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MicroScribe G2 Rhino XploRe 1*6 (Principal components analysis) 30 (cluster analysis) 30 (pooled covariance matrix) 2 2000 3 USING THRESHOLD OF SINGLE LINKAGE TO IMPROVE THE MATCH INDEX BETWEEN COMPUTER MODEL AND ARTIFACT Chuan-Shu Kao Jyh-Jeng Deng Chia-Lin Liu Chun-Ching Chen Shih-Hung Wang Yi-Ho Lai Department of Industrial Engineering and Technology Management DaYeh University ChangHua, Taiwan 515, R.O.C. ABSTRACT This paper investigates the relationship between geometric profile and measurement precision in repeated measurements of a wooden sculpture, christened as loving each other. Statistically, if the covariance matrix formed by the repeated measurement is dependent on the profile of the artifact, then given a large difference in the profile of two domains on 245 Journal of Technology, Vol. 23, No. 4, pp. 245-262 (2008) Key words : principal component, covariance matrix, repeated measure- ments, cluster analysis.

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Page 1: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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USING THRESHOLD OF SINGLE LINKAGE TO IMPROVE THE MATCH INDEX BETWEEN COMPUTER MODEL AND ARTIFACT

Chuan-Shu Kao� Jyh-Jeng Deng� Chia-Lin Liu� Chun-Ching Chen� Shih-Hung Wang� Yi-Ho Lai

Department of Industrial Engineering and Technology Management DaYeh University

ChangHua, Taiwan 515, R.O.C.

ABSTRACT

This paper investigates the relationship between geometric profile and measurement precision in repeated measurements of a wooden sculpture, christened as loving each other. Statistically, if the covariance matrix formed by the repeated measurement is dependent on the profile of the artifact, then given a large difference in the profile of two domains on

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Journal of Technology, Vol. 23, No. 4, pp. 245-262 (2008)

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Key words : principal component, covariance matrix, repeated measure- ments, cluster analysis.

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Page 2: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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the artifact, the corresponding covariance matrices would be significantly different. This study finds a rule of single linkage to discriminate outliers in repeated measurements based on similarities among them.

The measurement tool is MicroScribe G2; the display software is Rhino; and the analysis tool is XploRe. We convert a covariance matrix into a 1x6 vector and by using principal components analysis the vector is projected into a 2D point. Thirty points are chosen among the domains to explore different profiles of the artifact. Next cluster analysis is applied to the 30 points in the 2D graph to find points which share the common covariance matrix. This further renders a pooled covariance matrix for points within a 0.2 mm square which inscribes the most crowded points in the projected 2D principal components. Then 2000 pairs of simulated points from the pooled covariance matrix are generated, and the distance between the pair points is calculated. The sample mean and standard deviation of the distance are obtained and the upper limit, mean plus the three standard deviations, is used as an empirical single linkage threshold.

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Page 5: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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Page 6: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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Page 8: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

252 ./01� 23456� 278� 9:;4<=

� 14 ���������

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Page 9: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

���� ���� �� �� � ���� ��������������� !"#$%&'()*+,- 253

� 19 ���� 3���-�-SVD� 8�

� 20 ���� 3���-�-check� 1�

JK�-.b�c�¬q� single linkage�893[\>

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> 0.3 �²³�V/�}] eÅ��+��²�=>�

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linkage893��~ 0.45© 0.3$IqÄÅfg7��W

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linkage 893��¢ 0.6n0.45 © 0.3 ²qÄÅi��7

�W¬~xyz�0?@�^ Minitab Q"/�_`

Two-Way ANOVA�Q"���-.=>f 224 23�MN

­f 23-.=C treat1� p-value3~ 0.000��.¢

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]/�¦ treat2� p-value3~ 0.092�¢Q"8_�q�

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0.45©q 0.3qÄÅfg7��W

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Page 10: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

254 ./01� 23456� 278� 9:;4<=

�� �� ��!

� 22 "���!#$Minitab%

single linkage �893�Ò. SN 3�¨\_`|

}W~�0+IJ=C,123¦� single linkage�89

3v¨ SN 3mv¨!¥¢Q"8q�i���¬q[µ

¶ÅÖ��IJ� SN3�G�· single linkage��6,

JÄÅá��!¥(²�IJ�oiM�V�ìíÈî�

�� SN3�W{g'Ù_�� single linkage�89

3�?0?@-.­����B�ab�4��D�

j�]Ö��WW�B�ç single linkage�89qEFb

�����'¢��D�� 30eab4� 59eab�h

i���Å]Ö�����û7-.�­893���_

dM��.893�Ò�~�9¨(S�EF��89

3)�Ø®���~¤´qäB�@��'Ó\ªk�

m��bdM�!¥B\]{� �x=C��.��}

+����oIJx��3~ 0.45W­®��ìíî�ë

ð-.��893W¬893�¹® 0.3�<(T�.H�

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� 23 Two-Way ANOVA�!

���3X:8 0.45 x��)�¢`Q"GHµ¶}B�

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¢I�)·��)Zl�(�OP�û7~Ù�0?

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single linkage�893×¢ 0.6W0IJ3-¡.�?@Ù

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¥�?@�×-b�ò®U�12���56WØ®U

�� 0.6 0.45 0.3

1 8.3415 8.3415 8.3415

2 7.0832 7.0832 7.0832

3 11.5295 11.4424 11.0438

4 11.2838 11.4301 11.1010

5 12.9642 12.7486 12.8594

6 13.3151 13.0277 12.9498

7 14.7723 14.6225 13.9873

8 14.3846 14.3892 14.0305

9 13.8669 12.2116 12.0309

10 17.8782 17.8782 17.8782

11 14.2130 14.2130 14.2130

12 12.1373 12.2009 12.1967

13 9.4993 9.4993 9.4993

14 8.0961 8.0961 8.0961

15 19.0013 19.0013 18.8232

16 9.1542 9.4530 9.6745

QD�RS �ETUVDW��X 2700FP�RYP

�Z[ 270F\]^_`a��bOcF\]^_`d

e[ 1*6 f9gb�f9hij[k��l� 2D m

�=a56RYP����no�pq +,gbr+

,s Ptu \]^_`vw�cxay� zw\

]^_`��� 2000 F{| Pg}�{|P ~

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��� single linkage =>:#S���:�V� S�

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Page 11: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

���� ���� �� �� � ���� ��������������� !"#$%&'()*+,- 255

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4

Seco

nd P

C*E

-2

-1

0

1

2

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4

Seco

nd P

C*E

-2

-1

0

1

2

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4

Seco

nd P

C*E

-2

-1

0

1

2

3

4

=

0.00411560.000389470.0013748

0.000389470.004018106-2.3839e-

0.001374806-2.3839e-0.005589

1,3,1C

����

1. ���������� ����������

Statistical Analysis,” Springer, Berlin, Germany, (2003).

7. http://www.xplore-stat.de/index_js.html

8. ()*��+,-./01��23456789:

�L� �M�6N�OP#$%�QR (2008)

2008� 08� 11��

��

� ��

2008� 08� 19��

��

� ��

2008� 10� 21��

��

� ��

2008S 10T 24UV WX

���

�� ������ 8������� 9��

2

����!*+, 24- 25./�01234567-8

9�:;+'

� 24 ��� 1 � 2 ��� 0.02

=

3409.0

0638.0

1495.0

1,2,1M

� 25 ��� 1 � 2 ��� 0.01

#$( 1�) 3����!*+, 26- 27./�0

1234567-89�:;+'

� 26 ��� 1 � 3 ��� 0.02

1,2,2

0.1432

0.0606

0.3251

M

=

1,2,2

0.0043741 0.0010751 0.00065103

0.0010751 0.0042398 0.00055013

0.00065103 0.00055013 0.0043399

C

=

1,2,1

0.0053156 0.0018243 0.0011492

0.0018243 0.0057668 0.0015084

0.0011492 0.0015084 0.0045955

C

=

2. V., Raja and Fernandes Kiran J., “Reverse Engineering -

An Industrial Perspective,” Springer, Berlin, Germany,

(2008).

3

“Least-Squares Fitting of Two 3-D Points Sets,” IEEE

Transactions on Pattern Analysis and Machine

Intelligence, Vol. PAMI-9, pp. 698-700 (1987).

Orientation using Unit Quaternions,” Journal of the Optical

Society of America A,” Vol. 4, pp.629-642 (1987).

Control,” 3rd ed. Pearson, London, UK, (2005).

&

���Ç 3 à�/�yo���¸r����S 1 �T

����������� !�"#$�%&(2006)'

3. Arun, K. S., Huang, T. S., and Blostein, S. D.,

4. Horn, B. K. P., “Closed-form Solution of Absolute

5. Craig, John J., “Introduction to Robotics: Mechanics and

6. Hardle, W. and Simar, L., “Applied Multivariate

;<=>?@ABCDEFGHIJK�������

Page 12: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

256 ����� ���� ���� �����

30 points classified into into 3 groups by ���

First PC*E-2

0 1 2 3 4

Seco

nd P

C*E

-2

-1

0

1

2

3

4

=

0.00400970.000385830.0008278

0.000385830.003844505-5.0602e-

0.000827805-5.0602e-0.0045135

2,3,1C

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4

Seco

nd P

C*E

-2

-1

0

1

2

3

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4

Seco

nd P

C*E

-2

0

1

2

3

30 points classified into into 3 groups by ���

First PC*E-2

0 5 10

Seco

nd P

C*E

-2

5

� 27 ��� 1 � 3 ��� 0.01

��� 2 �� 1 ������ 28 29 ����

������� ��������

� 28 ��� 2 � 1 ��� 0.02

� 29 ��� 2 � 1 ��� 0.01

��

������� ��������

� 30 ��� 2 � 2 ��� 0.02

=

3578.0

0678.0

1543.0

1,3,1M

=

3253.0

0604.0

1440.0

2,3,1M

2,1,1

0.1901

0.0819

0.4359

M

=

=

0.0060030.000495680.00049415

0.000495680.0060460.00019167

0.000494150.000191670.0068467

1,1,2C

=

3884.0

0726.0

1707.0

2,1,2M

=

0.005442505-4.0074e-0.00014968

05-4.0074e-0.0051770.00096255

0.000149680.000962550.0057408

2,1,2C

2,2,1

0.0065227 -0.00034851 0.0018841

-0.00034851 0.0082153 -0.00062052

0.0018841 -0.00062052 0.0064286

C

=

��� 2 �� 2 ������ 30 � 31 �

Page 13: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

���� ���� �� �� � ���� ��������������� !"#$%&'()*+,- 257

30 points classified into into 3 groups by ���

First PC*E-2

0 5 10

Seco

nd P

C*E

-2

5

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4 5

Seco

nd P

C*E

-2

-2

-1

0

1

2

3

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3 4 5

Seco

nd P

C*E

-2

-2

-1

0

1

2

3

30 points classified into into 3 groups by ���

First PC*E-2

0 5

Seco

nd P

C*E

-2

-1

0

1

2

� 31 ��� 2 � 2 ��� 0.01

��� 2 �� 3 ������ 32 33 ����

������� ��������

� 32 ��� 2 � 3 ��� 0.02

� 33 ��� 2 � 3 ��� 0.01

��� 3 �� 1 ������ 34 35 ����

������� ��������

� 34 ��� 3 � 1 ��� 0.02

2,2,1

0.1784

0.0748

0.4027

M

=

2,2,2

0.1631

0.0698

0.3724

M

=

2,2,2

0.0048047 0.00063231 0.001278

0.00063231 0.0058554 0.00018118

0.001278 0.00018118 0.0050197

C

=

2,3,1

0.1667

0.0717

0.3819

M

=

2,3,1

0.006065 0.00068983 0.00088903

0.00068983 0.0057611 8.1088e-05

0.00088903 8.1088e-05 0.005092

C

=

2,3,2

0.1611

0.0679

0.3649

M

=

2,3,2

0.0046771 0.00058074 0.00055969

0.00058074 0.00606 5.4207e-05

0.00055969 5.4207e-05 0.0044616

C

=

3,1,1

0.0055526 0.00079981 0.0016149

0.00079981 0.0091883 0.00049773

0.0016149 0.00049773 0.0042121

C

=

Page 14: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

258 ����� ���� ���� �����

30 points classified into into 3 groups by ���

First PC*E-2

0 5

Seco

nd P

C*E

-2

0

1

2

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3

Seco

nd P

C*E

-3

-5

0

5

10

15

20

30 points classified into into 3 groups by ���

First PC*E-2

0 5

Seco

nd P

C*E

-2

-5

0

5

� 35 ��� 3 � 1 ��� 0.01

��� 3 �� 2 ������ 36 37 ����

������� ��������

� 36 ��� 3 � 2 ��� 0.02

� 37 ��� 3 � 2 ��� 0.01

��� 3 �� 3 ������ 38 39 ����

������� ��������

� 38 ��� 3 � 3 ��� 0.02

3,1,1

0.1765

0.0777

0.4095

M

=

3,1,2

0.1639

0.0726

0.3818

M

=

3,1,2

0.0064406 0.0011428 0.001828

0.0011428 0.005294 -0.00049378

0.001828 -0.00049378 0.0048041

C

=

30 points classified into into 3 groups by ���

First PC*E-2

-1 0 1 2 3

Seco

nd P

C*E

-3

-10

-5

0

5

10

15

20

3,2,1

0.1692

0.0729

0.3880

M

=

3,2,1

0.0058399 -0.0012273 0.0010597

-0.0012273 0.0064723 -0.00021175

0.0010597 -0.00021175 0.0044249

C

=

3,2,2

0.1559

0.0677

0.3589

M

=

3,2,2

0.0050191 -0.00081312 0.00057521

-0.00081312 0.004862 -9.6536e-05

0.00057521 -9.6536e-05 0.0041246

C

=

3,3,1

0.0097483 -0.00028952 -0.001362

-0.00028952 0.0081157 0.0029365

-0.001362 0.0029365 0.0054059

C

=

Page 15: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

���� ���� �� �� � ���� ��������������� !"#$%&'()*+,- 259

30 points classified into into 3 groups by ���

First PC*E-2

0 5

Seco

nd P

C*E

-2

-5

0

5

� 39 ��� 3 � 3 ��� 0.01

� � �

C1 C2 C3 C4 C5 C6

Rnadom_Ord Std_Ord Block1 Block2 Block3a Treatment

1 16 SVD6 Join_rep2 ��� check_rep2

2 7 SVD5 Join_rep1 ��� check_rep1

3 9 SVD8 Join_rep5 ��� check_rep5

4 14 SVD6 Join_rep3 ��� check_rep5

5 5 SVD5 Join_rep5 ��� check_rep3

6 6 SVD5 Join_rep3 �� check_rep2

7 12 SVD8 Join_rep2 �� check_rep3

8 10 SVD8 Join_rep3 �� check_rep1

9 2 SVD4 Join_rep3 ��� check_rep3

10 1 SVD4 Join_rep5 �� check_rep2

11 8 SVD5 Join_rep2 �� check_rep5

12 13 SVD6 Join_rep5 �� check_rep1

13 11 SVD8 Join_rep1 ��� check_rep2

14 4 SVD4 Join_rep2 ��� check_rep1

15 15 SVD6 Join_rep1 �� check_rep3

16 3 SVD4 Join_rep1 �� check_rep5

3,3,1

0.1949

0.0855

0.4514

M

=

3,3,2

0.1500

0.0674

0.3522

M

=

3,3,2

0.0042256 0.0010152 0.00090883

0.0010152 0.0050421 0.0010957

0.00090883 0.0010957 0.0038473

C

=

Page 16: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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; -----------------------------------------------------------------------------------------------------------------

; Description MVAclus8p performs cluster analysis for the 8 points example

;------------------------------------------------------------------------------------------------------------------

Iibrary(“xplore”)

Iibrary(“stats”)

X=read(“D:\position01.txt”)

;creates data

n=rows(x) ;rows of data

xs=string(“%1.0f”, 1:n) ;adds labels

; the second part

d=distance(x, “euclid”) ;Euclidean distance

;squared distance matrix

t=agglom(d, “ward”, 2) ;hier. Cluster anal., Single linkage

g=tree(t.g, 0, “CENTER”)

g=g.points

tg2=paf (g[,2], g[,2] ! = 0)

zg2=sort(tg2)

zg

1 = 5. *(1:rows(g)/5) = (0:4)1 – 4

setmask1 (g, 1, 0, 1, 1)

setmaskp (g, 0, 0, 0)

; show the dendrogram

tg=paf(t.g[,2], t.g[,2]!=0)

numbers=(0:(rows(x)-1))

numbers=numbers~((0)*matrix(rows(x)))

setmaskp(numbers, 1, 2, 3)

setmaskt(numbers, string(“%.of”, tg), 0, 0, 16)

dd2=createdisplay(1, 1)

show (dd2, 1, 1, g, numbers)

setgopt(dd2, 1, 1, “xlabel”, “”, “ylabel”, “Squared Euclidian Distance”)

setgopt(dd2, 1, 1, “title”, “Ward Linkage Dendrogram – 6 points”, “xoffset”, 10 10, “yoffset”, 10(3)

Page 17: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

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1 ��-�-check 1 ok ok ok

2 ��-�-check 2 ok ok ok

3 ��-�-check 3 ok ok ok

4 ��-�-check 4 ok ok ok

5 ��-�-check 5 ok ok ok

6 ��-�-check 6 ok ok ok

7 ��-�-check 7 ok ok ok

8 ��-�-check 8 ok ok ok

9 ��-�-SVD 1 ok ok ok

10 ��-�-SVD 2 ok ok ok

11 ��-�-SVD 3 ok 2,3 2,3

12 ��-�-SVD 4 ok 1 1

13 ��-�-SVD 5 ok ok ok

14 ��-�-SVD 6 ok ok 4

15 ��-�-SVD 7 ok 1 1

16 ��-�-SVD 8 ok 1,3 1,3,4

17 ��-�-check 1 ok 2 2

18 ��-�-check 2 ok ok ok

19 ��-�-check 3 ok ok ok

20 ��-�-check 4 ok ok ok

21 ��-�-check 5 ok ok ok

22 ��-�-check 6 ok ok 1,4

23 ��-�-check 7 ok 1,3 1,3

24 ��-�-SVD 1 ok ok ok

25 ��-�-SVD 2 ok ok ok

26 ��-�-SVD 3 ok ok ok

27 ��-�-SVD 4 ok ok ok

28 ��-�-SVD 5 ok ok 1,4

29 ��-�-SVD 6 ok 2,4 2,4

30 ��-�-SVD 7 4,5 1,4,5 1,4,5

31 ��-�-SVD 8 ok ok ok

32 �-check 1 ok ok ok

33 �-check 2 ok ok ok

Page 18: ir.lib.ntust.edu.twir.lib.ntust.edu.tw/bitstream/987654321/35694/1... · 246 the artifact, the corresponding covariance matrices would be significantly different. This study finds

262 ./01� 23456� 278� 9:;4<=

� �� �� 0.6 0.45 0.3

34 �-check 3 ok ok ok

35 �-check 4 ok ok ok

36 �-check 5 ok ok ok

37 �-check 6 ok ok ok

38 �-check 7 ok ok ok

39 �-check 8 ok ok ok

40 �-SVD 1 ok ok ok

41 �-SVD 2 ok ok ok

42 �-SVD 3 ok ok ok

43 �-SVD 4 ok ok 1

44 �-SVD 5 ok ok ok

45 �-SVD 6 ok ok ok

46 �-SVD 7 ok ok ok

47 �-SVD 8 ok ok 1,2

48 �-check 1 ok ok ok

49 �-check 2 ok ok ok

50 �-check 3 ok ok 1

51 �-check 4 ok ok ok

52 �-check 5 ok ok 1

53 �-check 6 ok ok ok

54 �-check 7 ok ok 2,5

55 �-SVD 1 ok ok ok

56 �-SVD 2 ok ok ok

57 �-SVD 3 ok ok ok

58 �-SVD 4 ok ok 5

59 �-SVD 5 3,4,5 3,4,5 3,4,5

60 �-SVD 6 ok ok 5

61 �-SVD 7 ok ok 1

62 �-SVD 8 ok ok ok