: 1329pt; left: 18pt; width: 7.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0015_136693515.PNG" id="generatedImage314" path=".\HitFinding2DXsq_images/IMG0015_136693515.PNG" style="width: 7.5pt;height: 12.75pt;"></img></div><div id="mc-region-315" style="position: absolute; top: 1328.25pt; left: 66pt; width: 292.25pt; "><a name="" /><p class="Style1">is the half width of the rectangular search window in the x direction</p></div><div id="mc-region-316" style="position: absolute; top: 1353pt; left: 18pt; width: 6.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0016_136693515.PNG" id="generatedImage316" path=".\HitFinding2DXsq_images/IMG0016_136693515.PNG" style="width: 6.75pt;height: 12.75pt;"></img></div><div id="mc-region-317" style="position: absolute; top: 1352.25pt; left: 66pt; width: 292.25pt; "><a name="" /><p class="Style1">is the half width of the rectangular search window in the y direction</p></div><div id="mc-region-494" style="position: absolute; top: 1395pt; left: 12pt; width: 37.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0017_136693515.PNG" id="generatedImage494" path=".\HitFinding2DXsq_images/IMG0017_136693515.PNG" style="width: 37.5pt;height: 16.5pt;"></img></div><div id="mc-region-496" style="position: absolute; top: 1394.25pt; left: 84pt; width: 344pt; "><a name="" /><p class="Style1">Probability that there are no hits with a smaller weighted chi-square separation closer to the center projection than the point at x,y</p></div><div id="mc-region-512" style="position: absolute; top: 1430.25pt; left: 18pt; width: 191pt; "><a name="" /><p class="Style1">That is the probability that there are no hits</p></div><div id="mc-region-513" style="position: absolute; top: 1431pt; left: 234pt; width: 32.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0018_136693515.PNG" id="generatedImage513" path=".\HitFinding2DXsq_images/IMG0018_136693515.PNG" style="width: 32.25pt;height: 16.5pt;"></img></div><div id="mc-region-511" style="position: absolute; top: 1430.25pt; left: 294pt; width: 45.5pt; "><a name="" /><p class="Style1">such that</p></div><div id="mc-region-506" style="position: absolute; top: 1456.5pt; left: 48pt; width: 108.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0019_136693515.PNG" id="generatedImage506" path=".\HitFinding2DXsq_images/IMG0019_136693515.PNG" style="width: 108.75pt;height: 48pt;"></img></div><div id="mc-region-515" style="position: absolute; top: 1472.25pt; left: 216pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 5</p></div><div id="mc-region-520" style="position: absolute; top: 1520.25pt; left: 12pt; width: 173.75pt; "><a name="" /><p class="Style1">We use eq. 5 to obtain the probability: </p></div><div id="mc-region-519" style="position: absolute; top: 1521pt; left: 216pt; width: 37.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0020_136693515.PNG" id="generatedImage519" path=".\HitFinding2DXsq_images/IMG0020_136693515.PNG" style="width: 37.5pt;height: 16.5pt;"></img></div><div id="mc-region-522" style="position: absolute; top: 1550.25pt; left: 18pt; width: 271.25pt; "><a name="" /><p class="Style1">by dividing both sides of eq. 5 by the right side of eq. 5 giving:</p></div><div id="mc-region-530" style="position: absolute; top: 1582.5pt; left: 6pt; width: 158.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0021_136693531.PNG" id="generatedImage530" path=".\HitFinding2DXsq_images/IMG0021_136693531.PNG" style="width: 158.25pt;height: 67.5pt;"></img></div><div id="mc-region-531" style="position: absolute; top: 1598.25pt; left: 264pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 6</p></div><div id="mc-region-533" style="position: absolute; top: 1670.25pt; left: 6pt; width: 286.25pt; "><a name="" /><p class="Style1">We can now replace the &lt; in eq. with an = to get the equation in</p></div><div id="mc-region-538" style="position: absolute; top: 1671pt; left: 306pt; width: 9.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0022_136693531.PNG" id="generatedImage538" path=".\HitFinding2DXsq_images/IMG0022_136693531.PNG" style="width: 9.75pt;height: 16.5pt;"></img></div><div id="mc-region-539" style="position: absolute; top: 1670.25pt; left: 342pt; width: 20.75pt; "><a name="" /><p class="Style1">and</p></div><div id="mc-region-540" style="position: absolute; top: 1671pt; left: 390pt; width: 11.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0023_136693531.PNG" id="generatedImage540" path=".\HitFinding2DXsq_images/IMG0023_136693531.PNG" style="width: 11.25pt;height: 16.5pt;"></img></div><div id="mc-region-545" style="position: absolute; top: 1702.5pt; left: 6pt; width: 158.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0024_136693531.PNG" id="generatedImage545" path=".\HitFinding2DXsq_images/IMG0024_136693531.PNG" style="width: 158.25pt;height: 67.5pt;"></img></div><div id="mc-region-546" style="position: absolute; top: 1718.25pt; left: 264pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 7</p></div><div id="mc-region-542" style="position: absolute; top: 1784.25pt; left: 6pt; width: 438.5pt; "><a name="" /><p class="Style1">that defines a closed curve that is the boundary containing all points that have a weighted chi-squared less than the point </p></div><div id="mc-region-554" style="position: absolute; top: 1797pt; left: 150pt; width: 21.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0025_136693531.PNG" id="generatedImage554" path=".\HitFinding2DXsq_images/IMG0025_136693531.PNG" style="width: 21.75pt;height: 12.75pt;"></img></div><div id="mc-region-551" style="position: absolute; top: 1838.25pt; left: 6pt; width: 157.25pt; "><a name="" /><p class="Style1">conveniently this curve is an ellipse</p></div><div id="mc-region-552" style="position: absolute; top: 1822.5pt; left: 174pt; width: 63.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0026_136693531.PNG" id="generatedImage552" path=".\HitFinding2DXsq_images/IMG0026_136693531.PNG" style="width: 63.75pt;height: 43.5pt;"></img></div><div id="mc-region-558" style="position: absolute; top: 1892.25pt; left: 6pt; width: 86.75pt; "><a name="" /><p class="Style1">which has an area:</p></div><div id="mc-region-559" style="position: absolute; top: 1892.25pt; left: 162pt; width: 58.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0027_136693531.PNG" id="generatedImage559" path=".\HitFinding2DXsq_images/IMG0027_136693531.PNG" style="width: 58.5pt;height: 13.5pt;"></img></div><div id="mc-region-565" style="position: absolute; top: 1928.25pt; left: 6pt; width: 402.5pt; "><a name="" /><p class="Style1">The probability for this area, defined by eq. 7, to be empty of hits is given by Poisson's formula, P(0):</p></div><div id="mc-region-564" style="position: absolute; top: 1971.75pt; left: 18pt; width: 271.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0028_136693531.PNG" id="generatedImage564" path=".\HitFinding2DXsq_images/IMG0028_136693531.PNG" style="width: 271.5pt;height: 42.75pt;"></img></div><div id="mc-region-567" style="position: absolute; top: 1988.25pt; left: 342pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 8</p></div><div id="mc-region-568" style="position: absolute; top: 2037pt; left: 18pt; width: 24.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0029_136693546.PNG" id="generatedImage568" path=".\HitFinding2DXsq_images/IMG0029_136693546.PNG" style="width: 24.75pt;height: 12.75pt;"></img></div><div id="mc-region-573" style="position: absolute; top: 2060.25pt; left: 24pt; width: 7.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0030_136693546.PNG" id="generatedImage573" path=".\HitFinding2DXsq_images/IMG0030_136693546.PNG" style="width: 7.5pt;height: 13.5pt;"></img></div><div id="mc-region-572" style="position: absolute; top: 2060.25pt; left: 84pt; width: 141.5pt; "><a name="" /><p class="Style1">is the mean area density of hits</p></div><div id="mc-region-220" style="position: absolute; top: 2096.25pt; left: 12pt; width: 191pt; "><a name="" /><p class="Style1">The integral, eq. 4, using eq. 3 and eq. 8 is</p></div><div id="mc-region-221" style="position: absolute; top: 2112.75pt; left: 12pt; width: 360pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0031_136693546.PNG" id="generatedImage221" path=".\HitFinding2DXsq_images/IMG0031_136693546.PNG" style="width: 360pt;height: 69pt;"></img></div><div id="mc-region-575" style="position: absolute; top: 2138.25pt; left: 396pt; width: 26.75pt; "><a name="" /><p class="Style1">eq. 9</p></div><div id="mc-region-576" style="position: absolute; top: 2204.25pt; left: 6pt; width: 352.25pt; "><a name="" /><p class="Style1">which, another convenience, separates into a product of two integrals of the form</p></div><div id="mc-region-577" style="position: absolute; top: 2238pt; left: 6pt; width: 69pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0032_136693546.PNG" id="generatedImage577" path=".\HitFinding2DXsq_images/IMG0032_136693546.PNG" style="width: 69pt;height: 40.5pt;"></img></div><div id="mc-region-233" style="position: absolute; top: 2306.25pt; left: 6pt; width: 329.75pt; "><a name="" /><p class="Style1">this solves to give the final result for the probability of picking the correct hit</p></div><div id="mc-region-584" style="position: absolute; top: 2346.75pt; left: 6pt; width: 265.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0033_136693546.PNG" id="generatedImage584" path=".\HitFinding2DXsq_images/IMG0033_136693546.PNG" style="width: 265.5pt;height: 57.75pt;"></img></div><div id="mc-region-583" style="position: absolute; top: 2378.25pt; left: 354pt; width: 32pt; "><a name="" /><p class="Style1">eq. 10</p></div><div id="mc-region-323" style="position: absolute; top: 2420.25pt; left: 12pt; width: 236.75pt; "><a name="" /><p class="Style1">For an infinite search window the probability becomes</p></div><div id="mc-region-328" style="position: absolute; top: 2456.25pt; left: 12pt; width: 98.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0034_136693546.PNG" id="generatedImage328" path=".\HitFinding2DXsq_images/IMG0034_136693546.PNG" style="width: 98.25pt;height: 32.25pt;"></img></div><div id="mc-region-329" style="position: absolute; top: 2462.25pt; left: 324pt; width: 32pt; "><a name="" /><p class="Style1">eq. 11</p></div><div id="mc-region-397" style="position: absolute; top: 2525.25pt; left: 6pt; width: 348.5pt; "><a name="" /><p class="Style1"><span style="font-size: 13.26315789473684pt; "><b>4. example comparing the two hit selection methods</b></span></p></div><div id="mc-region-586" style="position: absolute; top: 2564.25pt; left: 12pt; width: 189.5pt; "><a name="" /><p class="Style1">by the weighted chi-square method, eq 10:</p></div><div id="mc-region-589" style="position: absolute; top: 2598.75pt; left: 6pt; width: 265.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0035_136693562.PNG" id="generatedImage589" path=".\HitFinding2DXsq_images/IMG0035_136693562.PNG" style="width: 265.5pt;height: 57.75pt;"></img></div><div id="mc-region-596" style="position: absolute; top: 2630.25pt; left: 324pt; width: 32pt; "><a name="" /><p class="Style1">eq. 10</p></div><div id="mc-region-592" style="position: absolute; top: 2684.25pt; left: 6pt; width: 124.25pt; "><a name="" /><p class="Style1">by the <b><u><span style="color: #00f; text-decoration: underline ;"><a href="http://www-rnc.lbl.gov/~wieman/HitFinding2D.htm">closest hit method</a></span></u></b>:</p></div><div id="mc-region-598" style="position: absolute; top: 2750.25pt; left: 312pt; width: 32pt; "><a name="" /><p class="Style1">eq. 12</p></div><div id="mc-region-594" style="position: absolute; top: 2718.75pt; left: 6pt; width: 247.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0036_136693562.PNG" id="generatedImage594" path=".\HitFinding2DXsq_images/IMG0036_136693562.PNG" style="width: 247.5pt;height: 71.25pt;"></img></div><div id="mc-region-656" style="position: absolute; top: 2828.25pt; left: 6pt; width: 107.75pt; "><a name="" /><p class="Style1">Compare methods with </p></div><div id="mc-region-657" style="position: absolute; top: 2828.25pt; left: 114pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0037_136693562.PNG" id="generatedImage657" path=".\HitFinding2DXsq_images/IMG0037_136693562.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-658" style="position: absolute; top: 2828.25pt; left: 132pt; width: 77.75pt; "><a name="" /><p class="Style1">starting equal to </p></div><div id="mc-region-659" style="position: absolute; top: 2828.25pt; left: 216pt; width: 10.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0038_136693562.PNG" id="generatedImage659" path=".\HitFinding2DXsq_images/IMG0038_136693562.PNG" style="width: 10.5pt;height: 17.25pt;"></img></div><div id="mc-region-660" style="position: absolute; top: 2828.25pt; left: 246pt; width: 71.75pt; "><a name="" /><p class="Style1">and going up to</p></div><div id="mc-region-661" style="position: absolute; top: 2828.25pt; left: 318pt; width: 22.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0039_136693562.PNG" id="generatedImage661" path=".\HitFinding2DXsq_images/IMG0039_136693562.PNG" style="width: 22.5pt;height: 17.25pt;"></img></div><div id="mc-region-662" style="position: absolute; top: 2846.25pt; left: 6pt; width: 339.5pt; "><a name="" /><p class="Style1">An infinite search window is used for this example.</p></div><div id="mc-region-351" style="position: absolute; top: 2876.25pt; left: 12pt; width: 50.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0040_136693562.PNG" id="generatedImage351" path=".\HitFinding2DXsq_images/IMG0040_136693562.PNG" style="width: 50.25pt;height: 17.25pt;"></img></div><div id="mc-region-607" style="position: absolute; top: 2876.25pt; left: 90pt; width: 185.75pt; "><a name="" /><p class="Style1">Track pointing resolution in the x direction</p></div><div id="mc-region-601" style="position: absolute; top: 2912.25pt; left: 12pt; width: 40.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0041_136693562.PNG" id="generatedImage601" path=".\HitFinding2DXsq_images/IMG0041_136693562.PNG" style="width: 40.5pt;height: 33.75pt;"></img></div><div id="mc-region-613" style="position: absolute; top: 2918.25pt; left: 96pt; width: 73.25pt; "><a name="" /><p class="Style1">Area hit density</p></div><div id="mc-region-644" style="position: absolute; top: 2976pt; left: 6pt; width: 334.5pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0044_136693578.PNG" id="generatedImage644" path=".\HitFinding2DXsq_images/IMG0044_136693578.PNG" style="width: 334.5pt;height: 281.25pt;"></img></div><div id="mc-region-645" style="position: absolute; top: 3212.25pt; left: 216pt; width: 12pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0045_136693578.PNG" id="generatedImage645" path=".\HitFinding2DXsq_images/IMG0045_136693578.PNG" style="width: 12pt;height: 17.25pt;"></img></div><div id="mc-region-646" style="position: absolute; top: 3212.25pt; left: 234pt; width: 20.25pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0046_136693578.PNG" id="generatedImage646" path=".\HitFinding2DXsq_images/IMG0046_136693578.PNG" style="width: 20.25pt;height: 13.5pt;"></img></div><div id="mc-region-678" style="position: absolute; top: 3266.25pt; left: 12pt; width: 207.5pt; "><a name="" /><p class="Style1">The two methods are identical as expected for </p></div><div id="mc-region-677" style="position: absolute; top: 3266.25pt; left: 222pt; width: 33.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0047_136693578.PNG" id="generatedImage677" path=".\HitFinding2DXsq_images/IMG0047_136693578.PNG" style="width: 33.75pt;height: 17.25pt;"></img></div><div id="mc-region-680" style="position: absolute; top: 3290.25pt; left: 12pt; width: 397.25pt; "><a name="" /><p class="Style1">The minimum chi-squared method is nearly a factor of two better for the above example for </p></div><div id="mc-region-682" style="position: absolute; top: 3314.25pt; left: 12pt; width: 45.75pt; "><a name="" /><img border="0" src=".\HitFinding2DXsq_images/IMG0048_136693578.PNG" id="generatedImage682" path=".\HitFinding2DXsq_images/IMG0048_136693578.PNG" style="width: 45.75pt;height: 17.25pt;"></img></div><div></div></body></html>