P ojec i e Shape Analysis o Spa ial O ien a ion
in Vi ual En i onmen s
Mihaela P icop-Jecks ad 1*, Alexande Ga he2,
Vic Pa angena u3, Robe L. Paige4
1*Cen e o Resea ch and T aining in Inno a i e Techniques o Applied
Ma hema ics in Enginee ing “T aian Lalescu”, Na ional Uni e si y o
Science and Technology Poli ehnica Bucha es , Splaiul Independen ¸ei
no. 313, Bucha es , 060042, Romania.
2Ge man Cen e o Neu odegene a i e Diseases, Fe sche s aße 105,
D esden, 01307 , Saxony, Ge many.
3Depa men o S a is ics, Flo ida S a e Uni e si y, 222 S Copeland S ,
Tallahassee, FL 32306, Flo ida, US.
4Depa men o Ma hema ics and S a is ics, Missou i S&T Uni e si y,
300 W 13 h S , Rolla, MO 65409, Missou i, US.
*Co esponding au ho (s). E-mail(s): [email p o ec ed];
Con ibu ing au ho s: [email p o ec ed];
[email p o ec ed];[email p o ec ed];
Abs ac
In his a icle, we apply ex insic da a analysis o he s udy o cogni i e abili ies
e alua ed based on he lea ning beha iou in he D esden Spa ial Na iga ion Task
(DSNT) i ual na iga ional expe imen . Ou no el ma hema ical modelling o
he spa ial o ien a ion and spa ial lea ning is based on ecen concep s in objec -
o ien ed da a analysis like ex insic mean and ex insic co a iance as well as
no el s a is ical es ing me hods o andom objec s on mani olds. Finally, he
allocen ic o ien a ion pa e ns in pe sons exhibi ing mild cogni i e impai men
(MCI) and con ols a e de ec ed o he i s ime ia p ojec i e shape analysis.
Keywo ds: nonpa ame ic s a is ics, p ojec i e geome y, ex insic da a analysis,
neu oscience, spa ial o ien a ion
1
1 In oduc ion
55 million people we e li ing wi h demen ia in 2020 and i is a apidly g owing
heal h h ea . The numbe o hose li ing wi h demen ia is p ojec ed o almos iple
o 135 million by 2050 ([1]). Ou eseach b ings an impo an medical and social
con ibu ion o heal hie aging and o he unde s anding o some neu ological p o-
cesses undamen al o b ain unc ioning by he in e play be ween neu opsychology,
non-euclidean geome y, s a is ical me hods and applied ma hema ics (see [2,3]).
Mo eo e , he applica ion o ou p ojec i e shape app oach o spa ial o ien a ion go
beyond cogni i e neu oscience owa ds au oma ed o ien a ion o ca s, d ones and
sa elli es as well as obo ics. Addi ionaly, his app oach links spa ial o ien a ion o
quan um mechanics and cosmology ia p ojec i e geome y (see [4,5]). This se ing
migh allow us o connec loss o ime eeling in neu ological diseases and loss o
spa ial o ien a ion, and i migh suppo he hypo hesis ha he b ain unc ions as a
quan um compu e ([6,7]).
A decline in spa ial o ien a ion is one o he ea lies symp oms o demen ia, as
indi iduals lose he abili y o na iga e hei en i onmen ([8]). Spa ial lea ning - he
abili y o mo e e ec i ely om place o place - depends on e e ence memo y, a o m
o long- e m memo y ha s o es s able in o ma ion o e ime ([9]). The hippocampus,
loca ed in he medial empo al lobe, plays a cen al ole in bo h long- e m memo y
and spa ial na iga ion ([10]). Seminal heo ies ha e p oposed ha i unc ions as a
“cogni i e map,” encoding he layou o he en i onmen ([11,12]). E idence o his
comes om bo h human neu oimaging, such as he enla ged hippocampal olume
obse ed in London axi d i e s ([13]), and animal models. The Mo is Wa e Maze
demons a ed ha a s can lea n o loca e a hidden pla o m using dis al cues,
p o iding a classic pa adigm o allocen ic na iga ion ([14,15]). Allocen ic s a e-
gies, which ep esen spa ial ela ions independen o he obse e ’s posi ion, ely
hea ily on he hippocampus, and pa ien s wi h mild cogni i e impai men (MCI) o
Alzheime ’s disease (AD) show ma ked de ici s in such asks ([16,17]). Because he
hippocampus is among he i s b ain egions a ec ed in AD, impai men s in allocen-
ic spa ial na iga ion ep esen a sensi i e cogni i e ma ke o ea ly de ec ion ([18]).
A majo ad ance in s udying hippocampus-dependen spa ial na iga ion in humans
has been he de elopmen o i ual eali y (VR) echniques. These app oaches enable
he ansla ion o pa adigms such as he Mo is Wa e Maze in o compu e ized 3D
en i onmen s ([19,20]). In such asks, pa icipan s a e placed in a i ual ci cula
pool su ounded by dis al cues bu wi hou local ma ke s and a e ins uc ed o loca e
a hidden pla o m using a joys ick. VR o e s p ecise expe imen al con ol, objec i e
acking o beha io al esponses, and s ong ecological alidi y, as na iga ion pe o -
mance in VR co ela es well wi h na iga ion in eal-wo ld se ings ([21,22]). The
D esdne Spa ial Na iga ion Task (DSNT), de eloped a Uni e si ¨a sklinikum Ca l
Gus a Ca us D esden, is one such pa adigm, g ounded in he allocen ic p inciples
o he Mo is Wa e Maze ([14,23]). In he DSNT, pa icipan s mus loca e a hidden
pla o m wi hin one minu e, guided by isual landma ks placed a ound he pool’s
pe ime e . Thei ajec o ies a e eco ded as Ca esian coo dina es and eac ion
imes, p o iding ich quan i a i e da a o analyzing na iga ional s a egies. The
ask includes mul iple ial ypes: p e es s (assessing e lexes and amilia i y wi h he
2
joys ick), acquisi ion ials (assessing lea ning), and p obe ials (assessing memo y
when he pla o m is emo ed). Tes ing spans wo consecu i e days: on Day 1, pa -
icipan s comple e wo p e es s, one p obe ial, and 12 lea ning ials wi h a ixed
pla o m loca ion; on Day 2, wo p e es s a e ollowed by 12 lea ning ials in which
he pla o m loca ion is shi ed a e he ou h ial. Mild cogni i e impai men
(MCI) is widely ega ded as an in e media e s age be ween heal hy aging and demen-
ia, pa icula ly Alzheime ’s disease ([24,25]). In he DSNT da ase , 54 pa icipan s
we e es ed: 25 heal hy con ols, 26 wi h MCI, and 3 wi h mino demen ia o he
Alzheime ype (DAT). All pa icipan s also unde wen s anda d neu opsychological
assessmen s. P e ious s udies sugges ha spa ial na iga ion asks can e eal dis-
inc s a egies— e lec ed in ajec o y pa e ns— ha a e sensi i e o hippocampal
in eg i y and disease s a us ([8,18]). Building on his e idence, ou esea ch examines
how ajec o y pa e ns in he DSNT e eal he in e play be ween spa ial lea ning
and spa ial long- e m memo y, as well as landma k-based o ien a ion—speci ically,
which en i onmen al cues pa icipan s ely on when na iga ing.
2 P ojec i e Shape Da a Analysis
P ojec i e shape analysis has i s o igins in he pa ame ic s a is ics s udy o p ojec-
i e in a ian s. In his se ing, objec s a e obse ed subjec o an unknown p ojec i e
ans o ma ion, and i is usual o use p ojec i e in a ian s o ei he es ing o a
alse ala m o o classi ying an objec . Fo ou collinea poin s, he c oss- a io is
he simples s a is ic which is in a ian unde p ojec i e ans o ma ions ([26]). A
nonpa ame ic me hodology, ia p ojec i e ames in a bi a y dimensions, was de el-
opped in he pape [27] on high le el image analysis while [28] s udies asymp o ics o
p ojec i e in a ian s and ex insic and in insic sample means on p ojec i e mani olds
o unde s anding o 2Dscenes om hei digi al came a images. Fo a landma k
con igu a ion in R3, ”shape” deals wi h he esidual s uc u e o his con igu a ion
when ce ain ans o ma ions a e il e ed ou . The non-shape componen s consis s o
size, posi ion and o ien a ion. P opo ions, ela i e angles and he ela i e a ange-
men o pa s a e geome ic ea u es ha belong o shape. Hence, spa ial o ien a ion
as unde s ood by he allocen ic heo y can be be e s udied in e ms o p ojec i e
shape. Addi ionally, p ojec i e shape is in e p e ed in e ms o axial s a is ics, allow-
ing a di ec ional in e p e a ion o he es ima o s.
Ma hema ically, he shape o a con igu a ion consis s o i s equi alence class unde a
g oup o ans o ma ions. He e he g oup ac ion desc ibes he way in which an image
is cap u ed. Fo example, he pinhole came a model assumes ha a 2D−image is
he esul o he p ojec ion on o he image plane o an ideal pinhole came a, whe e
he came a ape u e is desc ibed as a poin and no lenses a e used o ocus ligh
([29]). I wo di e en images o he same scene a e ob ained using a pinhole came a,
he co esponding ans o ma ion be ween he wo images is he composi ion o wo
cen al p ojec ions om 3D o 2D, which is a p ojec i e ans o ma ion ([30]). The
p ojec i e shape o an objec consis s o he geome ic in o ma ion ha is in a ian
unde di e en came a iews. Mo eo e , hese images di e only by a p ojec i e
3
ans o ma ion be ween hemsel es and om he o iginal scene, e en i he images
a e aken wi h di e en came as. The pinhole came a model as a p ojec i e model
can be ex ended o gene al dimensions. The homogenous coo dina es a e unique up
o non-ze o scala s in he elegan p ojec i e geome ic amewo k. The e o e, he
homogenous coo dina es become poin s in p ojec i e spaces, and he pe spec i e
p ojec ion landma k model in he DSNT expe imen may now be s udied in e ms o
p ojec i e shape geome y om 3D o 3D.
2.1 P ojec i e shape space
We e iew in he ollowing he ma hema ical esul s necessa y o employ he p ojec i e
shape analysis ([2]). In p ojec i e geome y wo non-ze o ec o s xand yin (m+ 1)-
dimensional nume ical space Rm+1 a e equi alen i hey di e by a non-ze o scala
mul iple, whe e mis a abs ac dimension o he image space. The equi alence class o
x∈Rm+1 {0}is labeled [x].The se o all such equi alence classes is he p ojec i e
space P(Rm+1) associa ed wi h Rm+1
P(Rm+1) = {[x]; x∈Rm+1 {0}}.
and is o en deno ed as RPm. The p ojec i e space RPmis opologically an
m-dimensional uni sphe e Smwi h he an ipodal poin s iden i ied, and can be
ep esen ed as a disjoin union o o dina y and ideal p ojec i e poin s;
RPm=
x1
.
.
.
xm
1
∈RPm
| {z }
o dina y poin s
[
x1
.
.
.
xm
0
∈RPm
| {z }
ideal poin s
In his amewo k o ins ance, a anishing poin in a h ee-dimensional image co e-
sponds o an (ideal) poin in RP3.
We now e iew concep s o p ojec i e ans o ma ions and p ojec i e shape space. I
he calib a ions o he came as in he pinhole came a model a e unknown, i.e. i he e is
no in o ma ion a ailable on he came a pa ame e s such as ocal leng h, angle be ween
scene and ilm hype plane, loca ion o he came a, e c., hen an image elies only
on in o ma ion abou he scene which is in a ian unde p ojec i e ans o ma ions
([30]). Recons uc ion o a con igu a ion o poin s in 3D om wo ideal non-calib a ed
came a images ega ded as subse s in p ojec i e spaces wi h unknown came a pa am-
e e s in absence o occlusions is known as he 3D econs uc ion p oblem ([31]). This
leads o he p ojec i e ambigui y o he 3D econs uc ion p oblem: gi en wo came a
images o unknown ela i e posi ion and in e nal came a pa ame e s and wo ma ched
k−se s o labeled poin s in he p ojec i e space, ind all k−se s o poin s in space as
solu ions o bo h 3D econs uc ion p oblem. I could be p o en in [32] ha a land-
ma k co espondence is needed o ob ain a 3D econs uc ion om any pai o 2D
4
images. This leads o he concep o p ojec i e k-ads, as p esen ed below.
Ak-ad is an o de ed lis o klabeled poin s in Rm; i can be also ega ded as a k-ad
in RPm, ia he s anda d a ine embedding o Rmin RPmgi en by
p:Rm→RPm
x= (x1, . . . , xm)→p(x)=[x1:· · · :xm: 1] = [(x1...,xm,1)T].
One app oach o p ojec i e shape analysis ([33]), is based on he idea o a p ojec i e
ame selec ed om he poin s o a ini e gene ic k-ad in mdimensions. Such a ep e-
sen a ion has he ad an age ha i associa es o he ull se o p ojec i e in a ian s
([34]) o such a con igu a ion one poin on a p ojec i e shape mani old. The p ojec-
i e ame app oach can be used o iden i y he p ojec i e shape o a plana cu e in
he con ex o a scene ha con ains ou poin s in gene al posi ion, ha a e no nec-
essa ily on he cu e. The p ojec i e ame de e mined by such con ol poin s is used
o egis a ion. Ideally wo egis e ed images o he same scene should be iden ical;
ne e heless due o egis a ion e o s and depa u e om a plana scene, hey a e di -
e en , and his aises ques ions abou iden i ica ion o he mean p ojec i e shape o a
cu e, es ing o equali y o such mean p ojec i e shapes o cu es, e c ([35]). The e
a e a ew p oblems ha a ise in such a es ing p oblem om cu es in digi al images.
The iden i ica ion o he ac ual cu e in image p ocessing, e en om less noisy images,
is i s p oblem. Secondly, he cu e egis a ion p oblem p esen s challenges. Thi dly,
he classical simple null hypo hesis o equali y o wo mean cu es, e en when hey
a ise o m he same scene, is e y likely o be ejec ed because o inhe en egis a-
ion e o s; he e o e a neighbo hood null hypo hesis o unc ional da a is p e e ed
([36]). Finally once he es is es ablished one has o dwell wi h in ensi e compu a-
ional algo i hms in ol ed in unc ional da a analysis.
An o de ed lis o m+ 2 labeled poin s in RPmis said o o m a p ojec i e ame
(basis) i hey span RPm.An o de ed lis o k≥m+ 2 labeled p ojec i e poin s in
RPma e said o be in gene al posi ion i he i s m+ 2 o hese poin s o m a p o-
jec i e ame. G(k, m) is he space o all k-ads in gene al posi ion in RPm.
A p ojec i e ans o ma ion g=gPo RPmis he p ojec i e map associa ed wi h a
nonsingula ma ix P∈GL(m+ 1,R) and i s ac ion on RPm
gP([x]) = g([x1:· · · :xm+1]) = [P(x1· · · xm+1)T]
The p ojec i e ans o ma ions o RPm o m a g oup, deno ed by PGL(m).
Di ision by he las coo dina e yields a ep esen a ion o a p ojec i e ans o ma ion
gPin e ms o a ine coo dina es, as = (u), wi h
j=aj
m+1 + Σm
i=1aj
iui
am+1
m+1 + Σm
i=1am+1
iui,∀j= 1, ..., m
whe e de (P) = de ((aj
i)i,j=1,...,m+1)) = 0.
Two k-ads o poin s in Rmha e he same he p ojec i e shape i hey di e by a
p ojec i e ans o ma ion o Rm.
5
The p ojec i e shape o a k-ad X= ([x1],...,[xk]) ∈G(k, m) is he o bi o Xunde
he ac ion αo PGL(m) on G(k, m), gi en by
α(gP, X) = (gP([x1]), . . . , gP([xk])).
P ojec i e shape space PΣk
m, is he space o p ojec i e shapes o all k-ads in RPmin
gene al posi ion X= ([x1],...,[xk]), such ha X= ([x1],...,[xm+2]) is a p ojec i e
ame :
PΣk
m=G(k, m)/P GL(m).
PΣk
mis a mani old, homeomo phic wi h
RPm×RPm× · · · × RPm
| {z }
qcopies
= (RPm)q
whe e q=k−m−2. P ojec i e shape analysis ocuses on he p ope ies o a con ig-
u a ion o colinea o coplana poin s, as hey a e seen in a cen al p ojec ion by an
ex e nal obse e and i can be seen as a simpli ied analysis o ision in absence o
occlusions. The image using me hod is based on ex insic means o p ojec i e shapes
o con igu a ions o landma ks. P ojec i e shape analysis is iden i ied wi h mul i a i-
a e (axial) s a is ics and i can be in e p e ed in he con ex o di ec ional s a is ics.
The p ojec i e space RPmis opologically equi alen o a m-dimensional uni sphe e
Smwi h he an ipodal poin s iden i ied. The e o e, PΣk
mis a mani old which is home-
omo phic wi h polysphe e (Sm)qwi h he an ipodal poin s iden i ied. Hence, he
ex insic shape analysis has an elegan in e p e a ion as a di ec ional s a is ics.
2.2 Ex insic S a is ical Analysis in P ojec i e Shape Spaces
The no ions o ex insic mean and ex insic co a iance on mani olds a e a he cen e
o ou s a is ical analysis ([2]).
Le Qbe a p obabili y measu e on (M, ρ), a comple e me ic space wi h a mani old
s uc u e o dimension m. I j:M → RNis an embedding, a minimize µEo
F(x) = Z||j(x)−j(y)||2Q(dy).
is called he ex insic mean (se ) o Q.
A poin xo RNsuch ha he e is a unique pin M o which ρ0(x, j(M)) = ρ0(x, j(p))
is called j-non ocal whe e ρ0is he Euclidean dis ance in RN.Fcis he se o j-
non ocal poin s.
A p obabili y measu e Qon Mis said o be j-non ocal i he mean µo j(Q) is a
j-non ocal poin .
Ap ojec ion Pj:Fc→j(M) maps any x∈ Fc o he unique ysuch ha
ρ0(x, j(M)) = ρ0(x, y).
The ex insic sample mean is de ined acco dingly (see [37]).
6
Theo em 1. Assume Qis a non ocal p obabili y measu e on he mani old Mand
X={X1, . . . , Xn}a e i.i.d. .o.’s om Q.
(a) I he sample mean j(X)is a j-non ocal poin hen he ex insic sample mean is
gi en by XE=j−1(Pj(j(X))).
(b) XEis a s ongly consis en es ima o o µj,E(Q).
Popula ion ex insic co a iance ma ix can be de ined s a ing om he co a i-
ance o he embedded andom objec . Le (e1(y), e2(y), . . . , eN(y)) be an adap ed
ame o he embedding ja ound Pj(µ) = j(µE) i.e. i is an o hono mal ame
ield such ha (e (j(p)) = dpj( (p)), = 1, . . . , m, ∀p∈j−1(Uj(µE)) whe e p→
( 1(p), . . . , m(p)) is an o hono mal local ame ield on an open subse o M. Le
an( )=(e1(Pj(µ))T . . . em(Pj(µ))T )Tbe he angen ial componen o . The
ex insic co a iance ma ix o he j-non ocal dis ibu ion Qwi h espec o he
basis 1(µE), . . . , m(µE) is de ined as he co a iance ma ix o he andom a i-
able an(Pj(j(X))) wi h espec o he j−adap ed ame. Sample ex insic co a iance
ma ix is ela ed o he sample co a iance ma ix o he embedded andom objec in
he usual way.
The sample ex insic co a iance ma ix o he j-non ocal dis ibu ion Qwi h espec
o he basis 1(µE), . . . , m(µE) is de ined as
Sj,E,n =hXdj(X)Pj(eb)·ea(Pj(j(X)))ia=1,...,m·Sj,n
hXdj(X)Pj(eb)·ea(Pj(j(X)))ia=1,...,mT
whe e Sj,n is he sample co a iance ma ix o j(X). Sj,E,n is a consis en es ima o o
Σj,E.
Fo he p ojec i e space RPm, we will use he Ve onese-Whi ney embedding jde ined
in he ollowing. RPmcan be equi a ian ly embedded in he space S(m+ 1) o (m+
1) ×(m+ 1) o symme ic ma ices ia he Ve onese-Whi ney (V-W) embedding
j:RPm→S(m+ 1),
j([x]) = xxT, xTx= 1.
The V-W equi a ian embedding o he p ojec i e shape space PΣk
mjk:PΣk
m=
(RPm)q→(S(m+ 1))qis de ined by
jk([x1],...,[xq]) = (j([x1]), . . . , j([xq])),
whe e xs∈Rm+1, xT
sxs= 1,∀s= 1, . . . , q (see [28,38]).
I Y , = 1, . . . , n a e i.i.d. .o.’s o which he mean shape µjk exis s, i has a
mul i a ia e axial ep esen a ion
Y = ([X1
],...,[Xq
]),(Xs
)TXs
= 1; s= 1, . . . , q.
7
Le Jsbe he andom symme ic ma ix gi en by
Js=n−1
n
X
=1
Xs
(Xs
)T, s = 1, . . . , q,
and le ds(a) and gs(a) be he eigen alues in inc easing o de and he co esponding
uni eigen ec o o Js, a = 1, . . . , m + 1. Then he sample mean p ojec i e shape is
gi en by
Yjk,n = ([g1(m+ 1)],...,[gq(m+ 1)]).
F om a gene al consis ency heo em o ex insic means on mani olds in [37], i ollows
ha he ex insic sample mean [Y]jk,n is a s ongly consis en es ima o o µjk.
Theo em 2. In he case o he VW embedding jk, he sample ex insic co a iance
ma ix es ima o Sjk,E,n is gi en by he (mq)×(mq)symme ic ma ix Sj,E,n, wi h
he en ies in pai s o indices (s, a), s = 1, . . . , q;a= 1, . . . , m, in hei lexicog aphic
o de gi en by
Sj,E,n(s,a),( ,b)=n−1(ds(m+ 1) −d (a))−1(d (m+ 1) −d (b))−1·
n
X
=1
(gs(a)TXs
)(g (b)TX
)(gs(m+ 1)TXs
)(g (m+ 1)TX
).
Ho elling’s ype s a is ics will alow us o apply a one-sample s a is ical es o
andom objec s on p ojec i e shape spaces.
Le
Ds= (gs(1) . . . gs(m)) ∈ M(m+ 1, m;R), s = 1, . . . , q.
I µ= ([γ1],...,[γq]),whe e γs∈Rm+1, γT
sγs= 1, o s= 1, . . . , q, we de ine a
Ho elling’s T2- ype s a is ic
T(Yjk,n;µ) = n(γT
1D1, . . . , γT
qDq)S−1
j,E,n(γT
1D1, . . . , γT
qDq)T.
Theo em 3. Assume (Y ) =1,...,n a e i.i.d. .o.’s on (Rpm)q, and Y1is jk-non ocal,
wi h ΣE>0. Le λs(a)and γs(a)be he eigen alues in inc easing o de and co -
esponding uni eigen ec o s o E[Xa
i(Xa
i)T]. I λs(1) >0, o s= 1, . . . , q, hen
T(Yjk,n;µjk)con e ges weakly o χ2
mq.
Rema k 1. Assume (Y ) =1,...,n a e i.i.d. .o.’s om a jk-non ocal p obabili y dis i-
bu ion on (RPm)q, and ΣE>0. An asymp o ic (1−α)-con idence egion o µjk = [ ]
is gi en by Rα(V) = {[ ] : T(Yjk,n; [ ]) ≤χ2
mq,α}.
I he p obabili y measu e o Y1has a nonze o absolu ely con inuous componen w. . .
he olume measu e on (RPm)q, hen he co e age e o o Rα(V)is o o de O(n−1).
3 Spa ial O ien a ion in Vi ual En i onmen s
A majo p og ess in s udying hippocampus-dependen spa ial na iga ion in humans
is due o he de elopmen o i ual eali y echniques. This allowed es ing na iga-
ional skills in a ious en i onmen s by enabling comple e con ol o e he complexi y
8
o he ask and objec i e eco ding o beha iou al esponses. Vi ual en i onmen s
can also be used o in es iga e spa ial cogni ion since s ong co ela ions ha e been
obse ed be ween na iga ion in he eal wo ld and in he i ual eali y. In his case,
he assumed spa ial na iga ion p ocessing is he allocen ic e e ence ame whe e
he objec loca ions a e p ocessed in e e ence o each o he o ixed landma ks,
independen o he obse e ’s posi ion in he en i onmen . The medial empo al
lobe, especially he hippocampus, plays an impo an ole in he allocen ic spa ial
na iga ion, as indica ed in many s udies and con i med by he poo pe o mance o
pa ien s wi h AD and MCI in he es s equi ing he use o allocen ic memo y. The
D esdne Spa ial Na iga ion Task belongs o he amily o o spa ial es s con o ming
he allocen ic p inciples o Mo is Wa e Maze ha is designa ed as hidden goal o
hidden goal asks.
3.1 The MCI da a se
The mild cogni i e impai men (MCI) is conside ed o be he link be ween cogni i e
changes in he heal hy aging pe sons and hose who a e p edisposed o de eloping
demen ia, usually he Alzheime disease (AD). Hence, he decline in spa ial na iga-
ion skills can be seen as a cogni i e ma ke in diagnosing AD. The MCI da a se ,
whe e a g oup o 54 pa icipan s, 25 diagnosed heal hy, 26 diagnosed wi h MCI, and
3 p esen ing e symp oms o mino demen ia o he Alzheime ype (DAT) pe o med
his ask, and we e addi ionally subjec ed o a se o neu opsychological es s. The
g oup consis ed o 31 women and 23 men be ween 51 and 79 yea s old (mean age 69.1
yea s). Since he numbe o DAT pa ien s is no high enough o cons i u e a sepa a e
g oup, hey ha e been inco po a ed in o he MCI g oup.
The indi idual plana cu es a e i egula ( he s ep size a ound 10ms), and he s a ing
poin a ies be ween ials. The inal poin (aka goal) and he scien i ic ( opological)
landma ks a e ixed o all pa icipan s and all ials. The mo emen is cons ain by
he choice o he domain (ci cula in his case) and p esen s a d i in he di ec ion o
he goal (see Figu e 1). When he pa icipan s each he goal, hey s op, while some
pa icipan s can gi e up he es ea lie om o he easons, like lack o concen a ion
o mo i a ion.
3.2 3DEx insic P ojec i e Shape S a is ical Analysis
A da a poin in (Rm)kco esponding o he 3D− ela i e posi ion o he klandma ks
wi h espec o he pa h posi ion is ega ded as a k-ad in RPm, ia he s anda d a ine
embedding o Rmin RPm. Fo m= 3, he k−ad belongs o he g oup o p( ojec i e)-
qua e nions as an elemen o he p ojec i e space RP3. Mo eo e , he p ojec i e shape
space PΣk
3is homeomo phic o RP3k−5. The ex insic s a is ical analysis in he
p ojec i e shape space PΣk
3p esen ed in sec ion 2.2 is applied o he MCI da a se .
The p ojec i e ame consis s o he mos conspicuous landma ks i.e. he alles i e
landma ks. Fi s , we es ima e he ex insic means o he indi idual pa hs and he pop-
ula ion ex insic mean o he pa h ex insic means ha could be in e p e ed as he
9
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Fig. 2: The empi ical dis ibu ion o he pa h ex insic means and he popula ion
ex insic mean o ials 2,5,6,9,11 and o landma k 1 (le :con ol, igh :disease)
Fig. 3: The popula ion ex insic means o he landma k 1,3,11,15,21,71 o ials
1−12 (le :con ol, igh :disease)
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Fig. 4: The one sample es o equali y wi h he goal p ojec i e shape (le :con ol,
igh :disease)
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Fig. 5: The pai ed sample es o equali y o he ex insic means o he goal p ojec i e
shape be ween consecu i e ials (le :con ol, igh :disease)
