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Page 1: homepages.inf.ed.ac.uk · ! #"$ &%' (%' *),+.- /%/ 0 - 1 0 ),+.- /%/ 02 ),%' !+.-(+ 354 687 +.9: %/ ; !)*9:+< ! =+ 3 + >

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d

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S3

n3

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n1

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k (p)C

λ initialise p and

p0

p

λ λ 0

k (p)CΣk

C(p) =

find p minimizing F(p) + C(p) λ

kτ> λ λ + ∆λ

while

k=1..M

update p

0λ λ

τk

λ + ∆λλ

C(p) = C(p) + λ C (p) k

(p) >

C(p) = 0

initialise pp p0

For k =1 to K DO

λ initialise

while

Ck

update p

find p minimizing F(p) + C(p)

/* add new constraint */

k = k + 1

. ��/ . # /

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(b)

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P

axecyl

CylCyl

E

(a)(b)

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PP

P

Cyl

1

2 3

P

P

1

d

2

P

P P

P

PP1

2

3 4

5

6

(a) (b) (c)

d

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S1

S4S2

S3

S57 8 9 10 11 12 13

−14

−12

−10

−8

−6

−4

−2

log10

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log 10

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unit constraint n1

. ��/ . # /

7 8 9 10 11 12 13−15

−10

−5

log10

(λ3 )

log 10

(C_a

ngle

_2)

angle constraint (n1,n

5)

7 8 9 10 11 12 13−1.3

−1.2

−1.1

−1

−0.9

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log10

(λ1 )

log 10

(ang

le(n

1,n1 ex

act)

n1 estimation error

. / . � /

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7 8 9 10 11 12 13−4.7

−4.65

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log10

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step1: addition of the unit constraints

7 8 9 10 11 12 13−15

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log10

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step2: addition of the angle constraint (n1,n2)

. ��/ . # /

7 8 9 10 11 12 13−14.485

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log10

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7 8 9 10 11 12 13−14.4862

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( C_a

ngle

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step4: addition of the angle constraint (n2,n5)

. / . � /

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7 8 9 10 11 12 13−4.457

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step2: addition of the angle constraint (n1,n2)

. ��/ . # /

7 8 9 10 11 12 13−16

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log10

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step3: addition of the angle constraint (n1,n5)

7 8 9 10 11 12 13−14.365

−14.36

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−14.34

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log10

(λ4 )

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( C_a

ngle

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step4: addition of the angle constraint (n2,n5)

. / . � /

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7 8 9 10 11 12 13−4.75

−4.7

−4.65

−4.6

−4.55

−4.5

−4.45

−4.4

log10

(λ1 )

log 10

( C_a

ngle

_3 )

step1: addition of the unit constraints

7 8 9 10 11 12 13−4.4343

−4.4342

−4.4341

−4.434

−4.4339

−4.4338

−4.4337

−4.4336

−4.4335

log10

(λ2 )

log 10

( C_a

ngle

_3 )

step2: addition of the angle constraint (n1,n2)

. ��/ . # /

7 8 9 10 11 12 13−4.452

−4.45

−4.448

−4.446

−4.444

−4.442

−4.44

log10

(λ3 )

log 10

( C_a

ngle

_3 )

step3: addition of the angle constraint (n1,n5)

7 8 9 10 11 12 13−14

−12

−10

−8

−6

−4

log10

(λ4 )

log 10

( C_a

ngle

_3 )

step4: addition of the angle constraint (n2,n5)

. / . � /

{|/0@)��,"�J�Sª < ������� ���"�!� ���������������� .��&��� �(�������"� � �� ���� �8?LA$C E�F O .)I�" / � � ������� ����� � ���,� � ��������� �� .���!���"����, �������J�� /

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7 8 9 10 11 12 13−0.4248

−0.4248

−0.4248

−0.4248

−0.4248

−0.4248

log10

(λ1 )

log 10

(ang

le(n

1,n1 ex

act)

step1: addition of the unit constraints

7 8 9 10 11 12 13−0.52

−0.515

−0.51

−0.505

−0.5

−0.495

−0.49

−0.485

log10

(λ2 )

log 10

(ang

le(n

1,n1 ex

act)

step2: addition of the angle constraint (n1,n2)

. ��/ . # /

7 8 9 10 11 12 13−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

log10

(λ3 )

log 10

(ang

le(n

1,n1 ex

act)

step3: addition of the angle constraint (n1,n5)

7 8 9 10 11 12 13−1.21

−1.205

−1.2

−1.195

−1.19

−1.185

log10

(λ4 )

log 10

(ang

le(n

1,n1 ex

act)

step4: addition of the angle constraint (n2,n5)

. / . � /

{|/0@)��,"�ev��Sª < ������� ���"�!� ��� �������!���.� � �������.��� � �����!� �.� ����� �*� ���.,-� ���"�!� ��� . => � / � � ����� � ����� �(���,��� ��������� �� .���!���"����,��������� /

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S1

S2

S3

Tetrahedron

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4 5 6 7 8 9 10 11−9

−8

−7

−6

−5

−4

−3

log10

(λ1 )

log 10

( C_u

nit_

1 )

unit constraint 1

4 5 6 7 8 9 10 11−6

−5

−4

−3

−2

−1

0

log10

(λ2 )

log 10

( C_a

ngle

_1 )

angle constraint 1

. ��/ . # /

4 5 6 7 8 9 10 111.4

1.5

1.6

1.7

1.8

1.9

2

2.1

log10

(λ2 )

log 10

(Lea

st s

quar

es e

rror

func

tion)

least squares error function

4 5 6 7 8 9 10 11−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

log10

(λ2 )

log 10

( C(p

) )

sum of the constraint functions

. / . � /

{|/0@)��,"�Fv � ª . � / � Q �* ��� ����� ��������� �����"� &�!���(�������"� ��� ����&���.��� .JIDN / ' �"��� ��� ���%�*&� ���;: � /+. # / � Q �* ��� �����-�������� ������ .�+ ����� �������.� � � ����&���.��� .JI0S�/ ' �"��� ���*���%�*�� ��� :

�/ . / ����� . � / ������� ���.��� ��� �����+�!#��(�*&�������� ��������"�!�

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half cylinderS3

1

2

n

nn

S2

S1

p�-1t p�5�t p�&Qt{�/3@)��,+�Hv��Sª ��')�� �"�*' ����� ����� ���� �� �� .�"����� � ��� � ����� �B# �����!���� �+������� �� �*�*/

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8 9 10 11 12 13 14−20

−18

−16

−14

−12

−10

−8

log10

(λ2 )

log 10

( C_a

ng )

angle constraint

8 9 10 11 12 13 14−12

−11

−10

−9

−8

−7

−6

−5

log10

(λ1 )

log 10

( C_u

nit )

unit constraint

p�-1t9ª '�� � p�5�t«ª ' � ���

8 9 10 11 12 13 14−22

−20

−18

−16

−14

−12

−10

log10

(λ3 )

log 10

( C_a

xe )

cylinder axis constraint

8 9 10 11 12 13 14−14

−13

−12

−11

−10

−9

−8

−7

−6

−5

log10

(λ4 )

log 10

( C_c

irc )

cylinder circularity constraint

p�&Qt9ª '��/3 p�=�t«ª '�� � � �

8 9 10 11 12 13 145.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

log10

(λ2 )

log 10

(ls fu

nctio

n)

ls function

8 9 10 11 12 13 14−12

−11

−10

−9

−8

−7

−6

−5

−4

log10

(λ2 )

log 10

( sum

of t

he c

onst

rain

t fun

ctio

ns) sum of the constraint functions

pZ��t9ª0� � �8,+,"#), p~$_t«ª � '� ��� � 3 � � � p ( t{|/0@)��,"��vQ��ª � ���%� K �����", ���*� ���"�!� ��� ����� ���� �� ,� "�.�����*� ��#��(�*�� / . � /L2 . # /B2�. /B2�. � / � ���*&���*�!� � ��������� ����j%�*���*�!�&�!��� �������.�!��� ' �"������ ���%�*&� ��� ����� ��� �� ��� � : / . � /L2 . � / � ������� ���"�!� ��� .�*��� � � � �������*� � ����&���.�������� ����� ����� �������.� �� ����&���.��� /

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(a)(b)

(c) (d)

lateral plane

S1cylinder patch

S2cone patch

S5

S6

back planeS3

bottom plane

S4

lateral plane

{|/0@)��,"�Hv��>ª`{^#)��,id7/3�9jX!2#C$�*"���h:A��?0*"/�Y �7��-%=�,"/0&B#)5c6+�8&'* v)L

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8 9 10 11 12 13 14 15−16

−14

−12

−10

−8

−6

−4

log10

(λ2 )

log 10

( C_a

ng )

angle constraint

8 9 10 11 12 13 14 15−14

−13

−12

−11

−10

−9

−8

−7

−6

−5

log10

(λ1 )

log 10

( C_u

nit )

unit constraint

pz-1t pZ5�t

8 9 10 11 12 13 14 15−14

−12

−10

−8

−6

−4

−2

log10

(λ3 )

log 10

( C_c

irc_c

yl )

cylinder circularity constraint

8 9 10 11 12 13 14 15−16

−14

−12

−10

−8

−6

−4

log10

(λ4 )

log 10

(C_c

irc_c

onex

)

cone circularity constraint

pZ&�t p�=�t

8 9 10 11 12 13 14 157.075

7.08

7.085

7.09

7.095

log10

(λ2 )

log 10

( F(p

) )

objective function

8 9 10 11 12 13 14 15−14

−12

−10

−8

−6

−4

−2

log10

(λ2 )

log 10

Sum

(C(p

))

sum of the constraint functions

pZ��t p~$_t{|/0@)��,"�hv��>ª �����%� �������., ��� � ���"�!�-��� , �� ������� ���������� ��#��(� �� "�/ ��2 #X2 ,2 � � Q �* ��� ����� ��� ����� ���]j � ��� � �) ����� �������.� �� ����&���.����� ' � ��� ���*��� � �� ��� ����� ����� ��&��� ���*��: � /��,2 � �@< ������� ���.��� ��� .�*�!�(� � � �������*���*������� ����� ����� ����, ��� �� . �����&�!��� �������.�!� �� ����*� �������"��� ����� �������", ���*�����.��� /

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plane S1

plane S2plane S5 plane S6

plane S3

plane S4 sphere patch S8

cylinder patch S7

{|/0@)��,"�Hv�w>ª`{^#)��,id7/3�9jX!2#C$�*"���h:A��?0*"/�Y �7��-%=�,"/0&B#)5c6+�8&'* � L

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0 5 10 15 20 25 300.95

1

1.05

parameters

scale

d m

ax a

nd m

in

0 5 10 15 20 25 300

0.5

1

1.5

2

parameterssta

ndard

devia

tion (

%)

. ��/ . # /

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7.5 8 8.5

x 105

0

5

10

15

LS error distribution

{�/3@)��,+�Hv��Sª �����(�����.#������"�!� ��� ����� "� ��� � �k� ������� � �*���������*/

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−5 −4.8 −4.6 −4.4 −4.2 −4 −3.80

2

4

6

8

10

12

14

16

log10

(angle(nc,n ))

−5 −4.5 −4 −3.5 −30

2

4

6

8

10

12

14

log10

(|LSc − LS

i|/mean(LS

i))

. ��/ . # /

{|/0@)��,"� � �Sª . ��/ � ����� �����"#������.��� ��� ����� � �(���",-� ���"�!�-���]j � ��� � &��/ . # / � ���.� �����.#������.��� ��� ����� &( ���*���.��� �� �� ���]j � ��� � &��/

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6 7 8 9 10 11 12 13−14

−12

−10

−8

−6

−4

−2

0

log10

(λ2 )

log

10(

sum

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sum of the constraint functions

6 7 8 9 10 11 12 136.14

6.15

6.16

6.17

6.18

6.19

6.2

log10

(λ2 )

log

10(|

least

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least squares function

. ��/

6 7 8 9 10 11 12 13−12

−10

−8

−6

−4

−2

0

2

log10

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log

10(

sum

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sum of the constraint functions

6 7 8 9 10 11 12 136.22

6.24

6.26

6.28

6.3

6.32

6.34

6.36

6.38

6.4

log10

(λ2 )

log

10(|

least

square

s err

or|

)least squares function

. # /

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6 7 8 9 10 11 12 13−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

log10

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log

10(

C_

an

gl(p

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angle constraint function

6 7 8 9 10 11 12 13−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

log10

(λ2 )

log

10(

su

m (

C(p

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sum of the constraint functions

6 7 8 9 10 11 12 136.24

6.26

6.28

6.3

6.32

6.34

6.36

6.38

6.4

6.42

log10

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log

10(|

lea

st

sq

ua

res e

rro

r|)

least squares function

. ��/ . # / . /

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