Simulated annealing algorithm to detect North‐South opening hyperbolas

Chapter 3 Proposed System

3.2 Simulated annealing parameter detection system

3.2.2 Simulated annealing algorithm to detect North‐South opening hyperbolas

For seismic applications, patterns of reflection wave are North-South opening hyperbolas. Besides, patterns of direct waves are asymptotes of hyperbolas [3]-[5].

Equation of a North-South opening hyperbola is f the distance from a point to a pattern becomes

|

We consider these properties and modify the algorithm to be just for North-South opening hyperbolas. This algorithm proves the detected patterns have the properties of North-South opening hyperbolas.

Algorithm: SA algorithm to detect parameter vectors of K North-South opening

hyperbolas.

Input: N points in an image. Set K as the number of patterns.

Output: A set of detected K parameter vectors.

Step 1: Initialization.

Step 2: Randomly change parameter vectors and decide the new parameter vectors in the same temperature.

For m = 1 to Nt (Nt trials in a temperature) For k = 1 to K (k is the index of the pattern)

Start a trial, including (a), (b), and (c) in the following.

(a) Randomly change the center of the kth pattern:

Calculate the new energy E(P’) from N points to K patterns. Using Metropolis criterion decides whether or not to accept P’: for the new energy less than or equal to the original one, ∆E = E(P’) - E(P) ≤ 0, accept P’. Otherwise, new energy is higher than the original one, ∆E = E(P’) - E(P) > 0. In this case, compute prob = exp[-∆E/T(t)], and generate a random number x uniformly distributed over (0, 1). If prob ≥ x, accept P’; otherwise, reject it, and keep P.

(b) Randomly change the shape parameters:

abn

Similar to Step 2(a), calculate the new energy E(P’) from N points to K patterns. Using Metropolis criterion decides whether or not to accept P’.

End for k End for m

Step 3: Cool the System.

Decrease temperature T according to the cooling function (9), T(t)=

T

max×0.98^(t-1), for t = 1, 2, 3, …, and repeat Step 2, and 3 until the temperature is low enough, for examples, repeat 500 times.

Fig. 11 - Fig. 14 illustrate a possible procedure of SA algorithm. The example has some points on an ellipse. There is only one pattern K = 1, so P = p1. Fig. 11 shows the trial of the center.

In the step 1, at a certain temperature, the center (mx, my) is randomly displaced to (m’x, m’y), and the parameter vector changes from P = p1 = [m1,x, m1,y, a1, b1, f1]^Tto P’

= p1’ = [m’1,x, m’1,y, a1, b1, θ1, f1]^T. The resulting energy E(P’) is less than the original energy E(P). In this case, the trial parameter vector is accepted and set it as the starting parameter vector in the step 2, P← P’ = p1’ = [m’1,x, m’1,y, a1, b1, θ1, f1]^T.

In the step 2, as shown in Fig. 12, the shape parameters b and a are randomly changed to b’ and a’, and the parameter vector p1 becomes p1’, and this results in the energy E(P’) > E(P). In this case, Metropolis criterion decides whether or not to accept the trial parameter by comparing prob = exp[-∆E/T(t)] and random number x.

Assuming prob ≥ x, the trial parameter vector with a higher energy is still preserved and set it as starting parameter vector in step 3, P ← P’= p1’ = [m1,x, m1,y, a’1, b’1, θ1,

f

1]^T.

18 Fig. 12. A trial of shape parameters b and a. (a): original parameter. (b): trial parameter. (c): preserved parameter.

In the step 3, the trial of the rotation angle gives the trial parameter vector P’ = [m1,x, m1,y, a1, b1, θ’1, f1]^T, and the resulting energy E(P’) < E(P). The trial parameter vectors is preserved and set as the starting parameter vector in the step 4, P ← P’. Fig.

13 illustrates the step 3.

Fig. 13. A trial of rotation angle θ. (a): original parameter. (b): trial parameter. (c):

preserved parameter.

In the step 4, the trial of the size gives the trial parameter vector P’ = [m1,x, m1,y, a1,

b

1, θ1, f’1]^T, and the resulting energy E(P’) > E(P). Metropolis criterion decides the preserved parameter vector in the same way, and this time, prob < x, so the trial parameter vector P’ is rejected. The starting parameter vector in the next step is still the original one, P ← P. Fig. 14 illustrates the step 4.

E(P) E(P’)

Δ

=

(

' ) −

(

) > 0

prob=exp[-∆E/T(t)] < r

Reject P’; keep P

(a) (b) (c)

Fig. 14. A trial of f. (a): original parameter. (b): trial parameter. (c): preserved parameter.

Chapter 4 Implementation and Experimental Results

The experiments are first on simulated pattern detections in images with size 50 × 50. First, we use the general algorithm to detect hyperbolas, ellipses, and consider lines as asymptotes of hyperbolas. Then, we use the algorithm just for North-South opening hyperbolas. Experiments on determination of the number of patterns are also shown. In seismic applications, we detect line pattern of direct wave and hyperbolic pattern of reflection wave in the simulated and real seismic data.

4.1 Detection of ellipses, hyperbolas, and lines

The general algorithm can detect circles, ellipses, hyperbolas, and treats line as asymptote.

In initial stage, mx and my are randomly distributed over (0, 50), fk = 0, ak = 1, bk = 1, and θk = 0. The cooling function is as (9) with a high enough temperature, Tmax = 500. We have 100 trials in the same temperature. The temperature decreases 500 times to T = 0.0209, and this temperature is low enough. Constants αm = 1, αab = 1, αθ

= 2, and αf = 2.

Simulation 1: ellipses

Fig. 15 and Fig. 16 show the results of detecting ellipses. There are two ellipses in each figure and each ellipse has 50 points. Data are disturbed by Gaussian noise with zero mean and variance is 0.5, N(0, 0.5) × N(0, 0.5). The error vs. cooling cycles shows that the error oscillates at high temperature and goes toward lower energy and becomes stable as the temperature decreasing.

(a)

(b)

Fig. 15. Detection of ellipses – (a): 2 ellipses with noise. (b): error plot of (a) with cooling cycles.

(a)

(b)

Fig. 16. Detection of ellipses – (a): 1 ellipse and 1 circle with noise. (b): error plot of (a) with cooling cycles.

Simulation 2: hyperbolas

Result of detecting hyperbolas are shown in Fig. 17 where K = 2. Patterns are with Gaussian noise N(0, 0.5)×N(0, 0.5). Figures of energy vs. cooling cycles are also shown.

(a)

(b)

Fig. 17. Detection of hyperbolas − (a): 2 hyperbolas with noise. (b): error plot of (a) with cooling cycles.

(a)

(b)

Fig. 18. Detection of hyperbolas − (a): 2 hyperbolas with noise. (b): error plot of (a) with cooling cycles.

Simulation 3: an ellipse and a hyperbola

Result of detecting ellipses and hyperbolas are shown in Fig. 19. Patterns are with Gaussian noise N(0, 0.5) × N(0, 0.5). Figures of energy vs. cooling cycles are also shown.

(a)

(b)

Fig. 19. Detection of ellipses and hyperbolas − (a): 1 ellipse and 1 hyperbola with noise. (b): error plot of (a) with cooling cycles.

Simulation 4: a line and an ellipse

Result of detecting ellipses and hyperbolas are shown in Fig. 20 where K = 2.

Pattern are with Gaussian noise N(0, 0.5) × N(0, 0.5). Figures of energy vs. cooling cycles are also shown.

(a)

(b)

Fig. 20. Detection of ellipses and hyperbolas − (a): 1 ellipse and 1 hyperbola with noise. (b): error plot of (a) with cooling cycles.

Simulation 5: a line and a hyperbola

Result of detecting ellipses and hyperbolas are shown in Fig. 21 where K = 2.

Patterns are with Gaussian noise N(0, 0.5) × N(0, 0.5). Figures of energy vs. cooling cycles are also shown.

(a)

(b)

Fig. 21. Detection of line and hyperbola − (a): 1 line and 1 hyperbola with noise. (b):

error plot of (a) with cooling cycles.

Simulation 6: two lines

Detected line patterns in Fig. 20 and Fig. 21 have parameter f = 0.245 and f = 0.065 respectively, but the ideal result is f = 0. To make detections more precise, we can put a constraint, f = 0, on detection of lines. Fig. 22 and Fig. 23 show the result of detecting lines data are also disturbed by Gaussian noise N(0, 0.5). In Fig. 22, two lines are crossing at (0, 0). Since asymptotes of a hyperbola are two crossing lines, we can set K = 1 for Fig. 22. In Fig. 23, we set K = 2, so two additional lines appear.

(a)

(b)

Fig. 22. Detection of lines by setting f = 0 − (a): 2 lines with noise. (b): error plot of (a) with cooling cycles.

(a)

(b)

Fig. 23. Detection of lines by setting f = 0 − (a): 2 lines with noise. (b): error plot of (a) with cooling cycles.

4.2 Detection of North-South Opening Hyperbolas

North-South opening hyperbolas have the properties a < 0, b > 0, and θ = 0 in (5).

We put these constraints in the algorithm to meet the properties. The algorithm used here is just for North-South opening hyperbola. The detected parameter vector pk = [mk,x, mk,y, ak, bk, fk]^T.

In the initial step, mk,x and mk,y are randomly distributed over (0, 50), ak = -1, bk = 1, and fk = 0 for hyperbolic pattern detection. The cooling function is as (9) with a high enough temperature, Tmax = 500. We have 100 trials in the same temperature.

The temperature decreases 500 times to T = 0.0209, and this temperature is low enough. Constants αm = 1, αab = 1, and αf = 2.

Fig. 24 and Fig. 25 show the results of North-South opening hyperbolic pattern detection, where Fig. 24 (a) has 187 points and Fig. 25 (a) has 148 points. Each data is with Gaussian noise N(0, 0.5) × N(0, 0.5).

(a)

(b)

Fig. 24. Detection of hyperbolas − (a): 2 hyperbolas with noise. (b): Corresponding plot of error vs. cooling cycles of (a).

(a)

(b)

Fig. 25. Detection of hyperbolas − (a): 2 hyperbolas with noise. (b): Corresponding plot of error vs. cooling cycles of (a).

4.3 Determination of the Number of Patterns

In HTNN [8], the number of patterns was chosen by comparing the results from different number of patterns. Here we propose a method to determine the number of patterns, K, in the image. We define the detection error as

∑

where N is the number of input points. Equation (19) implies that the detection error is the average of the minimum distance from N points to their nearest patterns.

Algorithm runs from pattern number K = 1, 2, …, until the detection error has a minimum and no improvement or lower than a threshold. At that time, the best choice of K is determined. Fig. 26 has three circles and shows the result of getting K automatically. In Fig. 26 (e), the detection error greatly decreases and no significant improvement after K = 3. So we choose K = 3. Table II lists the detection error in Fig.

26 (a)-(d).

The algorithm runs on Matlab 7.2 with Intel Duo Core CPU 1.66GHz and 1G RAM. Time consumption of SA algorithm is shown in Fig. 27. When K = 1, we have 6 dimensional parameter space and in the case of K = 2, that has 12 dimension. CPU time grows with the size of parameter space.

Table II

DETECTION ERROR IN Fig. 26

K 1 2 3 4 5

Detection error 218.9 49.2 7.50 7.21 7.20

Table III CPU Time in Fig. 26

K 1 2 3 4 5

CPU time (seconds) 69.5 167.7 292.2 423.7 572.0

(a) (b)

(c) (d)

(e) (f)

Fig. 26. Determination of number of patterns K. (a): K=1. (b): K=2. (c): K=3. (d): K=4.

(e): K=5. (f): Detection error of (a), (b), (c), (d) and (e).

Fig. 27. CPU time (in seconds) vs. number of patterns K.

4.4 Seismic Applications on Simulated data

Experiments on simulated one-shot seismogram have two cases: horizontal reflection layer and dipping reflection layer. Two lines are the asymptote of the hyperbola [3]-[5], and the asymptote is a hyperbola with the same shape but size zero.

So a line can be treated as a hyperbola. Here, we use the algorithm just for North-South opening hyperbolas.

Fig. 28 is the simulated horizontal reflection layer where the depth of the

reflection layer is 500m and the velocity of the p-wave in the sedimentary rock is about 2,500m/sec [6]. There are 65 receiving stations on both side of explosion with 50m between each others. The sampling interval is 0.004 sec. The impulse response is 25 Hz Ricker wavelet. Reflection coefficient is 0.2 and noise is band-passed noise, 10.2539Hz ~ 59.5703Hz, with uniform distributed over (-0.2, 0.2).

Fig. 29 (a) shows a one-shot seismogram from horizontal reflection layer in Fig.

28. The horizontal axis in Fig. 29 is the trace number and the vertical axis stands for time t. The one-shot seismogram is first preprocessed by envelope processing in Fig.

29 (b) and thresholding [7] in Fig. 30 with the threshold 0.15. The image size is 512 × 65 where the origin is on the top-left corner with horizontal x-axis and vertical y-axis.

The points are then used as the input to the parameter detection system.

The initial parameter mk,x and mk,y are random between 0 and 50, ak = −1, bk = 1, and fk = 1. The cooling function is as (9) with a high enough temperature, Tmax = 600.

There are Nt = 100 trials in a temperature. The temperature decreases 500 times.

Constants αm = 1, αab = 0.5, and αf

= 5. Since lines of direct wave is asymptotes of a

Fig. 28. Illustration of horizontal reflection layer.

(a)

(b)

Fig. 29. Simulated seismic patterns − (a): Simulated one-shot seismogram (horizontal reflection layer). (b): After envelope processing.

(a)

(b)

Fig. 30. (a): Result of thresholding from Fig. 29 with the threshold 0.15. The origin is at the top-left corner. (b): Detected peak from (a).

(a)

(b)

Fig. 31. Detection of seismic patterns in Fig. 29 − (a): Detection result. (b): Error plot with the cooling cycles.

Fig. 32. Illustration of dipping reflection layer.

Fig. 32 illustrates the reflection layer, where the dipping angle is 10° and the depth of the reflection layer is 500m and the velocity of the p-wave in the sedimentary rock is about 2,500m/sec [6]. There are 65 receiving stations on both side of explosion with 50m between each others. The sampling interval is 0.004 sec. The impulse response is 25 Hz Ricker wavelet. Reflection coefficient is 0.2 and noise is band-passed noise, 10.2539Hz ~ 59.5703Hz, with uniform distributed over (-0.2, 0.2).

Fig. 33 (a) is the simulated one-shot seismogram. Fig. 33 (b) shows the envelope. Fig.

34 (a) is the result of threshold with the threshold 0.15 and Fig. 34 (b) plots the detect peaks.

Points in Fig. 34 (b) are the inputs to the algorithm. The initial parameters mk,x and

m

k,y are random between 0 and 50, ak = −1, bk = 1, and fk = 1. The cooling function is as (9) with a high enough temperature, Tmax = 600. There are Nt = 100 trials in a temperature. The temperature decreases 500 times. Constants αm = 1, αab = 0.5, and αf

= 5. Number of patterns is K = 2, and we set f1 = 0 for line patterns of direct wave. Fig.

35 (a) is the detection result of Fig. 34 (b) and Fig. 35 (b) is the corresponding error plot.

Table IV and Table V list the detected parameters for the simulated seismic data.

(a)

(b)

Fig. 33. Simulated seismic patterns − (a): Simulated one-shot seismogram (dipping reflection layer). (b): After envelope processing.

(a)

(b)

Fig. 34. (a): result of thresholding from Fig. 33(b) with the threshold 0.15. The original is at the top-left corner. (b): detected peak from (a).

(a)

(b)

Fig. 35. Detection of seismic patterns in Fig. 33 − (a): Detection result. (b): Error plot with the cooling cycles.

Table IV

Detected parameters in Fig. 31 (a) in image space 512×65

mx my a b f

Direct wave 33.01 8.21 -5.031 0.198 0 (preset)

Reflection wave 32.95 40.09 -4.412 0.226 1040.99

Table V

Detected parameters in Fig. 35(a) in image space 512×65

mx my a b f

Direct wave 33.00 8.96 -5.00 0.199 0 (preset)

Reflection wave 29.29 -1.46 -5.115 0.195 2300.17

4.5 Seismic Applications on Real Seismic Data

The system is also applied to detect direct wave and reflection wave in real seismic data. We obtain data from Seismic Unix System developed by Colorado School of Mine [5].

The real data with the size 3100×48 showed in Fig. 36 (a) is from Canadian Artic, which has 48 traces and 3100 samples per trace with sampling interval 0.002 seconds.

The horizontal axis is the trace number and the vertical axis is time t.

After envelope and threshold preprocessing [7], Fig. 36 (b) shows the result of envelope and Fig. 37 (a) shows the result of thresholding with the threshold 0.15. The result of peak detection is in Fig. 37 (b). We only choose points with y < 700 which includes points from direct wave, first reflection wave and second reflection wave as in Fig. 38 (a) where there are 88 points. Detected curves are plotted in Fig. 38 (b) with

K = 3 and we preset f

1 = 0. Here, the initial parameters mk,x and mk,y

are random

number between 0 and 50, ak

= −1, b

k = 1, and fk = 0. The cooling function is as (9) with a high enough temperature, Tmax = 1,000. There are Nt = 100 trials in a temperature, and the temperature decreases 500 times. Constants settings: αm = 1, αab

= 0. 5, and αf = 10.

Since the second reflection wave is not a hyperbola in theory [3], we remove the points from the second-layer reflection wave. That is, we remove the points nearest to the bottom pattern in Fig. 38 (b). Remaining 65 points are plotted in Fig. 39 (a). We redo the experiment, and this time K = 2. Fig. 39 (b) shows the result. The detected parameters in Fig. 38 (b) and Fig. 39 (b) are listed in Table VI and Table VII.

Table VI

Detected parameters in Fig. 38 (b) with fixed f1 = 0 in image space 3100×48

mx my a b f

Direct wave 24.48 8.59 -25.69 0.038 0 (preset)

Reflection wave 24.83 28.83 -22.91 0.044 2,441.7

Second Reflection wave 24.74 49.59 -23.10 0.043 8,942.7

Table VII

Detected parameters in Fig. 39 (b) with fixed f1 = 0 in image space 3100×48

mx my a b f

Direct wave 24.53 11.20 -25.59 0.039 0 (preset)

Reflection wave 24.62 -2.81 -24.13 0.041 2,978.05

As mentioned in [3]-[5], two lines of direct wave is a pair of asymptotes of the hyperbola and a pair of asymptotes is a hyperbola with size zero. From the detected parameters in Table VIII, we obtain the equation of the direct wave

)

Rearrange (20) and take square root on both sides, the equations of two lines in image space are

(a)

(b)

Fig. 36. Experiment on real data -- (a): Real seismic data from Canadian Artic. (b):

Plot of envelope.

(a)

(b)

Fig. 37. Experiment on real data -- (a): Threshold 0.15. (b): Detect peak.

(a)

(b)

Fig. 38. Experiment on real data -- (a): Choose peak with y < 700. (b): Detection result of (a).

(a)

(b)

Fig. 39. Experiment on real data -- (a): Remove points nearest to the bottom pattern.

(b): Detection result.

(a)

(b)

Fig. 40. Plot detected curve on the original data -- (a): Detection result from Fig. 38 (b). (b): Detection result from Fig. 39 (b).

The other real data is Gulf of Cadiz’s seismic data. There are 48 traces and 2050 samples in a trace with sampling interval 0.004 seconds. Fig. 41 (a) shows the real data. The horizontal axis is the trace number and the vertical axis is time t.

After envelope and threshold preprocessing [7], Fig. 41 (b) shows the envelope and Fig. 42 shows the thresholding result with threshold 0.5. The detected peak in Fig.

42 are plotted in Fig. 43 (a) where there are 66 points and the size of image is 2050×48, where the horizontal axis is x and the vertical axis is y. The initial parameters mk,x and mk,y

are random number between 0 and 50, a

k = −1, bk = 1, and fk

= 0. The cooling function is as (9) with a high enough temperature, Tmax = 1,000.

There are Nt = 100 trials in a temperature. The temperature decreases 500 times.

Constants settings: αm = 1, αab = 0.5, and αf = 10. Number of patterns is K = 2. The detection result is in Fig. 43 (b).

Since the points nearest to the pattern around y = 800 are from second-layer reflection wave. In theory, the second-layer reflection wave is not a hyperbola [3]. We remove those points and remaining 48 points are plotted in Fig. 44 (a). Fig. 44 (b) shows the detection result and Fig. 45 plots the detected curve in the original data.

Table VIII and Table IX list the detected parameters in image space in Fig. 43 (b) and Fig. 44 (b).

Table VIII

Detected parameters in Fig. 43 (b) in image space 2050×48

mx my a b f

Reflection wave 44.57 187.84 -6.93 0.144 2,116.8

Second-layer reflection wave

21.64 56.27 -25.13 0.040 12,537.7

Table IX

Detected parameters in Fig. 44 (b) in image space 2050×48

mx my a b f

Reflection wave 45.58 174.02 -7.00 0.143 2,519.30

(a)

(b)

Fig. 41. Experiment on real data -- (a): Real seismic data from Gulf of Cadiz. (b): Plot of envelope.

(a)

Fig. 42. Experiment on real data – Thresholding result of the envelope in Fig. 41 (b) with threshold 0.5.

(a)

(b)

Fig. 43. Experiment on real data -- (a): Detected peak from Fig. 42. (b): Detection result of (a), K = 2.

(a)

(b)

Fig. 44. Experiment on real data -- (a): Remove points nearest to the pattern around y

= 800 in Fig. 43 (b). (b): Detection result of (a), K = 1.

Fig. 45. Plot the detected curve in Fig. 44 (b) on the original data.

Chapter 5 Conclusions and Discussions

5.1 Conclusions

This paper is about a proposed system, which adopted the simulated annealing algorithm to detect patterns such as lines, circles, ellipses, and hyperbolas by finding their parameters in an unsupervised manner and global minimum fitting error related to points in an image. The iterative adjustment requires less memory space. Also, we define the distance from a point to a pattern and this makes the computation feasible, especially for hyperbola. Using four steps to adjust parameters from center, shape, angle to the size of the pattern can get fast convergence. Based on the average minimum distance from points to patterns, we have proposed a method to determine the number of patterns automatically. Experimental results on the detection of line, circle, ellipse, and hyperbola in images are successful. The detection results of line pattern of direct wave and hyperbolic pattern of reflection wave in one-shot seismogram are good, and can improve seismic interpretations and further seismic data processing.

5.2 Discussions

Parameter settings. In the cooling schedule, the value of T

max, and Nt, are set prior.

For a trial which includes a change of center, a change of b and a, a change of θ, and a change of f for every pattern, there are three possible results to accept or reject the change determined by Metropolis criterion:

1. The new parameter has smaller error and it is accepted.

2. The new parameter has larger error and it is still accepted.

3. The new parameter has larger error and it is rejected.

The determination of Tmax, we considered the accept ratio of the larger-error trials. If the Tmax is not high enough, the trial with larger error will almost reject, that is, it always accept trial with smaller error, so it is possible to reach local minimum. Fig. 46 shows this situation, where Tmax = 1,500 iterations, initial center (0,0), a = 1, b = 1, θ

= 0, and f = 1. Fig. 47 shows the result when Tmax = 10. In Fig. 47 (b), the accept ratio of canter, angle, and size has increased and shows the good result for this simple example. Fig. 48 show the result when Tmax = 100,000. After 500 iterations, T = 4.1, but this temperature is not low enough and the high accept ratio of larger-error parameters results in the instability. To solve this problem, we can increase the number of iterations to 1,000 and show the result in Fig. 49. This still shows good result, but it takes more time. In conclusions, for temperature Tmax, we have to choose a high enough temperature that gives a high accept ratio of larger-error parameters.

Besides, we need many enough iterations to cool the temperature to ensure stability.

Also we find the setting of Tmax is proportional to the scale of input points. In Fig.

47, Tmax = 10 provides good result. In Fig. 50, we enlarge the scale of data by two.

Fig. 50 (a) with Tmax = 10 cannot give good result, but Fig. 50 (a) with Tmax = 100 can give good result. In our simulation experiments, we choose Tmax = 500 and 500 iterations to ensure high enough initial temperature and lower final temperature T ≈ 0.02.

As for Nt, if trials are not many enough, we cannot get good result. Larger Nt takes more time but gives more chances. So we can have as many trials as possible if the computational power is strong enough. Fig. 51 shows too few trials cannot provide good result and for this simple example we need only Nt = 10 to obtain a good result.

(a)

(b)

Fig. 46. Illustration of low initial temperature Tmax: (a) Detection result of Tmax = 1. (b) Accept ratio of larger-error parameters.

(a)

(b)

Fig. 47. Illustration of low initial temperature Tmax: (a) Detection result of Tmax = 10.

(b) Accept ratio of larger-error parameters.

(a)

(b)

Fig. 48. Illustration of high initial temperature Tmax: (a) Detection result of Tmax = 100,000. (b) Accept ratio of larger-error parameters.

(a)

(b)

Fig. 49. Illustration of high initial temperature Tmax with more iterations: (a) Detection result of Tmax = 100,000. (b) Accept ratio of larger-error parameters.

(a)

(b)

Fig. 50. Enlarge the scale of points by two: (a) Detection result of Tmax = 10. (b) Detection result of Tmax = 100.

(a)

(b)

Fig. 51. Relationship between Nt and detection result Tmax = 500: (a) Detection result of Nt= 1. (b) Detection result of Nt = 10.

Time consumption. As for the time consumption, Table III shows that the CPU

time is proportional to the number of patterns or the number of parameters. The larger number of parameters, the algorithm takes more time to obtain the solution.

Time consumption. As for the time consumption, Table III shows that the CPU

time is proportional to the number of patterns or the number of parameters. The larger number of parameters, the algorithm takes more time to obtain the solution.

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