Using Symme y E alua ion o Imp o e Robo ic Manipula ion
Pe o mance
P. J. Sanz, R. Ma ´ın and S. Dabi´c
Depa men o Compu e Science & Enginee ing. Uni . Jaume I. Campus Riu Sec (12071-Cas ell´on)
Key wo ds: Vision based Robo ic Manipula ion; 2D G asp De e mina ion; Geome ic Reasoning.
PACS:
1. In oduc ion
P esen ly, he obo ics domain is inc easing i s
pe o mance possibili ies mainly by using all kind
o ecen senso based echnology, join ly wi h e y
e icien so wa e and ha dwa e o p ocess he in-
o ma ion in a sui able manne . In pa icula , new
obo ic applica ions in se ice con ex a e s a ing
o be a ailable now (e.g. space and unde wa e ac-
i i ies, elesu ge y, e c.), as can be obse ed in all
he mos impo an con e ences a ound he wo ld
and in he eal li e scena ios. These eme gen ac-
i i ies in obo ics, ou side he well s uc u ed and
p edic able indus ial domains, would be impossi-
ble wi hou he app op ia e use o senso y in o -
ma ion.
In ou case, a e some yea s o p e ious esea ch
in he obo ic manipula ion a ea, by using com-
pu e ision o guide he g asping ac ions o 2D
objec s [Sanz e al., 98], we ha e disco e ed he im-
po ance o implemen some algo i hms ha make
easie he unde lying geome ic easoning neces-
sa y o imp o e he inal obo ic manipula ion pe -
o mance. In pa icula , he knowledge abou sym-
me y o plana shapes (i.e. 2D images o he ob-
jec s a e a ailable) has been success ully used by
he au ho s [Sanz e al., 99], and some o he e-
sea che s be o e [Blake, 95], wi hin he g asping
de e mina ion domain.
2. P oblem Desc ip ion and P e ious
Resul s
As i has been commen ed be o ehand, in p e i-
ous wo ks he geome ic easoning necessa y o de-
e mine sui able egions o one objec , in o de o
be g asped, was suppo ed by he symme y knowl-
edge associa ed o he con ou o his objec . Wi h
he aim o quan i y his symme y deg ee a new
concep was in oduced by he au ho s [Sanz e al.,
99] he .no malized global symme ic de iciency..
In he ollowing we cla i y his concep . Fo a pla-
na shape wo p i ileged di ec ions exis ela ed o
i s mass dis ibu ion (ine ia), hese di ec ions a e
Imin and Imax, and hey ep esen he eigen ec o s
o he momen o ine ia co a iance ma ix. I mi -
o symme y exis s, hese di ec ions a e he i s
candida es o ha . As a esul o ha he nex
objec i e is e alua e he symme y deg ee associ-
a ed wi h hese wo di ec ions. F om he cu a u e
desc ip ion Kki, he symme y deg ee is compu ed
wi h espec o he Imin associa ed wi h he con-
ou , using he in e sec ion poin s {CTImin },
compu ing ∆i, such as:
∆i=Kk,P3−i−Kk,P3+i;i= 1, K, N
2,
whe e P3∈ {CTImin } a e sing he con ou
clockwise om he ini ial poin , PI, be ween P0
1
and P0
2. Tha is o say, ∆iis compu ed as indica ed,
o each couple o poin s equidis an o P3, co e ing
all he con ou . Using hese quan i ies, we de ine
he no malized global symme ic de iciency as:
Φ = 1
NPi=N
2
i=1 ∆i
whe e N is he o al numbe o poin s in he con-
ou . The same p ocess is u ilized o compu e Φ
o he Imax di ec ion, bu now, ins ead o P3, we
compu e ∆iwi h P0
1∈ { CTImax }, as in oduced
abo e. No e ha Φ = 0, i.e. pe ec mi o symme-
y wi h espec o he conside ed axis, exis s only
o an ideal mi o symme ic objec . Some esul s
in ela ion wi h he use ulness o Φ a e shown in
20 h EWCG Se ille, Spain (2004)
20 h Eu opean Wo kshop on Compu a ional Geome y
Table 1
No malized global symme ic de iciency Φ compu ed o
di e en images in bo h di ec ions Imin and Imax. I shows
he mean,νΦ, and he s anda d de ia ion, ρΦ, o each one.
Table 1.
The da a shown in Table 1 a e he mean and
s anda d de ia ion compu ed om ou digi iza-
ions o each objec a di e en loca ions (posi ion
and o ien a ion) o e he wo k a ea. F om ha a-
ble some empi ical esul s can be obse ed:
Imin di ec ion. Looking a he able a gap be-
ween ”plie s” and ”pince s” is obse ed, Φ =≤3.
A e many ials we ha e ollowed a mi o sym-
me y app oach o hose images ha p esen Φ <
5, named Φc= 5 o his c i ical alue.
Imax di ec ion. In his case he gap appea jus
be ween ”nu ” and he es o shapes. So a alue
Φc= 3 ep esen s a good c i ical alue.
A p ima y classi ica ion o he objec s p esen
in Table 1 would be he ollowing: ”mi o symme-
y o Imin and Imax di ec ions” {nu }; ”mi o
symme y in Imin di ec ion” {nu ..pince s}; and
”wi hou symme y” {Allen w ench}. No e ha
only ”nu ” has mi o symme y in bo h di ec ions
in co espondence wi h i s inhe en adial symme-
y.
These esul s we e applied success ully o he
g asping de e mina ion domain [Sanz e al., 99],
whe e he inpu we e con ou s ex ac ed om 2D
images.
Ne e heless, as i has been ema ked abo e, a
p oblem was de ec ed wi h some shapes, o in-
s ance, he pince s. The di e ences obse ed be-
ween he hopped (i.e. mi o symme y along he
Imin di ec ion) and eal esul s has been he s a -
ing poin o he p esen esea ch con ibu ion. To
Fig. 1. The pe o mance analysis in he case o pince s.
Ma ked wi h a ci cle is obse ed he bad si ua ion be ween
he Imin axis and he ex e nal con ou o his shape (i.e.
P3).
cla i y his si ua ion is con enien o obse e he
Fig 1, in which an image o pince s is p ocessed.
When he cu a u e-symme y usion [Sanz e al.,
99] diag am is used o analyze he pe o mance, we
ound ha he in e sec ion be ween one o he ex-
emes o he Imin axis, and he ex e nal con ou
(i.e.P3), is no well si ua ed (i.e. see he ci cle in
his Fig.1). And when he compu a ion o e alu-
a e Φ is ca ied ou , he inal esul is ha shown
in Table1, namely a highe alue ha he heo e i-
cally hopped o his kind o shape, ha as we can
obse e i is symme ic in ha di ec ion.
3. How o sol e his p oblem?
Well, looking a li e a u e we ind he Ley-
on’s Theo em [Ley on, 87], abou ”symme y-
cu a u e duali y”, we e a local co espondence
be ween a symme y axis and a cu a u e poin
con ou is es ablished. Thus, i a symme y axis
exis s in a shape, his axis in e sec he shape al-
ways in a local ex eme o cu a u e. Ou p esen
con ibu ion has been o implemen his Theo-
em in ou algo i hms in o de o sol e he ini ial
p oblems ound.
And, as wo k in p og ess, we a e now es ing
he use o his new Φ, inco po a ing he Ley on.s
Theo em, as a new desc ip o in au oma ic objec
ecogni ion.
Finally, i is no iceable ha wi h his imp o e-
men we ha e go a e y obus solu ion o e alua e
he symme y deg ee associa ed o a plana shape
in a p ede ined di ec ion, and in a e y as way,
Ma ch 25-26, 2004 Se ille (Spain)
making easible eal applica ions in he obo ics
domain.
Re e ences
[1] [Sanz e al., 98] Sanz PJ, del Pobil AP, Ies a
JM, Reca al G. Vision-Guided G asping o Unknown
Objec s o Se ice Robo s . In IEEE P oc. on Robo ics
and Au oma ion (ICRA.98), pp. 3018-3025. Leu en,
Belgium. May 1998.
[2] [Sanz e al., 99] Sanz PJ, Ies a JM, del Pobil
AP. Plana G asping Cha ac e iza ion Based on
Cu a u e-Symme y Fusion . Applied In elligence 10,
pp. 25-36. Kluwe Academic Pub. 1999.
[3] [Blake, 95] Blake A. A Symme y Theo y o Plana
G asp. The In e na ional Jou nal o Robo ics
Resea ch, Vol.14, No. 5, pp. 425-444. Oc obe , 1995.
[4] [Ley on, 87] Ley on M. Symme y-Cu a u e Duali y,
Compu . Vision G aphics Image P ocess. 38, pp. 327-
341. 1987