It is im p o rta n t to reco rd a c c u ra te ly d ifferences a n d sim ilarities
b etw een face sh a p e s . M ost of th e m a th e m a tic a l m e th o d s for
d e scrib in g th e s h a p e of th e face a re d e p e n d e n t o n a n a to m ic a lly
d efined la n d m a rk p o in ts. A p art from th e p ro b lem of iden tify in g
in d iv id u a ls, la n d m a rk a n a ly sis also ex clu d es a m ajo rity of w h a t we
perceive a s th e face, i.e. th e su rfa ce b etw een la n d m a rk p o in ts.
1.3.1 In te r la n d m a rk d ista n c e s
L a n d m a rk c o -o rd in a te s c a n be u s e d to d e te rm in e th e
re la tio n sh ip b etw een p o in ts u s in g b a sic m a th e m a tic s p rin cip le s:
If L a n d m a rk A is ( x \ y i, z^) a n d L a n d m a rk B is (x^, y^, z^) th e n th e
d ista n c e b etw een th e two la n d m a rk s c a n be c a lc u la te d a s follows:
d ista n c e 12 = + (z, - f
W here d ista n c e 12 is th e d ista n c e b etw een two p o in ts; 1 a n d 2.
Sim ilarly th e a n g u la r re la tio n sh ip b etw een p o in t c a n be c a lc u la te d by
first c a lc u la tin g th e d ista n c e b etw een th e p o in ts a n d th e n u s in g
trig o n o m etry :
If L a n d m a rk A is (x^, y i, z^) a n d L a n d m a rk B is (x^, y2, z^) a n d
L a n d m a rk C is (x^, y^, z^) th e n th e angle b etw een th e th re e la n d m a rk s
c a n be c a lc u la te d a s follows:
' — ' 2 2 2
(12 + 2 3 - 3 1 ) an g le 123 = --- = — =
2 (1 2 x 2 3 )
A te c h n iq u e ap p lied for a n a ly sis of facial a sy m m e try u s in g
b a sic trig o n o m etry h a s b e en d e scrib e d w h ere a se t of 13 la n d m a rk s
a re p lac ed o n th e face a n d u s e d to m ak e 13 tria n g le s e a c h sid e of th e
tria n g le s is th e n c a lc u la te d a n d c o m p a red w ith th e c o rre sp o n d in g
tria n g le on th e o p p o site side of th e face (Moss e t al 1991).
T he m ajo rity of w ork a n a ly sin g facial d a ta h a s b e e n d e p e n d e n t
on th e d e sc rip tio n of th e in te g ra te d difference of a se t of h o m ologous
la n d m a rk s betw een two d a ta sets. P rin cip al c o m p o n e n t a n a ly sis,
P ro c ru ste s a n a ly sis a n d E u clid e a n d is ta n c e m a trix a n a ly sis (EDMA)
are m e th o d s com m only u s e d to explore c o n fig u ra tio n al c h a n g e s
b etw een la n d m a rk d a ta sets.
T h ese m e th o d s of a n a ly sis provide a s ta tis tic for th e d e sc rip tio n
of s h a p e v a ria tio n w h e n tiy in g to se p a ra te d a ta s e ts in to g ro u p s, on
th e b a s is of pathology, sex, age or e th n ic v a ria tio n (D ean et a l 2000;
S ingh et al 1998; H a n ih a ra 1997; R ich tsm eier a n d Lele 1990). The
m ajo r d ra w b a c k w ith th e s e m e th o d s is th e difficulty of a c c u ra te a n d
rep ro d u c ib le p la c e m e n t of la n d m a rk s in h o m o lo g o u s p o sitio n s on
different su b je c ts. T his is very h a rd to acc o m p lish w ith a n extrem ely
v ariab le s h a p e s u c h a s th e face a n d a d d itio n a l m e th o d s h a v e b een
in tro d u c e d to cope w ith th is p ro b lem th ro u g h c re a tio n of ‘p s e u d o ’
la n d m a rk s , e sse n tia lly in te rp o la te d from th e a n a to m ic a l la n d m a rk s ,
to in c re a s e th e rep e rto ire of p o in ts (B ookstein 1997). H ow ever a n
a p p ro a c h relying on overall c o n fig u ra tio n al d ifferen ces is p ro b lem atic
w h e n try in g to a s s e s s in d iv id u al v a ria tio n in te rm s of d is c re te facial
1.3.2 E igen faces a n d N eural N etw orks
M uch of th e w ork in c o m p u te r reco g n itio n of faces h a s focused
on d e te ctin g in d iv id u al fe a tu re s s u c h a s th e eyes, n o se , m o u th , a n d
h e a d o u tlin e, a n d defining a face m odel by th e p o sitio n , size, a n d
re la tio n sh ip s a m o n g th e s e fe a tu re s. T h ese s y ste m s u s e a u to m a te d or
se m i-a u to m a te d face reco g n itio n stra te g ie s w h ich m o d el a n d classify
faces on th e b a s is of n o rm alised d is ta n c e s a n d ra tio s a m o n g p o in ts
s u c h a s th e eye c o rn e rs, m o u th c o rn e rs, n o se tip , a n d c h in p o in t
(Craw et al 1987).
R ecently, a u to m a te d a p p ro a c h e s h av e fo cu sed m o re o n th e
overall co n fig u ratio n , o r g estalt-lik e n a tu r e of facial reco g n itio n w ith
th e d ev elo p m en t of te c h n iq u e s s u c h a s eigenfaces a n d n e u ra l
n e tw o rk s (Haxby et al 2000). The eigenface te c h n iq u e b re a k s u p th e
face in to c o m p o n e n t p a rts (these c a n be u p to 100 iso la te d reg io n s of
th e face or c o m b in a tio n s of featu res) a n d s e a rc h e s a d a ta b a s e of faces
for a m a tc h . A n e u ra l n etw o rk is a p ro c e ss in g device, e ith e r a n
alg o rith m , or a c tu a l h a rd w a re , d esig n ed to w o rk sim ilarly to a n im a l
b ra in s . M ost n e u ra l n etw o rk s have som e s o rt of "training" ru le
w h ereb y th e w eig h ts of c o n n ec tio n s a re a d ju s te d o n th e b a s is of
p re s e n te d p a tte r n s so t h a t th e n e u ra l n e tw o rk s effectively le a rn from
ex am p les. N eu ral n e tw o rk s u s e d for face id en tificatio n h av e
p ro p e rtie s sim ila r to th o se of n e u ro n e s in reg io n s of th e p rim a te
o n th e availability of a large n u m b e r of labelled e x am p les of face ty p es
t h a t c a n be u s e d for tra in in g th e n etw o rk .
B oth n e u ra l n e tw o rk s a n d eigenface te c h n iq u e s h av e b een
su c c e ss fu l in pilot s tu d ie s for finding a n d reco g n isin g faces, lip-
read in g , sex classificatio n , a n d e x p re ssio n reco g n itio n . T he eigenface
te c h n iq u e h a s b e en su c ce ssfu lly ap p lied to d a ta b a s e s c o n ta in in g u p
to 8 ,0 0 0 face im ag es a n d is re p o rte d to achieve reco g n itio n ra te s
g re a te r th a n 90% (Turk a n d P e n tla n d , 1991). N eu ral n e tw o rk s have
also b e en re p o rte d to p erform well in ta s k s s u c h a s h u m a n sex
d e te rm in a tio n from face im ag es (Golomb et al 1991).
1.3.3 S u rfac e se g m e n ta tio n
W ith th e d ev elo p m en t of 3D d a ta c a p tu re te c h n iq u e s s u c h a s
th e o p tical su rfa c e s c a n n e r th e re w as a n o p p o rtu n ity to d e scrib e
co m p lete su rfa c e m orphology w ith o u t losing in fo rm a tio n b etw een
la n d m a rk p o in ts. S u rface type a n a ly sis m e th o d s w ere developed th a t
co u ld b re a k u p a su rfa ce in to c o m p o n e n t s h a p e ty p es. M ethods
d e sc rib e d u s e d ifferen tial geom etry a n d p a tc h -fittin g a lg o rith m s
(Stokely a n d W u 1992) a t a selected p a tc h s u r r o u n d in g e a c h d a ta
p o in t to c a lc u la te m e a n (H) a n d G a u s s ia n c u rv a tu re (K) p ro p e rtie s
a c ro s s th e su rfa ce . T he m e a n a n d g a u s s ia n c u rv a tu re s re la te to th e
in trin s ic a n d ex trin sic p ro p erties, th e s tre tc h in g a n d b e n d in g
p ro p e rtie s, of a su rface. T he th e o iy is well d e sc rib e d by several
a u th o r s (Besl a n d J a in , 1985; K o en d erin k a n d v a n D o o m , 1992). The
th e su rfa c e o n th e b a s is of eig h t prim itive su rfa c e ty p es t h a t c a n be
co lo u r coded to facilitate in te rp re ta tio n (Besl a n d J a i n 1986).
T his h a s b e en ap p lied to 3D facial d a ta o b ta in e d from th e la s e r
s c a n n e r w ith th e aim of providing a q u a n tita tiv e d e sc rip tio n of th e
face (C oom bes et al 1992) a n d also to d e scrib e d ifferences in p a tie n ts
faces p re- a n d p o st- su rg e iy (Coom bes e t al 1990). T he x,y,z co
o rd in a te s from th e la s e r s c a n c a n be u s e d to c o m p u te m e a n (H) a n d
g a u s s ia n (K) c u rv a tu re s a c ro ss th e su rfa ce of th e face to p ro d u c e th e
su rfa c e type a t e ac h d a ta point. T he d a ta p o in t is th e n c o lo u r coded
on th e b a sis of th e su rfa c e type v alu e a n d se g m e n ts th e face into
reg io n s of sim ilar sh a p e .
The S h a p e In d ex valu e ta k e s th is c lassificatio n sc h e m e fu rth e r
to c a p tu re th e n o tio n of th e local sh a p e of a su rfa c e a n d a ttr ib u te a
n u m e ric a l v alu e (Dorai a n d J a in 1997). T he S h a p e In d ex scale ra n g e s