cMinMax: A Fast Algorithm to Find the Corners of an N-dimensional
Convex Polytope
Dimitrios Chamzas
1,2 a
, Constantinos Chamzas
3 b
and Konstantinos Moustakas
1 c
1
Department of Electrical and Computer Engineering, University of Patras, Rio Campus, Patras 26504, Greece
2
McCormick School of Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, U.S.A.
3
Department of Computer Science, Rice University, Houston, TX 77251, U.S.A.
Keywords:
A ugmen t ed Rea lity En vir onmen t s , I mage Reg i s tr a ti on , Con v e x Po ly gon Co r ne r D e t ec ti on A l go rit hm ,
N-dimensional Convex Polyhedrons.
Abstract:
D u ri ng t he l as t y ea r s , t he eme r g i ng fi e l d o f A ugmen t ed & Virt ua l Rea lity ( A R -V R ) has seen tr emendous g r ow t h .
A t t he same ti me t he r e i s a tr end t o de v e l op l ow cos t h i gh - qua lity A R s y s t ems whe r e compu ti ng powe r i s
i n demand . Fea t u r e po i n t s a r e e xt ens iv e ly used i n t hese r ea l-ti me fr ame -r a t e and 3 D app li ca ti ons , t he r e f o r e
e ffi c i en t h i gh - speed f ea t u r e de t ec t o r s a r e necessa ry. Co r ne r s a r e such spec i a l f ea t u r es and o ft en a r e used as t he
fir s t s t ep i n t he ma r ke r a li gnmen t i n A ugmen t ed Rea lity ( A R ). Co r ne r s a r e a l so used i n i mage r eg i s tr a ti on and
r ecogn iti on , tr ack i ng , S L A M , r obo t pa t h fi nd i ng and 2 D o r 3 D ob j ec t de t ec ti on and r e tri e v a l. T he r e f o r e t he r e i s
a l a r ge numbe r o f co r ne r de t ec ti on a l go rit hms bu t mos t o f t hem a r e t oo compu t a ti ona lly i n t ens iv e f o r use i n
r ea l-ti me app li ca ti ons o f an y comp l e xity. Man y ti mes t he bo r de r o f t he i mage i s a con v e x po ly gon . Fo r t h i s
spec i a l, bu t qu it e common case , we ha v e de v e l oped a spec ifi c a l go rit hm , cM i nMa x. T he p r oposed a l go rit hm
i s f as t e r, app r o xi ma t e ly b y a f ac t o r o f 5 compa r ed t o t he w i de ly used Ha rri s Co r ne r D e t ec ti on a l go rit hm . I n
add iti on i s h i gh ly pa r a ll e li zab l e . T he a l go rit hm i s su it ab l e f o r t he f as t r eg i s tr a ti on o f ma r ke r s i n augmen t ed
r ea lity s y s t ems and i n app li ca ti ons whe r e a compu t a ti ona lly e ffi c i en t r ea l ti me f ea t u r e de t ec t o r i s necessa ry. T he
algorithm can also be extended to N-dimensional polyhedrons.
1 INTRODUCTION
A ugmen t ed & Virt ua l Rea lity ( A R -V R ) s y s t ems and
app li ca ti ons ha v e seen mass iv e de v e l opmen t and ha v e
been s t ud i ed e xt ens iv e ly o v e r t he l as t f ew decades
( B illi nghu r s t e t a l., 2 015). A l so w it h t he de v e l op -
men t o f t h r ee - d i mens i ona l measu ri ng t echno l og i es ( 3 D
Scanne r s ) it i s poss i b l e t o acqu ir e t h r ee - d i mens i ona l
da t a us i ng i ne x pens iv e t h r ee d i mens i ona l scanne r s r a i s -
i ng t he e x pec t a ti on t ha t t h r ee - d i mens i ona l da t a and
i n t e rf aces w ill be used . A t t he same ti me t he r e i s a
tr end t o de v e l op l ow cos t h i gh - qua lity 3 D A R s y s -
t ems whe r e compu ti ng powe r i s i n demand . F i gu r e 1
shows such a l ow cos t 3 D A ugmen t ed Rea lity s y s t em
us i ng a t ang i b l e i n t e rf ace and cons tr uc t ed us i ng com -
mod ity ha r dwa r e ( Chamzas and Mous t akas , 2 0 2 0). It s
cen tr a l p r ocess i ng un it i s a Raspbe rry P i 4 equ i pped
w it h a Raspbe rry came r a . Mo r eo v e r sma rt phones a r e
con ti nuous ly e v o lvi ng , add i ng mo r e compu t e r powe r,
a
https://orcid.org/0000-0002-4375-5281
b
https://orcid.org/0000-0001-5830-5542
c
https://orcid.org/0000-0001-7617-227X
F i gu r e 1: 3 D A ugmen t ed Rea lity T ang i b l e Use r I n t e rf ace
us i ng Commod ity Ha r dwa r e Us i ng cM i nMa x t o r eg i s t e r Rea l
World to AR World.
mo r e senso r s , and h i gh - qua lity d i sp l a y. Mu lti came r as
and dep t h senso r s a r e some o f t he ir r ecen t add iti ons .
T he r e f o r e , we e x pec t t ha t it w ill be poss i b l e t o i m -
p l emen t a ll t he f unc ti ona liti es o f an A R s y s t em j us t
i n a sma rt phone . I n t h i s cases , compu ti ng powe r w ill
be i n demand and we w ill need t o de v e l op new f as t
and e ffi c i en t a l go rit hms . O ne o f t he ma i n p r ob l ems
Chamzas, D., Chamzas, C. and Moustakas, K.
cMinMax: A Fast Algorithm to Find the Corners of an N-dimensional Convex Polytope.
DOI: 10.5220/0010259002290236
In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 1: GRAPP, pages
229-236
ISBN: 978-989-758-488-6
Copyright
c
 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
229

i n t hese s y s t ems i s t he r eg i s tr a ti on o f t he Rea l and
Virt ua l wo rl d , whe r e we need t o map t he r ea l- wo rl d
3 D coo r d i na t es (x
r
,y
r
,z
r
) t o t he d i g it a l wo rl d coo r d i-
na t es (x
v
,y
v
,z
v
) . O ne common ly used t echn i que i s t he
i mage ma r ke r. We p l ace an ob j ec t, t he ma r ke r, w it h
a known shape i n t he r ea l wo rl d and we wan t t o fi nd
a p r o j ec tiv e tr ans f o r ma ti on t ha t w ill map t h i s ob j ec t
t o it s virt ua l wo rl d coun t e r pa rt. T h i s tr ans f o r ma ti on
has t o be r eca l cu l a t ed e v e ry ti me t he came r a changes
pos iti on w it h i n t he r ea l wo rl d en vir onmen t and f o r r ea l
ti me s y s t ems t h i s r equ ir es a subs t an ti a l amoun t o f t he
s y s t ems compu t e r r esou r ces . T h i s becomes e v en wo r s t
when we a r e dea li ng w it h ma r ke rl ess A R s y s t ems . A
common app r oach t o add r ess t h i s r eg i s tr a ti on p r ob l em
i s fi nd i ng f ea t u r es on t he r ea l wo rl d ma r ke r and s i nce
we know t he ir pos iti on i n t he Virt ua l wo rl d , we can ca l-
cu l a t e t he r equ ir ed p r o j ec tiv e tr ans f o r ma ti on . Co r ne r s
are such features.
D e t ec ti ng Co r ne r s i s a l so t he fir s t s t ep i n man y
Compu t e r Vi s i on and O b j ec t i den tifi ca ti on and r e tri e v a l
t asks . It i s a l so i mpo rt an t t o a r eas such as med i c i ne ,
eng i nee ri ng , en t e rt a i nmen t and so on t ha t a r e i nc r eas -
i ng ly r e lyi ng i n p r ocesses t ha t r equ ir e t h i s k i nd o f
i n f o r ma ti on . I n t h i s wo r k we p r esen t a s i mp l e and f as t
a l go rit hm t ha t add r esses t he abo v e p r ob l em when t he
border of the image is a convex polygon.
2 PREVIOUS WORK
T he p r ob l em t o fi nd t he co r ne r s i n an i mage was e x am -
i ned i n t he pas t. Mos t o f t he me t hods p r esen t ed we r e
based on t he o ri g i na l a l go rit hm p r oposed i n ( Ha rri s
e t a l., 1 988 ), whe r e t he y compu t e a co r ne r b y e x p l o iti ng
sudden changes i n i mage b ri gh t ness . SUS A N ( Sm it h
and B r ad y, 1 99 7) i s ano t he r a l go rit hm w i de ly used
f o r edge and co r ne r de t ec ti on . Us i ng mo r pho l og i ca l
ope r a t o r s was ano t he r app r oach (Li n e t a l., 1 998 ) used
t o fi nd t he co r ne r s i n an i mage . A d iff e r en t app r oach
us i ng mach i ne l ea r n i ng was a l so p r oposed i n ( Ros t en
and Drummond, 2006).
W it h t he de v e l opmen t o f t h r ee - d i mens i ona l t echno l-
og y and t he usage o f V R & A R and Robo ti c s y s t ems ,
another field that is growing fast over the last years is
3 D o r mu lti d i mens i ona l da t a . F i nd i ng po i n t s o f i n t e r-
es t i n 3 D c l ouds ( Nous i as e t a l., 2 0 2 0 a ; Nous i as e t a l.,
2 0 2 0 b ) o r decompos i ng mu lti d i mens i ona l wo r kspaces
i n t o l oca l p ri m itiv es ( Chamzas e t a l., 2 01 9 ), becomes
i mpo rt an t and aga i n co r ne r s (v e rti ces ) a r e one o f t hem .
A n e xt ens i on o f Ha rri s Co r ne r D e t ec ti on a l go rit hm t o
3 D was p r oposed i n ( G ł omb , 2 00 9 ; S i p ir an and Bus -
t os , 2 011). A n e x amp l e o f e xt end i ng SUS A N t o 3 D
po i n t c l ouds i s desc ri bed i n ( Wa lt e r e t a l., 2 00 9 ) wh il e
i n ( Ka t sou l as and Be r gen , 2 001) t he r e i s an i nd ir ec t
me t hod t ha t e xtr ac t s edges fr om a 3 D po i n t c l oud , and
t hen r ega r ds t hese i n t e r sec ti on po i n t s as co r ne r s . I n
( A be e t a l., 2 017), a t echn i que i s p r esen t ed t ha t es -
ti ma t es t he v e rti ces i n a 3 D Po i n t C l oud on con v e x
po ly hed r a su rf aces us i ng D e l auna y T e tr ahed r a li za ti on .
Con v e x Hu ll a l go rit hms ( Be r g e t a l., 2 01 3 ; T o t h e t a l.,
2017) could also be used to determine the corners.
A ll o f t he abo v e a l go rit hms ha v e a cons i de r ab l e
p r ocess i ng cos t as compa r ed t o t he p r oposed t echn i que ,
wh i ch i s s i mp l e , r obus t and app li cab l e t o an y d i men -
s i on . Mo r eo v e r i s h i gh ly pa r a ll e li zab l e . T he i npu t i n
t he p r oposed me t hod i s a po i n t c l oud con t a i ned i n a
convex polytope acquired by an appropriate scanner.
3 THE ALGORITHM
I n i mage r eg i s tr a ti on we o ft en need t o fi nd t he co r ne r s
o f t he i mage . Man y ti mes t he bo r de r o f t he i mage i s
a con v e x po ly gon . Fo r t h i s spec i a l, bu t qu it e common
case , we ha v e de v e l oped a spec ifi c a l go rit hm , r e f e rr ed
as cM i nMa x. T he a l go rit hm u tili zes t he f ac t t ha t if we
fi nd t he x- coo r d i na t es o f t he p ix e l s t ha t be l ong t o t he
i mage , t hen t he ir ma xi mum , x
max
, i s a co r ne r’ s coo r d i-
na t e . S i m il a rly f o r x
min
, y
min
and y
max
. T he p r oposed
a l go rit hm i s app r o xi ma t e ly 5 ti mes f as t e r t han t he Ha r-
ri s Co r ne r D e t ec ti on A l go rit hm , bu t it s app li cab ility i s
limited only to convex polygons.
3.1 The Algorithm Steps for 2D
The basic steps of the algorithm are
1. P r epossess i ng : G ene r a t e a b i na ry v e r s i on o f t he
image.
2. If φ
max
= 2ω
max
i s t he e x pec t ed ma xi mum ang l e
of the polygon, choose M >
π
2(π−φ
max
)
,
3. Fo r k = 0,1,..,M −1 , r o t a t e t he i mage b y ∆θ =
k ∗π/(2M) = k(π −φ
max
)
4. P r o j ec t t he i mage on t he v e rti ca l and ho ri zon t a l
a xi s and fi nd t he (x
min
,x
max
,y
min
,y
max
) . T hese a r e
coo r d i na t es o f f ou r co r ne r s o f t he r o t a t ed con v e x
polygon.
5. Ro t a t e t he i mage backwa r ds b y −∆θ t o t he i n i-
ti a l pos iti on and fi nd t he coo r d i na t es o f t he f ou r
corners.
6. A t t he end , we ha v e f ound 4M po i n t s wh i ch i s
g r ea t e r t han t he numbe r o f e x pec t ed po ly gon co r-
ne r s . Hence , t he r e a r e mo r e t han one p ix e l s a r ound
each co r ne r. We e v a l ua t e now t he cen tr o i d f o r each
o f t hese bunches and t hese a r e t he es ti ma t ed co r-
ners of the convex polygon.
GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications
230

F i gu r e 2 : D e t ec t ed co r ne r s i n a he x agon f o r M = 3 r o t a ti ons .
In each rotation we detect 4 corners.
I n F i gu r e 2 we app ly t he a l go rit hm i n a he x agon .
We have φ
max
= 150
◦
, thus M >
180
◦
2(180
◦
−150
◦
)
= 3
3.2 The Proof
I n t h i s sec ti on we p r esen t t he t heo r e ti ca l backg r ound
for the algorithm.
The Problem: Find the N-corners in an image that
con t a i ns an ob j ec t w it h a bounda ry t ha t i s a con v e x
polygon.
D e fi n iti on : L e t us ha v e a con v e x po ly gon w it h N -
co r ne r s w it h coo r d i na t es I = (x
i
,y
i
)
i=1,N
. O ne co r ne r
w it h coo r d i na t es (x
k
,y
k
) i s ca ll ed d i sco v e r ab l e , if one
o f it s coo r d i na t es i s ma xi mum o r m i n i mum i n t he se t
I, that is
x
k
= max or min o f (x
1
,x
2
,...,x
N
) or (1)
y
k
= max or min o f (y
1
,y
2
,...,y
N
)
E x amp l e : I n F i gu r e 3 , t he co r ne r s A,C,D o f t he
pen t agon (ABCDE) a r e d i sco v e r ab l e , wh il e B,E a r e
not.
Figure 3: Discoverable Corners.
P r opos iti on 1: We ha v e t wo connec t ed li nes (OO
0
)
and (O
0
A) w it h t he ang l e φ = ∠OO
0
A t o be cons t an t
( see F i gu r e 4). If we r o t a t e (OO
0
) i n i nc r emen t s o f ∆θ
a r ound O , t hen (O
0
A) w ill r o t a t e a l so i n i nc r emen t s o f
∆θ around O
0
.
P r oo f: L e t us r o t a t e (OO
0
) b y ∆θ = θ
2
−θ
1
, fr om
pos iti on P
1
t o pos iti on P
2
. T he li ne segmen t (O
0
A)
r o t a t es a r ound O
0
fr om θ
1
+φ t o θ
2
+φ , and t he change
i s aga i n ∆θ = (θ
2
+φ) −(θ
1
+φ) . T he r e f o r e if t he li ne
Figure 4: Rotating Rigidly Connected Lines.
segmen t (OO
0
) r o t a t es a r ound O
0
i n i nc r emen t s o f ∆θ ,
t hen , neg l ec ti ng tr ans l a ti ons , t he li ne segmen t (O
0
A)
rotates also in increments of ∆θ around O
0
.
P r opos iti on 2 : I n a con v e x po ly gon , B i s one o f it s
co r ne r s , ∠B i s it s ang l e and φ = 180
o
−∠B it s e x p l e -
men t a ry. If we r o t a t e t he po ly gon a r ound B i n i nc r e -
men t s o f ∆θ < φ , t hen i n a t mos t M ≥
2π
∆θ
r o t a ti ons , t he
ad j acen t po i n t s A and C w ill be a t l eas t once t o t he l e ft
side of vertical line yy
0
.
Figure 5: Invariant Rotation Center.
P r oo f: L e t us assume t ha t we r o t a t e t he po ly gon
coun t e r c l ockw i se a r ound B , i n i nc r emen t s o f ∆θ s t a rt-
i ng fr om pos iti on P
1
( F i g . 5 ( a )). I n M s t eps , BC
w ill make a f u ll r o t a ti on a r ound B . Now l e t us con -
s i de r pos iti on P
2
when BC mo v es f o r t he fir s t ti me
t o t he l e ft o f yy
0
, ( F i g . 5 ( b )). T hen ω < φ and
ω + ∠CBA < φ + ∠CBA = 180
o
. T he r e f o r e po i n t A
i s t o t he l e ft o f yy
0
. S i nce t he po ly gon i s con v e x, a ll
it s co r ne r s a r e t o t he l e ft o f yy
0
, t he r e f o r e it s coo r d i-
na t e x
B
, w ill be a t l eas t once t he ma xi mum o f a ll t he
x-coordinates of the polygon angles.
Co r o ll a ry 1: Fo r t he co r ne r B t o be discoverable , it
i s su ffi c i en t t o r o t a t e t he po ly gon a r ound B i n M ≥
pi
2∆θ
with ∆θ < φ (see Fig. 5).
P r oo f: I n o r de r f o r co r ne r B t o be d i sco v e r ab l e , it
i s enough f o r it s t wo ad j acen t edges t o be t o t he l e ft
cMinMax: A Fast Algorithm to Find the Corners of an N-dimensional Convex Polytope
231

o f yy
0
, o r t o t he ri gh t o f yy
0
, o r abo v e xx
0
o r be l ow xx
0
.
Therefore we need only M ≥
pi
2∆θ
rotation steps.
T heo r em 1: We ha v e a con v e x po ly gon see F i g . 5
( a )), and l e t φ be t he e x p l emen t a ry o f it s ma xi mum
ang l e . We se l ec t a po i n t O and we r o t a t e t he po ly gon
a r ound it i n i nc r emen t s o f ∆θ < φ , t hen i n a t mos t
M ≥
π
2∆θ
, a ll it s co r ne r s w ill be d i sco v e r ab l e a t l eas t
once.
P r oo f: L e t B a co r ne r o f t he po ly gon . T he ang l e
be t ween OB and it s ad j acen t edges i s cons t an t du ri ng
t he r o t a ti on a r ound O . T hen because o f P r opos iti on
1, as we r o t a t e t he po ly gon a r ound O i n i nc r emen t s
o f ∆θ , a ll it s edges a r e r o t a ti ng i n i nc r emen t s o f ∆θ
a r ound t he ir ad j acen t co r ne r s . Consequen tly acco r d i ng
t o Co r o ll a ry 1 if we r o t a t e t he con v e x po ly gon a r ound
a po i n t O i n i nc r emen t s o f ∆θ < φ , t hen i n a t mos t
M ≤
π
2∆θ
s t eps , a ll it s co r ne r s w ill be d i sco v e r ab l e a t
least once.
3.3 Extension to N-dimensional Convex
Polyhedrons
T he a l go rit hm can a l so be e xt ended t o N - d i mens i ona l
polyhedrons.
D e fi n iti on 2 : A se t C i s con v e x if f o r an y po i n t s
x,y ∈ C t he segmen t [x,y] j o i n i ng t hem be l ongs t o
C . A con v e x po ly hed r on i s a po ly hed r on t ha t, as a
so li d , f o r ms a con v e x se t. A no t he r de fi n iti on i s : A
con v e x po ly hed r on can a l so be de fi ned as a bounded
i n t e r sec ti on o f fi n it e ly man y ha lf- spaces ( G r unbaum
and Shephard, 1969; Gr
¨
unbaum, 2013).
D e fi n iti on 3 : L e t us ha v e a con v e x po ly hed r on w it h
N -v e rti ces w it h coo r d i na t es I = (x
i
,y
i
,z
i
)
i=1,N
. O ne
v e rt e x w it h coo r d i na t es (x
k
,y
k
,z
k
) i s ca ll ed d i sco v e r-
ab l e , if one o f it s coo r d i na t es i s ma xi mum o r m i n i mum
in the set I, that is
x
k
= max or min o f (x
1
,x
2
,...,x
N
) or
y
k
= max or min o f (y
1
,y
2
,...,y
N
) or (2)
z
k
= max or min o f (z
1
,z
2
,...,z
N
)
D e fi n iti on 4: L e t O be a v e rt e x o f t he con v e x po ly-
hed r on and OA,OB,OC,OD,OE it s edges ( F i gu r e 6 .
We de fi ne O
1
OO
2
as t he M i n i mum Bound i ng Cone f o r
t he v e rt e x O , t he sma ll es t cone t ha t it s t op i s t he v e rt e x
O and it con t a i ns a ll t he edges o f t he v e rt e x O . T h i s
M i n i mum Bound i ng Cone w ill ha v e a t l eas t t wo o f t he
v e rt e x edges on it s su rf ace and t he r es t i ns i de . L e t a l so
OK be it s a xi s o f s y mme try. T h i s wa y we can assoc i a t e
w it h each v e rt e x o f a con v e x po ly hed r on an ang l e , t he
ang l e ∠O
1
OO
2
= 2ω o f t he M i n i mum Bound i ng Cone .
S i nce t he po ly hed r on i s con v e x, we ha v e φ > 0
o
and
0
o
< ω = ∠KOO
2
< 90
o
.
P r opos iti on 3 : L e t R
x
(θ),R
y
(θ),R
z
(θ) be t he r o -
t a ti on ma tri ces b y θ a r ound a xi s x,y,z r espec tiv e ly
Figure 6: Minimum Bounding Cone of a vertex.
and N
θ
= d
2π
∆θ
e and N
φ
= d
2π
∆φ
e , whe r e ∆θ and ∆φ a r e
i nc r emen t a l r o t a ti on angu l a r s t eps a r ound t he a xi s z
and z . We mu lti p ly a v ec t o r
−→
OK b y t he r o t a ti on ma -
trix R
zy
= R
y
(k∆φ)R
z
(m∆θ) f o r k = 0, 1, 2, ..., N
θ
−1
and m = 0,1,2,...,N
φ
−1 . T he N
θ
N
φ
pos iti ons o f t he
r o t a t ed v ec t o r a r e shown i n F i gu r e 7. A t l eas t f o r one
o f t hem , it s d i s t ance d
x
fr om t he a xi s x i s l ess t han
√
∆θ
2
+∆φ
2
2
. T he same i s a l so tr ue f o r d
z
t he d i s t ance o f
a grid point from the axis z .
F i gu r e 7: 3 D G ri d o f a v ec t o r pos iti on r o t a t ed a r ound a xi s z
and axis y with ∆θ = ∆φ =
π
10
rads.
P r oo f: We fir s t r o t a t e a un it v ec t o r
−→
OK a r ound a xi s
z b y k∆θ un til it goes t o it s nea r es t pos iti on t o p l ane
(x, z ). T h i s i s pos iti on
−−→
OK
1
i n F i gu r e 8 . It s d i s t ance
fr om p l ane (x z ) w ill be l ess t han
∆θ
2
. T hen we r o t a t e it
a r ound a xi s y i n s t eps o f ∆φ un til it goes t o it s nea r es t
pos iti on t o p l ane (x,y). T h i s i s pos iti on
−−→
OK
f
i n F i gu r e 8 .
It s d i s t ance fr om p l ane (xy) w ill be l ess t han
∆φ
2
. T hus
t he r e i s a pa ir (k
x
,m
x
) f o r wh i ch t he v ec t o r
−→
OK goes t o
t he g ri d pos iti on
−−→
OK
x
and f o r t h i s pos iti on it s d i s t ance
d
x
fr om t he a xi s x i s d
x
<
√
∆θ
2
+∆φ
2
2
. S i m il a rly f o r
ano t he r pa ir (k
z
,m
z
) t he v ec t o r
−→
OK goes t o v ec t o r
−−→
OK
z
t he c l oses t g ri d pos iti on t o a xi s z . Howe v e r, we can
ne v e r go c l ose t o second a xi s o f r o t a ti on , a xi s y, s i nce
for any position in the grid its angle to axis y remains
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232

greater or equal to 90
o
−φ (see Figure 7) QED.
Figure 8: 3D Rotation of a vector around axis z and axis y.
P r opos iti on 4: I n a con v e x po ly hed r on , t he ang l e
o f t he M i n i mum Bound i ng Cone o f v e rt e x O i s 2ω .
If we r o t a t e t he po ly hed r on fir s t a r ound a xi s z and
t hen a r ound a xi s y, i n i nc r emen t s o f k∆θ and m∆φ , k =
0,...,N
θ
−1, m = 0,2,...,N
φ
−1 w it h ∆θ < ω,∆φ < ω
and N
θ
≥
2π
∆θ
,N
φ
≥
2π
∆φ
, t hen t he M i n i mum Bound i ng
Cone o f v e rt e x O w ill f a ll a t l eas t once i n t he uppe r
s i de o f t he p l ane v e rti ca l t o a xi s z , pass i ng fr om O ( see
F i gu r e 5). A l so s i m il a rly w ill be once be ll ow and once
above the plane passing from O and vertical to axis x.
Figure 9: 3D Rotation of a Convex Polyhedron.
P r oo f: S i m il a r t o P r opos iti on 2 . A s we mu lti p ly
t he po i n t s o f t he po ly hed r on w it h t he r o t a ti on ma trix
R
z
y t he d ir ec ti on o f v ec t o r
−→
OK w ill go t h r ough a ll it s
co rr espond i ng pos iti on i n it s 3 D g ri d . A s p r o v ed i n
P r opos iti on 3 , a t l eas t one pos iti on o f t he 3 D g ri d co rr e -
spond i ng t o t he a xi s o f s y mme try OK o f t he M i n i mum
Bound i ng Cone o f v e rt e x O w ill f a ll i ns i de t he cone
(OAB). This cone is perpendicular to the plane Q and
the angle of its sides to Q is ω
T heo r em 2 : We ha v e a con v e x po ly hed r on , and l e t
φ
max
= 2ω
max
be t he ma xi mum ang l e o f a ll t he M i n -
i mum Bound i ng Cones co rr espond i ng t o it s v e rti ces .
We se l ec t t wo a xi s o f t he coo r d i na t e s y s t em . i. e . z
and y and b y mu lti p lyi ng a ll t he po i n t s w it h t he r o t a -
ti on ma trix R
z
y , we r o t a t e t he po ly gon a r ound t hem
w it h k∆θ and m∆φ whe r e k = 1,...,N
θ
,m = 1,2,...,N
φ
,
∆θ <
π
2
−ω
max
, ∆φ <
π
2
−ω
max
and N
θ
≥
2π
∆θ
,N
φ
≥
2π
∆φ
.
Then all its vertices will be discoverable at least once
in the axis x and axis z.
Proof: It follows from Proposition 4.
No t e : A d iff e r en t li ne o f p r oo f cou l d be based on
t he f ac t t ha t t he p r o j ec ti on o f con v e x po ly hed r on on a
p l ane i s a con v e x po ly gon . T hus we r o t a t e t he po ly he -
d r on a r ound an a xi s , we p r o j ec t it on t he p l anes t ha t
con t a i n t he a xi s and t hen we app ly t he 2 D a l go rit hm
to the obtained convex polygons.
T he e xt ens i on o f t he a l go rit hm t o N - d i mens i ona l
Convex polytopes is straight forward.
4 IMPLEMENTATION
A c r uc i a l pa r ame t e r i n t he abo v e a l go rit hm was t he
choice of M. With φ
max
the expected maximum angle
o f t he po ly gon , it was shown t ha t if M ≥π/(π −φ
max
) ,
t hen each co r ne r o f t he po ly gon w ill appea r a t l eas t
once i n t he se t o f t he de t ec t ed co r ne r s . Fo r e x amp l e ,
f o r an o rt hogona l pa r a ll e l og r am , φ
max
= 2π/4 , M ≥ 2
and if we chose M = 2 , t he r o t a ti on s t ep i s π/4 . Fo r a
he x agon w it h equa l ang l es , φ
max
= 2π/3 , M ≥ 3 and
if we chose M = 3 t he r o t a ti on s t ep i s π/6 . Howe v e r,
when an edge becomes nea rly v e rti ca l t o an a xi s , due t o
nume ri ca l accu r ac y and no i s y da t a , man y ti mes t he r e
a r e mo r e t han one max o r min po i n t s i n t he p r o j ec ti on
on one a xi s . I n t h i s case we dec i ded t o neg l ec t a ll o f
t hem and go t he ne xt r o t a ti on s t ep . T hus , we mus t
make mo r e r o t a ti on s t eps t han t he one p r ed i c t ed b y t he
t heo r e ti ca l ana ly s i s . A no t he r pa r ame t e r i s t he cen t e r
of the image rotation. A gain, as it was shown, we can
choose an y po i n t as t he i mage r o t a ti on cen t e r, bu t it i s
e x pec t ed t ha t if t he r o t a ti on cen t e r i s t he cen tr o i d o f
the convex polygon, the algorithm to be less sensitive
to numerical errors.
4.1 Examples
4.1.1 2D Case
Fo r t he 2 D case , we used a 2 040x10 8 0 b i na ry i mage
o f a con v e x po ly gon w it h se v en co r ne r s F i gu r e 10 and
φ
max
≈ 158
◦
. T he r equ ir ed numbe r o f r o t a ti ons mus t
be a t l eas t M ≥
180
◦
180
◦
−158
◦
= 8.53
. T hus we used N = 9
rotations with ∆θ =
90
◦
9
= 10
◦
cMinMax: A Fast Algorithm to Find the Corners of an N-dimensional Convex Polytope
233

Figure 10: Estimated Corners in a heptagon (red dots).
4.1.2 3D Case
I n t h i s e x amp l e we used a dodecahed r on po i n t c l oud
ob t a i ned fr om Mesh L ab , w it h 145 3 5 po i n t s . T he l eng t h
o f it s edge i s 3 . 236 1 . T he r esu lt s f o r t h i s dodecahe -
d r on w it h ∆θ = ∆φ =
π
20
a r e shown i n F i gu r e 11. We
d i d 2 0x 2 0=400 r o t a ti ons , f o r e v e ry r o t a ti on we fi nd
6 co r ne r s , 2 i n each a xi s , bu t on ly 282 o f t hem we r e
accep t ed as v a li d and t he y we r e c l ass ifi ed as co r ne r s .
Fo r t he o t he r cases , due t o nume ri ca l accu r ac y we had
mo r e t han one ma x o r m i n i n one a xi s and t he y we r e
r e j ec t ed . T hese 282 po i n t s we r e c l us t e r ed t o 2 0 g r oups ,
and t he ir cen tr o i ds we r e t he es ti ma t ed co r ne r s . T he
a v e r age accu r ac y o f t he es ti ma ti on was app r o xi ma t e
2 % o f t he edge l eng t h . T he ma xi mum ang l e o f t he
m i n i mum bound cone i s ω
max
= 69.095
◦
, and t heo r e ti-
ca lly we cou l d use ∆θ = ∆φ =
π
9
, bu t due t o nume ri ca l
errors and noise we need less than half of it.
F i gu r e 11: Es ti ma t ed Co r ne r s i n a dodecahed r on (r ed do t s ).
Circles indicate the weighted position of corners.
4.2 Evaluation: Computational
Complexity
L e t us assume we ha v e a po i n t c l oud w it h n po i n t s
o f a N - d i mens i ona l po lyt ope and we know t ha t t he
φ
max
= 2ω
max
i s t he e x pec t ed ma xi mum ang l e o f a ll
t he M i n i mum Bound i ng Cones co rr espond i ng t o it s v e r-
ti ces . T hen , f o ll ow i ng T heo r em 2 , w it h ∆θ <
π
2
−ω
max
and N
θ
≥
2π
∆θ
, we ha v e t o pe rf o r m (N −1)N
(N−1)
θ
r o -
t a ti ons o f n po i n t s , i n o r de r a ll it s v e rti ces t o be d i s -
co v e r ab l e a t l eas t once i n one a xi s . T he r e f o r e i n e v e ry
s t ep o f t he a l go rit hm we pe rf o r m one r o t a ti on and t hen
fi nd t he max o f t he n po i n t s x- coo r d i na t es . Bo t h ope r a -
ti ons a r e o f comp l e xity O(n) and we ha v e t o pe rf o r m
a t l eas t L = (N −1)N
(N−1)
θ
s t eps , t hus t he a l go rit hm
compu t a ti ona l comp l e xity i s LO(n) . A t t h i s po i n t we
ha v e t o obse rv e t ha t each s t ep i s i ndependen t fr om t he
o t he r s , t he r e f o r e t he y can compu t ed i n pa r a ll e l and t he
p r oposed a l go rit hm i s h i gh ly pa r a ll e li zab l e . A ssum i ng
t ha t t he a l go rit hm i s r unn i ng i n a compu t e r w it h a t l eas t
L G PUs t hen we can c l a i m t ha t it s comp l e xity i s O(n) .
Con v e x Hu ll and Ha rri s co r ne r D e t ec ti on a l go rit hms
can a l so be used t o add r ess s i m il a r p r ob l ems . Con v e x
Hu ll a l go rit hms a r e d iffi cu lt t o be pa r a ll e li zab l e and
t he ir sequen ti a l v e r s i on i s o f O(nlog(n)) comp l e xity
( Be r g e t a l., 2 01 3 ; T o t h e t a l., 2 017). T o compa r e it w it h
F i gu r e 1 2 : Ra ti o o f e x ecu ti on ti me f o r cM i nMa x and Ha rri s
Co r ne r D e t ec ti on a l go rit hm app li ed t o r egu l a r po ly gons w it h
3 to 25 corners.
t he comp l e xity o f Ha rri s co r ne r de t ec ti on a l go rit hm
i n 2 D ( Chen e t a l., 2 00 9 ), we d i d r un bo t h o f t hem
i n Ma tL ab
®
, us i ng t he detectHarrisFeatures() com -
mand . Fo r 2 D space we ha v e N = 2 and t he comp l e xity
o f cMinMax i s N
θ
O(n) . Fo r i mages w it h 3 -1 2 co r ne r s ,
cM i n i Ma x i s on t he a v e r age 5 ti mes f as t e r t han Ha rri s
Co r ne r de t ec ti on a l go rit hm ( see F i gu r e 1 2 ). I n add iti on
t he p r oposed a l go rit hm appea r s t o be l ess sens itiv e t o
sampling quantization errors.
5 RANDOM SAMPLING
Mos t o f t he ti mes t he numbe r o f unknown co r ne r s i s
no t g iv en and i n add iti on we do no t ha v e a good es ti ma -
ti on o f φ
max
. T hus we canno t es ti ma t e a p r ope r r o t a ti on
s t ep f o r t he app li ca ti on o f cM i nMa x. O ne app r oach
w ill be t o s t a rt w it h an i n iti a l r o t a ti on s t ep . Ne xt we
r educe it and try aga i n , un til t he numbe r o f de t ec t ed
GRAPP 2021 - 16th International Conference on Computer Graphics Theory and Applications
234

co r ne r s r ema i n cons t an t. A n a lt e r na tiv e app r oach i s t o
r o t a t e t he po lyt opes w it h ang l es se l ec t ed r andom ly, I n
t h i s case it i s i mpo rt an t t o ha v e un if o r m ly d i s tri bu t ed
rotations. The 2D case is simple, but we have to to be
ca r e f u l when we dea l w it h ob j ec t s i n w it h d i mens i on -
ality higher than two.
2-D : We se l ec t a r andom ang l e ∆θ i n t he c l osed i n t e rv a l
[−π,π] . We r o t a t e t he con v e x po ly gon b y ∆θ and we
fi nd t he e xtr emes o f t he coo r d i na t es i n t he x- a xi s and
y- a xi s . We con ti nue un til no mo r e d iff e r en t co r ne r s a r e
detected.
F i gu r e 1 3 : ( a ) Un if o r m ly d i s tri bu t ed po i n t s on a sphe r e .
Red dots the 20 dodecahedron vertices. (b) A vector and its
r o t a t ed pos iti ons . ( c ) H i s t og r am o f ( a ) a r ound t he 2 0 r ed do t s .
(d) Histogram of (b) around the 20 red dots.
3-D : We se l ec t t wo r andom ang l es ∆θ and ∆φ . ∆θ i s
un if o r m ly d i s tri bu t ed i n t he i n t e rv a l [−π/2,π/2] . ∆φ i s
r andom ly d i s tri bu t ed i n t he i n t e rv a l [−π,π] w it h a den -
s ity d i s tri bu ti on f (φ) = sin(φ)/2 . T h i s wa y we ha v e
mo r e po i n t s a r ound t he equa t o r φ = 0 , gene r a ti ng t hus
un if o r m ly d i s tri bu t ed pa ir s ∆θ,∆φ on a sphe r e
1
( see
F i gu r e 1 3 ( a )). We r o t a t e now t he con v e x po ly hed r on
b y ∆θ and ∆φ and we fi nd t he e xtr emes o f t he coo r d i-
na t es i n t he x- a xi s , y- a xi s and z - a xi s . T o make t he fi na l
pos iti on o f t he r o t a t ed po i n t s as r andom as poss i b l e , i n
e v e ry s t ep we peak r andom ly one o f t he poss i b l e s ix
poss i b l e a xi s r o t a ti ons . T he r o t a t ed pos iti on o f a v ec t o r
a r e shown i n F i gu r e 1 3 ( a ). I n F i gu r e 1 3 ( c ) and ( d )
we show t he h i s t og r ams f o r ( a ) and ( b ). Each o f t he 2 0
b i ns con t a i n t he po i n t s t ha t c l ose t o t he co rr espond i ng
vertex of dodecahedron (red dots in (a)).
T o s i mp lify ou r ana ly s i s we w ill e x am i ne t he case
whe r e we fi nd t he ma x O N LY i n t he x- a xi s . It i s c l ea r
1
h tt p :// co ry s i mon . g it hub .i o / a rti c l es / un if o r md i s t n - on -
sphere/
t ha t i n e v e ry r o t a ti on we de t ec t on ly O NE co r ne r. T he
ques ti on we wan t t o answe r i s , how man y ti mes do we
ha v e t o r o t a t e a po lyt ope w it h N co r ne r s , i n o r de r t o
de t ec t a ll it s co r ne r s . T h i s p r ob l em i s equ iv a l en t t o t he
f o ll ow i ng D i e p r ob l em (I saac , 1 996 ), irr espec tiv e o f
t he d i mens i ona lity o f t he po lyt ope space , ” Ro ll a d i e
w it h N -f aces . Wha t i s t he e x pec t ed numbe r o f r o ll s t o
get all its N faces?.
2
I n F i gu r e 14, we show t he r equ ir ed numbe r o f r o t a -
ti ons f o r 2 D and 3 D canon i ca l po lyt opes w it h N v e r-
tices. For the case of rotating in equal angle steps, for
t he 2 D canon i ca l po ly gons we ha v e N
rot
=
2π
π−2ω
max
=
N . Fo r 3 D po lyt opes we ha v e N
rot
=

2π
π−2ω
max

2
. Fo r
t he p l a t on i c so li ds w it h N = 4,6,8,12,20 , we ha v e t ha t
t he ir ang l e ω
max
o f t he ir M i n i mum Bound Cones a r e
35.26
◦
,45.00
◦
,54.74
◦
,58.28
◦
69.09
◦
r espec tiv e ly. Fo r
t he r andom case , a canon i ca l po lyt ope i s equ iv a l en t t o
a f a ir d i e w it h N equ i p r obab l e f aces and it i s known
t ha t t he e x pec t ed numbe r o f r o ll s t o ge t a ll it s N f aces i s
t he ha r mon i c mean o f N , i. e . m
N
rot
= N
∑
N
n=1
1
n
. F r om
F i gu r e 14 we conc l ude t ha t f o r 2 D it i s p r e f e r ab l e t o
use t he r o t a ti on i n equa l s t eps , f o r 3 D , r o t a ti on i n equa l
s t eps and r andom r o t a ti on a r e equ iv a l en t bu t f o r h i ghe r
d i mens i ons t he r andom r o t a ti on case i s e x pec t ed t o be
preferable.
F i gu r e 14: Mean o f r equ ir ed Ro t a ti ons f o r a canon i ca l po ly-
t ope w it h N co r ne r s (r ed li ne ). W it h b l ue dashed li ne and
w it h dash - do t b l ack li ne a r e t he r equ ir ed r o t a ti ons f o r t he
deterministic case for 2D polygon and 3D polyhedron.
6 CONCLUSIONS
A new co r ne r es ti ma ti on t echn i que on N - d i mens i ona l
po i n t c l ouds o f con v e x po lyt opes was p r oposed i n t h i s
con tri bu ti on . T he p r oposed a l go rit hm i s based on t he
f ac t t ha t t he m i n and ma x o f p r o j ec t ed coo r d i na t es i n
an y a x es be l ong t o a co r ne r. Fo r 2 D we compa r ed it
w it h t he Ha rri s co r ne r de t ec ti on a l go rit hm (i mp l emen -
t a ti on a t Ma tl ab ) and it was app r o xi ma t e ly 5 ti mes
2
h tt p :// www . c i s .j hu . edu /
∼
xy e / pape r s and pp t s / pp t s /
SolutionsToFourProblemsOfRollingADie.pdf
cMinMax: A Fast Algorithm to Find the Corners of an N-dimensional Convex Polytope
235

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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