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f
as
t
e
r f
o
r
ob
j
ec
t
s w
it
h
l
ess
t
hen
10
co
r
ne
r
s
.
We de
-
fi
ned
t
he so
li
d ang
l
e o
f
a
v
e
rt
e
x
o
f
an N
-
d
i
mens
i
ona
l
con
v
e
x
po
ly
hed
r
on b
y i
n
tr
oduc
i
ng
t
he concep
t
o
f t
he
MinimumBoundingCone
and we p
r
o
v
ed
t
ha
t t
he a
l
go
-
rit
hm
t
e
r
m
i
na
t
es
i
n
fi
n
it
e s
t
eps and
t
he numbe
r
o
f
s
t
eps
depends on
t
he ma
xi
mum so
li
d ang
l
e o
f t
he con
v
e
x
po
lyt
ope
.
We s
t
ud
y
2 d
iff
e
r
en
t t
echn
i
ques
f
o
r r
o
t
a
ti
ng
t
he po
i
n
t
c
l
oud o
f t
he ob
j
ec
t,
e
it
he
r r
o
t
a
ti
ng b
y i
nc
r
e
-
men
t
a
l
ang
l
e s
t
eps
(
de
t
e
r
m
i
n
i
s
ti
c
)
o
r
b
y
choos
i
ng
t
he
ang
l
es o
f r
o
t
a
ti
on
r
andom
ly.
We conc
l
uded
t
ha
t f
o
r
2
D
if
p
r
e
f
e
r
ab
l
e
t
o use de
t
e
r
m
i
n
i
s
ti
c app
r
oach
, f
o
r
3
D
t
he
t
wo me
t
hods a
r
e equ
iv
a
l
en
t
bu
t f
o
r
h
i
ghe
r
d
i
mens
i
on
we e
x
pec
t t
he
r
andom
t
o be p
r
e
f
e
r
ab
l
e
.
A
no
t
he
r
ad
-
v
an
t
age o
f t
he a
l
go
rit
hm
i
s
t
ha
t it
can be
i
mp
l
emen
t
ed
us
i
ng pa
r
a
ll
e
l
p
r
ocess
i
ng s
i
nce a
ll t
he
r
o
t
a
ti
ons can be
e
x
ecu
t
ed s
i
mu
lt
aneous
ly.
A
li
m
it
a
ti
on o
f t
he p
r
oposed
a
l
go
rit
hm
i
s
t
ha
t it r
equ
ir
es a p
ri
o
r
es
ti
ma
ti
on o
f t
he
ma
xi
mum so
li
d ang
l
e o
f t
he con
v
e
x
po
lyt
ope and
i
n
f
u
t
u
r
e wo
r
k we w
ill try t
o add
r
ess
t
h
i
s p
r
ob
l
em
.
Usage
i
n
r
ea
l ti
me mu
lti
d
i
mens
i
ona
l
app
li
ca
ti
ons
i
s ano
t
he
r
one
,
so we p
l
an
t
o de
v
e
l
op a
f
as
t
e
r v
e
r
s
i
on o
f t
he a
l
go
-
rit
hm su
it
ab
l
e
f
o
r
g
r
aph
i
cs ca
r
ds us
i
ng mu
lti
p
l
e
G
PUs
b
y
e
x
p
l
o
iti
ng
it
s pa
r
a
ll
e
l i
mp
l
emen
t
a
ti
on
.
F
i
na
lly,
us
-
i
ng mo
r
pho
l
og
i
ca
l
ope
r
a
t
o
r
s we w
ill try t
o e
xt
end
it
s
applicability to non-convex objects,
ACKNOWLEDGMENTS
T
he au
t
ho
r
s w
i
sh
t
o
t
hank
t
he membe
r
s o
f t
he
Vi
sua
l-
i
za
ti
on
& Virt
ua
l
Rea
lity
G
r
oup o
f t
he Un
iv
e
r
s
ity
o
f
Pa
-
tr
as as we
ll
as
D
r.
A
.
Kou
t
soud
i
s and
D
r.
G
. I
oannak
i
s
fr
om
t
he Mu
lti
med
i
a Resea
r
ch
L
ab o
f t
he
X
an
t
h
i’
s
D
i-
vi
s
i
on o
f t
he
”
A
t
hena
”
Resea
r
ch and
I
nno
v
a
ti
on Cen
t
e
r,
f
o
r t
he
ir
use
f
u
l
commen
t
s and d
i
scuss
i
ons du
ri
ng
t
he
i
n
iti
a
l
p
r
epa
r
a
ti
on o
f t
h
i
s wo
r
k
.
Cons
t
an
ti
nos Chamzas
f
o
r t
h
i
s wo
r
k was suppo
rt
ed b
y
NSF
-
G
RFP
1
8
4
2
4
9
4
and Kons
t
an
ti
nos Mous
t
akas b
y t
he Eu
r
opean Un
i
on
’
s
Ho
ri
zon 2
0
2
0 r
esea
r
ch and
i
nno
v
a
ti
on p
r
og
r
amme un
-
der Grant Agreement No 871738 - CPSoSaware.
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