ParSA Results - Revision history

DemasMa at 19:02, 2 June 2008

2008-06-02T19:02:14Z

DemasMa: /* MIR Performance */

2008-06-02T19:00:24Z

MIR Performance

DemasMa: /* Optimizing Run Length */

2008-06-02T18:58:22Z

Optimizing Run Length

DemasMa: /* Optimizing Run Length */

2008-06-02T18:47:02Z

Optimizing Run Length

DemasMa at 18:45, 2 June 2008

2008-06-02T18:45:47Z

DemasMa at 18:36, 2 June 2008

2008-06-02T18:36:04Z

DemasMa at 18:30, 2 June 2008

2008-06-02T18:30:30Z

DemasMa at 17:45, 2 June 2008

2008-06-02T17:45:46Z

DemasMa at 17:00, 2 June 2008

2008-06-02T17:00:41Z

DemasMa at 16:11, 2 June 2008

2008-06-02T16:11:46Z

@@ Line 31: / Line 31: @@
-{|width="50%"
+{|width="80%"
 |align="right"|
 <math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>
@@ Line 39: / Line 39: @@
 where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters.  ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept.  The expression for the Bayesian estimator <math> \hat{p} </math> is given by
-{|width="50%"
+{|width="80%"
 |align="right"|
 <math> \hat{p} = \frac{n_f + 1}{N + 2} </math>
@@ Line 68: / Line 68: @@
 The optimal run length can be determined now that the ''K'' and <math>\alpha</math> parameters are known.  Given that the the total number of steps ''T'' is given by the following expression
-{|align=center
+{|width="80%"
-|<math> T=Rn </math>
+|align="right"|
-|(3)
+<math> T=Rn </math>
 |}
 where ''R'' is the number of runs and ''n'' is the run length.  The probability of obtaining one successful run out of ''R'' runs is given by
-{|align=center
+{|width="80%"
-|<math> P_s = 1 - \left(\frac{K}{n}\right)^{R\alpha} </math>
+|align="right"|
-|(4)
+<math> P_s = 1 - \left(\frac{K}{n}\right)^{R\alpha} </math>
 |}
 Using the method of Lagrange multipliers, the optimal step number can be found to be
 <math> n = K e </math>
 and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.

@@ Line 30: / Line 30: @@
 The ParSA library documentation gives the following equation which can be used to estimate performance of an MIR run
-{|align=center
-|<math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>
+{|width="50%"
-|(1)
+|align="right"|
 |}
 where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters.  ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept.  The expression for the Bayesian estimator <math> \hat{p} </math> is given by
-{|align=center
+{|width="50%"
-|<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>
+|align="right"|
-|(2)
+<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>
 |}

@@ Line 79: / Line 79: @@
 Using the method of Lagrange multipliers, the optimal step number can be found to be
-{|width="80%"|align="right"|<math> n = K e </math>|align="center"| width="80"|(5)|}
+<math> n = K e </math>
 and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.

@@ Line 79: / Line 79: @@
 Using the method of Lagrange multipliers, the optimal step number can be found to be
-{|width="80%"
+{|width="80%"|align="right"|<math> n = K e </math>|align="center"| width="80"|(5)|}
 and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.

@@ Line 79: / Line 79: @@
 Using the method of Lagrange multipliers, the optimal step number can be found to be
-{|align=center
+{|width="80%"
 |<math> n = K e </math>
 |(5)
 |}
 and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.

@@ Line 2: / Line 2: @@
 === Interferograms ===
-The set of images in Fig. 1 and Fig. 2 depict the test interferogram analyzed using ParSA and the best interferogram solution found by ParSA.  When compared visually, the two interferograms are nearly indistinguishable.  This comparison shows that the solution found by the ParSA algorithm is a good one.
+The set of images in Fig. 1 and Fig. 2 depict the test interferogram analyzed and the best interferogram solution found by simulated annealing.  When compared visually, the two interferograms are nearly indistinguishable.  This comparison shows that the solution found by simulated annealing is a good one.
 {|align=center
@@ Line 17: / Line 17: @@
 |}
-The solutions found by simulated annealing feature two ambiguities that are a result of the nature of light.  The first ambiguity results from the fact that a mirror transformation of the surfaces about a vertical plane do not change the interferogram.  The second ambiguity is a result of the periodic nature of light and can be seen in differences present in the distances between surfaces between the test and solution surfaces.
+The solutions found by simulated annealing feature two ambiguities.  The first ambiguity results from the fact that a mirror transformation of the surfaces about a vertical plane do not change the interferogram.  The second ambiguity is a result of the periodic nature of light and can be seen in differences present in the distances between surfaces between the test and solution surfaces.
 {|align=center

@@ Line 32: / Line 32: @@
 <math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>
-where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters.  ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept.  The expression for the Bayesian
+where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters.  ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept.  The expression for the Bayesian estimator <math> \hat{p} </math> is given by
 {|align=center
@@ Line 41: / Line 45: @@
 ''' ''alpha'' = 0.5 '''
-Fig. 7 shows the log plot of probability of non-convergence versus run length.  The
+Fig. 7 shows the log plot of probability of non-convergence versus run length for a MIR run with the ''alpha'' cooling parameter set to 0.5.  The slope and y-intercept of the linear least-squares fit are <math>-0.0273 \pm 0.0103</math> and <math>0.297 \pm 0.127</math> with a reduced <math>\chi^2</math> value of 1.0.  ''K'' and <math>\alpha</math> were then determined to be <math>(5.31 \pm 2.4) \times 10^4</math> and <math>0.0273 \pm 0.0103</math>.
 ''' ''alpha'' = 0.9 '''
 === Sequential Performance ===

@@ Line 28: / Line 28: @@
 === MIR Performance ===
-The ParSA library documentation gives an equation which can be used to determine
+The ParSA library documentation gives the following equation which can be used to estimate performance of an MIR run
 {|align=center