Numerical analysis of differential operators on raw point clouds

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clouds

Julie Digne, Jean-Michel Morel

To cite this version:

Numeri al analysis of dierential operators on raw

point louds

Julie Digne

·

Jean-Mi helMorel

A eptedforpubli ation-preprintversion

Abstra t 3Da quisitiondevi esa quireobje tsurfa eswithgrowinga ura yby

obtaining3Dpointsamplesofthesurfa e.Thissamplingdependsonthegeometry

of the devi e and of the s anned obje t and is therefore very irregular. Many

numeri als hemeshavebeenproposedforapplyingPDEstoregularlymeshed3D

data.Nevertheless,forhighpre isionappli ationsitremainsne essaryto ompute

dierentialoperatorsonrawpoint loudspriortoanymeshing.Indeeddierential

operatorssu has themean urvatureor theprin ipal urvaturesprovide ru ial

informationforthe orientationandmeshingpro ess itself.

This paper reviews a half dozen lo al s hemes whi h have been proposed to

omputedis rete urvature-likeshapeindi atorsonrawpoint louds.Allofthem

willbeanalyzedmathemati allyinauniedframeworkby omputingtheir

asymp-toti formwhenthesizeoftheneighborhoodtendstozero.Theyaregiveninterms

oftheprin ipal urvaturesorofhigherorderintrinsi dierentialoperatorswhi h,

inreturn, hara terizethe dis reteoperators. All onsideredlo als hemesare of

twokinds:eithertheyperformapolynomiallo alregression,orthey ompute

di-re tly lo al moments. But the polynomial regression of order 1 is demonstrated

to play a spe ial role,be auseits iterations yielda s ale spa e. This analysis is

ompletedwithnumeri alexperiments omparingthea ura iesoftheses hemes.

Wedemonstrate thatthisa ura y isenhan edforalloperatorsbyapplying

pre-viouslythes alespa e.

Keywords Point louds

·

s alespa e

·

MovingLeastSquaressurfa es

·

urvature omputation

J.Digne

LIRIS-Geomod,UniversitédeLyon,CNRSUMR52053boulevarddu11novembre1918

F-69622Villeurbanne

J.-M.Morel

CMLA,E oleNormaleSupérieuredeCa han

61AvenueduPrésidentWilson,

Introdu tion

The outputoflaser s anners oranysurfa e a quisitionsystem isa setof points

sampledwithvariabledensityonthesurfa e.Thes annersoftendeliverdire tlya

mesh,i.e.,asetoftriangleslinkingpointsamples.Butthe basi rawinformation

isanunorganizedpoint loudwhi h anbelo allysparseorover- luttered.Inthis

paperwefo usonthemathemati alanalysisandpro essingofsu hrawirregularly

sampled surfa es. Indeed, they ontain the most a urate information, before it

is altered or smoothed by any re-sampling and meshing. We shall interpret in

termsofintrinsi dierentialoperators(the urvatures) the mostinterestingand

simplelo alsurfa eestimators. Iterativesurfa eregularization pro esseswillalso

beanalyzedandthes alespa emethodusedto omputereliablylo aldierential

features.In oneword,the hallengeis:how to omputeintrinsi operatorswhi h

ideallyonly dependon theunderlying surfa e,not on its sampling?Surprisingly

enough,weshallseethatthisispossibleand thatthereliabilityofsu hoperators

anbe enfor edandevaluated.

Theeldofnumeri alsurfa eanalysishasbeenwidelystudiedoverthefteen

pastyears,duetothedevelopmentof omputergraphi s.Yet,moststudiestakeas

startingsurfa erepresentationamesh.Meshesaremu heasiertohandlethanraw

point louds,beingalreadyoriented,havingusuallyauniformorregularlyvarying

sampling,andhavingadenitesurfa etopology.Onthe ontrary,rawdatapoint

setsare ompletelyunstru tured heapsofpoints, knownonlybytheirEu lidean

oordinates. Nevertheless the onstru tion of a mesh and the onstitution of its

topology involve,impli itlyorexpli itly,the omputationofdierentialoperators

ontherawdata.

Themostpopularmeshre onstru tionmethodsfromarawpoint louddenea

signedfun tionover

R

3

representingthedistan etotheobje t,andthen extra t

the

0

level set whi h approximates the obje t surfa e. See (e.g.) [15℄, [22℄, [9℄, [27℄, [4℄, and the well established Poisson method [28℄, whi h solves a Poisson

equationto buildthe indi atorfun tion ofthe solid obje t. Thesemethods vary

intheapproa hto omputethedistan efun tion,butallextra titszero-levelset

byusingthe mar hing ubes algorithm [35℄, [29℄. In su hmeshing pro esses,the

initial raw points are irremediably lost. This in ursinto a lossof resolution and

explainsthe relevan eofpro essingdire tlyan unstru turedraw point loud.

The reminder of this paper is divided as follows: se tion 1 reviews surfa e

operatorsandsurfa emotions previouslydened onmeshesandonpoint louds,

se tion2givesthene essarydenitionsandtoolsforrawsurfa esandtheir

under-lyingsmoothmodel.Se tion3analyzestherstkindoflo aldierentialoperator

omputable on a raw loud: these are simplylo al order 2 moments, whi h will

beshowntoasymptoti ally omputefun tionsofthesurfa eprin ipal urvatures.

Se tion 4analyzes and ompares thesurfa e motions givenby theproje tion on

simpleregressionsurfa es:aplaneandadegree

2

polynomialsurfa e.Finally, se -tion5shows omparativenumeri alexperiments omparingthe a ura iesofthe

mesh freemethodsto ompute lo alpseudo-dierentialoperators,and alsoshow

1Computing urvatures onsampledsurfa e: state oftheart

1.1 Curvatures omputedonmeshes

Reviewsfor urvatureestimationon meshedsurfa es anbe foundin[44℄or[36℄.

CurvaturetensorestimationmethodswerepioneeredbyTaubin[46℄whopresented

asimpleapproximationfor omputingthedire tional urvatureinanytangent

di-re tion.The urvatureis omputedinallin identedgedire tionsanda ovarian e

matrixoftheedgedire tionsweightedbytheirdire tional urvaturesandthearea

ofthetwo in identtriangles isbuilt.Eigenve tors and eigenvalues ofthis

ovari-an e matrix yielda simple expression ofthe prin ipal urvatures and urvature

dire tions.

Other urvature tensor estimation methods in lude [44℄ where the tensor is

estimated bybuildinga linearsystem bindingthetensor oe ients. This linear

systemexpressesthe onstraints thatmultiplyingthetensorbyanedge dire tion

should give the dieren e ofthe edge's endpoints normals. The same method is

applied tond the urvature derivatives. Normals are also usedin [48℄ to give a

pie ewiselinear urvatureestimation (seealso[49℄).

Toavoidthe omputationofderivativeswithirregularsamples,anewkindof

integral urvatureestimationmethodhasbeenproposedin[52℄,[43℄and[42℄.The

interse tionof the surfa e with either a sphereor a ball entered at a vertex is

analyzed: the ovarian e matrix ofthisdomain is omputed and eigenvaluesare

expressedintermsofprin ipal urvatures.Byin reasingtheneighborhoodradius,

the urvatureestimate anbemademultis ale.Averyinterestingfeatureofthese

methodsisthattheydonotrelyonhighorderderivativesandarethereforemore

stable.

Surfa emotionshavealsobeenstudiedaspartofameshfairingpro ess.Akey

methodwasintrodu edbyTaubinin[47℄who onsideredadis reteLapla ianfor

amesh

V

withverti es

v

i

,

∂V

∂t

= λ

L(V )

,

L

beingadis retizationoftheLapla ian

L(v

i

) =

_{card N (v}

1

i

)

P j ∈ N(v

i

)(v

j

− v

i

)

where

N (v

i

)

isthesetofverti eslinked by an edge to

v

i

(

1

-ring neighborhood). This formulation is widely used. For example, [16℄ uses a similar umbrella operator. [20℄ also omputes the dis rete

Lapla ianforall mesh verti esitseigenve torsand eigenvalues.Byremovingthe

smallesteigenvalues,afair mesh(i.e.adenoised mesh)isobtained.

A well known formulation of the Lapla e Beltrami operator is the famous

otangentformula[38℄,

∆v

i

=

1

2

X

j∈N v

i

(cotanα

ij

+ cotanβ

ij

)

where

v

i

is avertexofthe mesh,

N

1

(v

i

)

its onering neighborhood,

α

ij

and

β

ij

are the angle opposite to edge

v

i

v

j

in the two triangles adja ent to

v

i

v

j

. This hasbeenusedto omputethesurfa eintrinsi equation.Anotherdenitionofthe

1.2 Curvatureestimationand surfa emotionsdenedon point louds

Wenowexaminetherareapproa hesdealingdire tlywithpoint louds.[50℄

intro-du edas alespa ede ompositionmethodforpoint louds.Themethodbuildsan

adja en y graph fromtheinput points (inorder to ompute easily thegeodesi s

onthesurfa e).Thegeodesi sareusedto omputeadensitynormalizationkernel

that regularizesthedensity. Thes alespa e operatoristhe operator that moves

ea h pointto the bary enter of all points weighted by the regularization kernel

andthedistan etothe enterpoint.Thenthes alespa emethodisusedtosele t

s ale-spa e extrema. At ea hs ale the pointmotion is onsidered.Introdu ing

a s alarfun tion on the displa ementnorm, the authors laim a re overyofthe

hara teristi s ales of the surfa e (the introdu ed fun tion is extremal at the

hara teristi s ales).

Estimating urvatures often ne essitates the omputation of surfa e

deriva-tives. Yet derivating a potentially noisysurfa e angenerate unstable estimates.

Instead,itwasnoti edthatlo alintegralquantities ontainalltheinformationof

the dierentialoperators. Thisidea wasused forexample in[37℄, with amethod

to ompute urvatures and normals basedon dening Voronoi ovarian e

matri- es.Morepra ti ally,aseriouseort wasmadefordeningintegralinvariantsfor

surfa es.

Ourstudywill onsiderthesimplestlo alintegralinvariants.Themostfamous

prin ipalintegralinvariantsweredenedasfollows: all

D

theinteriorofasurfa e

M

, then the area

A

r

of the interse tion of

D

with a sphere of radius

r

is an invariant.The se ond invariantis the volume

V

r

of the interse tionof a ball of radius

r

with

D

.Bothinvariantswereprovedtoberelatedtothemean urvature ([24℄,[11℄) sothat:

V

r

=

2π

3

r

3

−

πH

₄

r

4

+ O(r

5

)

A

r

= 2πr

2

− πHr

3

.

Su h invariants wereusedin [19℄ forsurfa e registration,orfor featuredete tion

([13℄, [12℄).Neverthelessa seriousnumeri aldrawba kofsu hintegralinvariants

expansionsisthat the dominant termnever ontainsthe a tualsurfa e

informa-tion. The dominantrst term isa tually ompletely independentof the surfa e

lo us.This makes themethod impra ti albe ause theterm ofinterest(here the

mean urvature

H

) is obtained as the dieren e of two lower order terms. Yet, sin e

V

r

and

A

r

are notexa tbutapproximate volumesandareas,

H

annot a -tuallybeobtaineda uratelyfromsu hformulas.Themethodswewillanalyzein

thispapera tuallysolvetheproblembydesigningthelo aloperatorinsu haway

that the dierential operator ofinterestisthe dominantterm inthe asymptoti

expansion.

Intermsofmathemati alanalysis,theanalysiswhi hgoes losesttothepresent

oneisdueto Pottmanet al.in[43℄ and [52℄.These authorsanalyzedthe

asymp-toti behavior of several integral invariants, parti ularly the moments of inertia

ofvariouslo alintrinsi neighborhoods.Yet,on eagain,thequantitytoestimate

is not ontained in their dominant terms, thus making the obtained asymptoti

formulas numeri allyimpra ti al. Forexample Theorem2 of[43℄ showsthat the

withaballofradius

r

have theTaylorexpansion

M

r

1

=

2π

15

r

5

−

π

48

(3k

1

+ k

2

)r

6

+ O(r

7

)

M

r

2

=

2π

15

r

5

−

₄₈

π

(3k

2

+ k

1

)r

6

+ O(r

7

)

M

r

3

=

19π

480

r

5

−

₅₁₂

9π

(k

2

+ k

1

)r

6

+ O(r

7

)

where

k

1

and

k

2

are the prin ipal urvatures of the surfa e at the onsidered position.Theauthorsthenbypassthedi ultyofnothavingtheestimatesinthe

dominanttermbytaking thedieren e

M

2

r

− M

r

1

=

24

π

(k

1

− k

2

)r

6

_{+ O(r}

7

₎

. Yet

thisonlyyields thesquareoftheprin ipal urvaturedieren e.

Amorepra ti alresultwasprovedin[43℄intheorem6:thebary enterofthe

surfa epat h (interse tionofa ballofradius

r

withthe surfa e

M

)isprovedto have oordinates

(0, 0,

k

1

+k

2

8

r

2

_{) + O(r}

3

₎

.Inthisexpressionthe signsare notlost

and the urvature isindeed thedominanttermoftheexpansion. Forthe sake of

ompletenesstheproofofthisresultwillbere alledinLemma2.

In[45℄, theproposedframeworkfor urvatureestimationataparti ularpoint

isbasedonasetof urvesrepresentingthelo alneighborhoodofthepointunder

onsideration.

Forea hpair

(p

i

, p

j

)

ofneighborsof

p

,the setoftriplets

(p

i

, p, p

j

)

is built. Ea h of those triplets an be used to dene a parametri spa e urve

p(t)

by quadrati polynomialinterpolationwith

p(0) = p

i

,

p(1) = p

j

and

p(t) = p

where

t =

|p−p

i

|

|p−p

i

|+|p

j

−p|

.This allowsfor theapproximation ofmaximum andminimum

urvature valuesas the minimum and maximum normal urvature valuesfor all

possiblepointtriplets.Thismethod anbeusedeitheronmeshesorpoint louds.

In[25℄, theauthorsproposedastatisti alestimationofthe urvatureofpoint

sampledsurfa esbasedonM-estimators 1

.Thepositiondieren e ve tor

∆p

and normal dieren e ve tor

∆n

are used to dene a linear system yielding a rst estimate ofthe urvaturetensor.Thenresiduals are omputedandused toweigh

the samples and the obje tive fun tion is minimized by iterative reweighing of

pointsamples. Thisyieldsthenal urvaturetensorestimate.

Finallyin[6℄,analgorithmto omputetheLapla ianofafun tiondenedon

point louds in

R

d

was proposed along with onvergen e proofs. Yet the model

is not tested on real surfa es. Neighborhood ovarian es being used already for

normalestimation,theideatoexpressfundamentalformsas ovarian esmatri es

wasintrodu ed.Thenextse tionreviewsthe ovarian e te hniques onsideredin

theliterature.

1.3 Curvatureestimationusing ovarian ete hniques

Thereare few ovarian e approa hes and theyhave seldombeen analyzed

math-emati allyyet,(withthenotableex eptionof[43℄and [52℄whi hwillbedetailed

inthisse tion).Nevertheless, ovarian emethods anbeanelegantalternativeto

1

surfa eregression.Three papers[7℄, [33℄ and[40℄ use ovarian e matri es forthe

urvatureestimation.

The rstone[7℄ onsiderstheneighbors

(p

i

)

ofa point

p

.These ond funda-mental form analog is then dened as the ovarian e matrix of the ve tors

pp

i

proje tedontothetangentplaneofthesurfa eat

p

.AnanalogoftheGaussmapis alsointrodu ed:itisthe ovarian ematrixoftheneighborsunitnormalsproje ted

onto the surfa etangentplane at point

p

. Theeigenve tors are said togive the prin ipaldire tions.In fa tthese two ovarian e matri es are inspiredfrom[33℄.

Indeed,[33℄rstproposedto omputethe ovarian ematrixofthenormalsatthe

neighborsof

p

,and toextra t the prin ipaleigenvalueswhi h orrespond tothe prin ipal urvaturesofthe surfa e at

p

. Thelast ovarian e method,introdu ed in[40℄ isnot laimedtobe expli itlylinkedtosurfa e urvaturesorfundamental

forms,yetitisusedtoa ountforthe surfa egeometri variations.Considerthe

ovarian e matrix ofve tors

p

i

where is the bary enter ofthe neighborhood of

p

. The surfa e variation is dened as the ratioof the least eigenve torover the sumofalleigenve torsofthis ovarian ematrix. Thisquantityhastheni e

prop-ertythatitisboundedbetween

0

(at ase)and

1/3

(isotropi ase). Allofthese methodswillbedetailedand analyzedinse tion3.

1.4 MovingLeastSquaresSurfa es

MLS(Movingleastsquare)surfa eswereintrodu edin[30℄asfollows.Givenadata

setofpoints

{p

i

}

i

(possiblya quiredbya3Ds anning devi e)andbelongingto asmooth surfa e

M

,thegoalistorepla ethe points

p

dening

M

byaredu ed set

R =

{r

i

}

dening a so alledMLS

M

′

surfa e whi happroximates

M

.The surfa e

M

isassumedtobea

C

∞

2-manifold. Theauthorsxaboundingerror

ε

su hthat

d(

M, M

′

_{) < ε}

,where

d

istheHausdordistan e.

The proje tion of a point on the MLS surfa e is dened as follows: given a

point

p

,ndalo alreferen edomain(plane) for

p

.Thelo alregressionplane

H

isobtainedbyminimizingalo alweightedsumofsquaredistan esofthe points

p

i

totheplane.Theweightsatta hedto

p

i

aredenedasfun tionsofthedistan e of

p

i

totheproje tionof

p

onplane

H

,ratherthantheirdistan eto

p

.

Assume

Q

istheproje tionof

p

onto

H

,then

H

isfoundbylo allyminimizing withrespe tto

n

and

D

thequadrati ost

N

X

i=1

(< n, p

i

>

−D)

2

θ(

kp

i

− Qk)

where

θ

isasmooth,monotonede reasingpositivefun tion.We anset

Q = p+tn

forsome

t

∈ R

,whi hleadstotheminimization of

N

X

i=1

(< n, p

i

− p − tn >)

2

θ(

kp

i

− p − tnk).

Thelo alreferen edomainisthengivenbyanorthonormal oordinatesystemon

let

(x

i

, y

i

)

be the oordinates of

Q

i

on

H

.The oe ients ofthepolynomialare omputedbyminimizingtheleast squareerror

P

N

i=1

(g(x

i

, y

i

)

− f

i

)

2

θ(

kp

i

− Qk)

Theproje tion of

p

onto

M

isdened bythe polynomialvalueattheorigin,i.e.

Q + g(0, 0)n = p + (t + g(0, 0))n

.Thus,givenapoint

p

andits neighborhood,its proje tion onto theMLSsurfa e anbe omputed. The approximationpower of

MLSsurfa eswasevaluated in [31℄ and therst appli ations wereintrodu edin

[2℄,[5℄ and[32℄.

MLSsurfa esarenotonlytheoreti allypowerful;theyalsoprovidene

imple-mentationsforrendering,up-samplingordown-samplingpointsets[3℄,[40℄. Finer

variants of MLS were subsequently proposed for a better preservation of sharp

edgesinsurfa esdenedbypoint louds[39℄, [34℄,[18℄, [21℄and [1℄.

The sameframeworkwas usedto builda s alespa e for point loudsin [41℄.

The surfa e is evolved through a diusion pro ess

∂p

∂t

− λ · ∆p = 0

, where

p

is a pointof the surfa e,

λ

a diusion parameter and

∆p = Hn

is the Lapla e BeltramiOperator (

H

isthe urvature and

n

the normal at point

p

, thisisthe de omposition pro ess). By rememberingthe setofdispla ements

D

i

(p)

of ea h point

p

wehaveare onstru tionoperator.The hoi eoftheLapla ian dis retiza-tion is very important: a rst possibilityis to use the standard mesh Lapla ian

te hniques [47℄ adaptedfor point loudsusing the

k

-nearest neighborsinstead of theoneringneighborhood.Anotherpossibilityistousetheweightedleastsquares

proje tion[23℄, [26℄: thesurfa eisiterativelyproje tedontotheplanedenedby

theweightedbary enter

o

andthenormalestimatedusingtheweighted neighbor-hood ovarian e matrix. The weights are a Gaussianfun tion of the distan e to

p

,andthesizeoftheGaussiankernelisaparameterthat ontrols theamountof smoothing.Thisproje tionpro essisinfa tanorder

1

proje tionmotion(MLS1) that willbeanalyzedinthe followingse tions.

Tomake the proje tion moree ient,[41℄ proposedto sub-samplethepoint

loud. This yieldsa s ale spa e de omposition where at ea h levelthe surfa e is

smoothed andsub-sampled.Thes alespa ede omposition isthenapplied tothe

multi-s alefreeformdeformation and tothemorphingproblem,withsatisfa tory

results.

Themovingleast squares(MLS)wereused toestimate urvatures.For

exam-ple,in [51℄, the authorsuse the MLS frameworkto builda losed formsolution

for urvatureestimation. Indeed,surfa es impliedbypoint louds anbe seenas

thezerolevelsetofanimpli itfun tion

f

whosegradientandHessianMatrixare built.Finally,usingformulasfortheGaussianandthemean urvaturedepending

ontheHessian andgradientof

f

,those urvatures anbe omputed.

In[10℄, theproblemofestimating dierentialquantitieson point loudsis

re- ast to that oftting the lo al representation of the manifold by a jet.A jet is

simplya trun atedTaylor expansion.A

n

jetisaTaylor expansiontrun ated at order

n

.Ajetoforder

n

ontainsdierentialinformationuptothe

n

-thorder.In parti ularitisstatedthatapolynomialttingofdegree

n

estimatesany

k

th

_order

dierential quantityto a ura y

O(h

n−k+1

₎

.This impliesthat the oe ient of

therstfundamentalformandunitve tornormalare estimatedwith

O(h

n

₎

pre- ision and the oe ients of the se ond fundamental form and shape operator

are approximatedwith a ura y

O(h

n−1

₎

,and soare theprin ipaldire tions.In

orderto hara terize urvatureproperties,themethod resortstothe Weingarten

the Gaussmap. Re all that the rstand se ond fundamental forms

I

,

II

and

A

satisfy

II(t, t) = I(A(t), t)

for any ve tor

t

of the tangentspa e. Se ond order derivatives are omputed by building the Weingarten map of the os ulating jet

whoseeigenvaluesare the prin ipal urvatures.Note that the des ribedmethods

anbeusedeitherwithameshorwithapoint loud.Jetsareinfa tveryrelated

to MLS surfa es. Indeed, to estimate dierential quantities a polynomial tting

of degree

n

is done, whi h is exa tly what MLS does. Therefore the analysis in se tion4givingtheequationgoverningMLS1andMLS2motionsarevalidforthe

jetstoo.

Nevertheless,weshallprove thatnoneoftheabove mentionedmomentbased

methods for omputing the urvatures without a surfa e regression gives ba k

the signed urvatures. We shall also prove experimentally that in order to be

stable,those integral estimatesaswellasthesurfa eregressionestimatesrequire

alargeneighborhood, whi hleadstolarger omputationtime.Intermsof

Signal-to-Noise Ratio, it willturn out tobe better to onsidera s ale-spa eapproa h:

applyingthes alespa eiterationswithasmallneighborhoodandthenextra ting

thedierentialoperatoranalogue.

This se tionhas reviewedthemainmethods aimingatestimating lo allythe

surfa eshape,thusimpli itly omputinglo alequivalentsoftheinnitesimal

ur-vature tensor. We have seen that two sorts of methods, logi ally, dominate: the

polynomialregressionsononeside,andthelo almomentsontheother.(Itis

a -tuallydi ulttoimagineotherkindsoflo almethodsonarawpointset).These

kinds have very dierent te hniques, but we shall be able to ompare them in

two unifying frameworks. We shall rst give their asymptoti equivalents, whi h

are fun tions of the surfa e prin ipal urvatures. Then we shall ompare their

reliabilitybyanumeri alsetupintheexperimentalse tion.

In parti ular se tion 3 nds theform ofthe dierential operators underlying

the four mentioned dis rete s hemes based on lo al loud point statisti s, and

proposingdis reteanaloguesofthese ondfundamentalformsoroftheprin ipal

urvatures.Thesedis retes hemeshaveverysimpleandrobustform,beingbased

on the omputation of lo al moments and eigenvalues of the point loud. The

nextse tion2providesthetoolstoanalyzenumeri allypoint loudmotions.The

analysisisin spirit losetothe imagelteranalysisperformedin[8℄.

2Toolsfor numeri alanalysis ofpoint loud surfa e motions

Wealways assumetheexisten eofasmoothsurfa e

M

supportingthepointset. These surfa esare the boundariesofsolid obje tsand an thereforebe assumed

tobelo allyLips hitzgraphs.However,foramathemati alanalysisofsmoothing

algorithmsand urvatureestimationsonthesurfa e,weshallalwaysassumethat

thesurfa eisa

C

∞

embeddedmanifold,knownfromitssamplesdenotedby

M

S

. This is not a limitation, in the sensethat any nite sample set an be anyway

interpolatedbyanarbitrarilysmooth surfa e.Let

p

= p(x

p

, y

p

, z

p

)

be apointof the surfa e

M

. Atea h non umbili al point

p

, onsiderthe prin ipal urvatures

k

1

and

k

2

linkedtotheprin ipaldire tions

t

1

and

t

2

,with

k

1

> k

2

where

t

1

and

t

2

areorthogonalve tors.(Atumbili alpoints,anyorthogonalpair

(t

1

, t

2

)

anbe taken.)Set

n

= t

1

× t

2

sothat

(t

1

, t

2

, n)

isanorthonormalbasis.Thequadruplet

Spherical Neighborhood

Regression Plane

Cylindrical Neighborhood

P

M

Fig.1 Comparisonbetween ylindri alandspheri alneighborhoods

expresslo allythesurfa easa

C

2

graph

z = f (x, y)

.ByTaylorexpansion 2 ,

z = f (x, y) =

1

2

(k

1

x

2

+ k

2

y

2

) + o(x

2

+ y

2

).

(1)

Noti ethatthe signofthe pair

(k

1

, k

2

)

depends onthearbitrarysurfa e orienta-tion.Points where

k

1

and

k

2

have thesamesign are alledparaboli ,and points wheretheyhave opposite signsarehyperboli .

Considertwokindsofneighborhoodsin

M

for

p

denedinthe lo alintrinsi oordinatesystem

(p, t

1

, t

2

, n)

:

 thespheri al neighborhood

B

r

= B

r

(p)

∩ M

isthe setofall points

m

of

M

with oordinates

(x, y, z)

satisfying

(x

− x

p

)

2

_{+ (y}

− y

p

)

2

+ (z

− z

p

)

2

< r

2

 the ylindri alneighborhood

C

r

= C

r

(p)

∩ M

isthesetofallpoints

m(x, y, z)

on

M

su hthat

(x

− x

p

)

2

_{+ (y}

_{− y}

p

)

2

< r

2

.

Thespheri alneighborhood inthesampledsurfa e

M

2

istheonlyneighborhood to whi h there is a dire t numeri al a ess. It serves for dening all numeri al

s hemes onsideredhere.Nevertheless, forthe forth omingasymptoti numeri al

analysis,the ylindri alneighborhoodwillprovemu hhandierthanthespheri al

one.Thenextte hni allemmajustiesitsuse intheoreti al al ulations.

Lemma 1 Integrating on

M

anyfun tion

f (x, y)

su h that

f (x, y) = O(r

n

₎

on

a ylindri al neighborhood

C

r

insteadofa spheri alneighborhood

B

r

introdu esan

o(r

n+4

)

error.Morepre isely:

Z

B

r

f (x, y)dm =

Z

x

2

_+y

2

_<r

2

f (x, y)dxdy + O(r

4+n

).

(2)

Proof The surfa eareaelementofapoint

m(x, y, z(x, y))

onthe surfa e

M

, ex-pressedasafun tionof

x

,

y

,

dx

and

dy

is

dm(x, y) =

p1 + z

2

x

+ z

y

2

dxdy

.Onehas

z

x

= k

1

x + O(r

2

)

and

z

y

= k

2

y + O(r

2

₎

.Thus

dm(x, y) =

q

(1 + k

2

1

x

2

+ k

2

2

y

2

+ O(r

3

))dxdy

2 We oulduse

z

= f (x, y) = −

1

2

(k1x

2

+ k2y

2

) + o(x

2

+ y

2

)

atthe ostof hangingthe

whi hyields

dm(x, y) = (1 + O(r

2

))dxdy.

(3) Using(3) ,theintegralsweare interested inbe ome

Z

B

r

f (x, y)dm = (1 + O(r

2

))

Z

§

2

_+y

2

_+z

2

_<r

2

_,(x,y,z)∈M

f (x, y)dxdy;

(4)

Z

C

r

f (x, y)dm = (1 + O(r

2

))

Z

x

2

_+y

2

_<r

2

_{, (x,y,z)∈M}

f (x, y)dxdy.

(5)

Therighthandformsareamenabletoanalyti omputations.Considerpolar

oor-dinates

(ρ, θ)

su hthat

x = ρ cos θ

and

y = ρ sin θ

with

−r ≤ ρ ≤ r

and

0

≤ θ ≤ π

. Then for

m(x, y, z)

belonging to the surfa e

M

, we have

z =

1

2

ρ

2

_(k

1

cos

2

θ +

k

2

sin

2

θ)+O(r

3

)

.Thus

z =

1

2

ρ

2

k(θ)+O(r

3

)

,where

k(θ) = k

1

cos

2

_θ+k

2

sin

2

θ

.The onditionthat

(x, y, z)

belongstotheneighborhood

B

r

anthereforeberewritten as

ρ

2

_{+ z}

2

_{< r}

2

,thatis

ρ

2

+

1

4

k(θ)

2

_ρ

4

_{< r}

2

_{+ O(r}

5

_).

Computingtheboundaries

±ρ(θ)

ofthisneighborhoodyields

ρ(θ)

2

₊

1

4

k(θ)

2

_ρ(θ)

4

−

r

2

+ O(r

5

) = 0

.Thus

ρ(θ)

2

=

−1 +

p1 + k(θ)

2

_(r

2

_{+ O(r}

5

₎₎

1

2

k(θ)

2

.

Thisyields

ρ(θ) = r

−

1

8

k(θ)

2

_r

3

_{+ O(r}

4

_).

Weshall usethisestimate forthe error

term

E

appearingin

Z

B

r

f (x, y)dxdy =

Z

[0,2π]

Z

[0,ρ(θ)]

f (x, y)ρdρdθ

=

Z

[0,2π]

Z

[0,r]

f (x, y)ρdρdθ

− E

=

Z

C

r

f (x, y)dxdy

− E,

with

E =:

R

[0,2π]

R

[ρ(θ),r]

f (x, y)ρdρdθ

.Thus

|E| ≤ 2π

sup

x

2

_+y

2

_≤r

2

|f(x, y)| max(k

2

1

, k

2

2

)r

4

+ O(r

5

).

Inparti ular if

f (x, y) = O(r

n

₎

,then

|E| ≤ O(r

4+n

_).

Finallywe have

Z

B

r

f (x, y)dxdy =

Z

C

r

f (x, y)dxdy + O(r

4+n

).

(6)

Combining(4),(5)and (6)yields(2) .

A methodologi al obje tion to the asymptoti analysis Lemma 1, as well as all

theorems in the remainder of this paper will assume that the surfa e is a

uni-formLebesguemeasure.Thusthetheoreti alanalysiswillbeperformedasthough

the surfa e were a very smooth obje t with dense uniform Lebesgue sampling.

This is very far fromreality, and ould ast doubtson the pertinen y ofsu h a

theoreti al analysis.However,as willbe explained later,the obje tion willprove

invalid, in that the hosen lo al integral operators will always be robust to

ir-regular sampling. For example the lo al area ould denitely not be omputed

by a lo al sample density. In the same way the lo al bary enter of the existing

sampled will be heavily biased by the irregular sampling and would have little

to do with the a tual lo al bary enter of the underlying surfa e. Nevertheless,

the moments we shall onsiderarefar morerobust, in theoryand inpra ti e, to

irregularsampling. This isthe ase for example for the normal when estimated

as the normal tothe lo al regression plane or, as we shall see, the mean

urva-tureve torestimated asthe proje tionofthe sampleonits regressionplane.On

the numeri al side, however, it is re ommended to ompensate for the irregular

samplingbyanadequatesamplereweighinginthe omputedlo almoments.This

intrinsi densityissimplyapproximatedondis retedatabyweightingea hpoint

bya weight inversely proportional to its initial density. More pre isely, let

p

be a point and

N

r

(p) =

M

s

∩ B

r

(p)

. Ea h point

q

should ideally have a weight

0

_{≤ w(q) ≤ 1}

su h that

P

q∈N

r

(p)

w(q) = 1

. Thisamounts tosolvea hugelinear

system. For this reason, we shall be ontented in the experimental se tion with

ensuring

P

q∈N

r

(p)

w(q)

≃ 1

bytaking

w(p) =

1

♯(B

p

(r))

,asproposedin[50℄.

3Curvatureestimatesby ovarian e matrix methods

Thisse tion ontainssomeofthemain ontributionsofthepresentpaper.Itnds

the formof the dierential operators underlying four dierent dis rete s hemes

basedonlo al loudpointstatisti s,andproposingdis reteanaloguesofthe

se -ond fundamental form matrix or of the prin ipal urvatures. These dis rete

s hemes have a very simple and robust form, being based on the omputation

of lo al moments and eigenvaluesof the point loud or ofits normals. We shall

seethat all ofthe methods asymptoti ally ompute nonlinear dierential

opera-tors linkedto the prin ipal urvatures. Their prin iple is to repla e the matrix

ofthe se ondfundamental formbysomesymmetri matrix that anbe dedu ed

fromthelo alstatisti softhe point loud. We shall onsiderfourmatri es(

2

or

3

-dimensional)that arethesimplestofsu h ovarian ematri es:

 2D ovarian ematrixoftheproje tionsof

p

−−→

i

p

onthetangentplanewhere

p

i

arethepoints oftheneighborhood(se tion3.1) ;

 2D ovarian ematrixoftheproje tionsofthe unitnormals

n(p

i

)

on the tan-gentplane(se tion 3.2);

 3D ovarian ematrix oftheunitnormals

n(p

i

)

(se tion3.3);

3.1 Adis retese ondfundamentalform [7℄

Let

(p

i

)

i∈1···N

be the set of neighbors of a point

p

with normal

n

. This paper proposestobuildthese ondfundamentalformmatrixasfollows.(Althoughthis

ovarian e matrixisnot,asweshallsee, onsistentwiththe se ondfundamental

form, itisthus alledinthispaper, and a tuallyhasasymptoti ally, aswe shall

see,theprin ipaldire tionsaseigenve tors.) Let

s

i

= (p

i

− p)

T

· n

,let

t

1

,

t

2

be two orthonormalve tors inthetangentplaneof

p

,and

α

i

= s

i

·

(p

i

− p) · t

1

(p

i

− p) · t

2



= ((p

i

− p)

T

· n) ·

(p

i

− p) · t

1

(p

i

− p) · t

2



.

The

α

i

are the proje tions ofthe ve tors

(p

i

− p)

ontothe tangent planeto

p

, weightedbytheir distan eto thisplane. These ond fundamental formmatrix

isthe ovarian eoftheseve tors,namely

Σ

d

=

N

X

i=1

(α

i

− α

m

)

· (α

i

− α

m

)

T

(7) where

α

m

=

1

N

P

N

i=1

α

i

and in

Σ

d

the

d

stands for dis rete. To ompute the underlyingdierentialoperators, two assumptions willbe made throughout this

paper.Therstoneisthatthesurfa esamplingisuniformwithrespe ttothearea

measureonthesurfa e. These ondoneisthatthissamplingisdenseenough,so

that the averagestakenon neighborhoods anbe interpreted asintegrals.Under

thisinterpretation,we anreinterpretthesumin(7)asanintegralona ylindri al

neighborhood of

p

,assuming thedatapointsettobea lo allysmoothmanifold. Inthelo alintrinsi surfa e oordinatesystematpoint

p

,

(p, t

1

, t

2

, n)

,thesurfa e an be written as a graph

z =

1

2

(k

1

x

2

+ k

2

y

2

) + o(r

2

)

.Thus the ve tors

α

i

are repla edbya ontinuousve tor

α(x, y)

denedby

α(x, y) =

1

2

(k

1

x

2

_{+ k}

2

y

2

)

·

x

y



=

1

2

·

k

1

x

3

+ k

2

y

2

x

k

1

x

2

y + k

2

y

3



+ o(r

3

).

(8)

Undertheinterpretationtakenabovethese ondfundamentalmatrix rewrites

Σ =

Z

B

r

(α(x, y)

_{− α}

m

)

· (α(x, y) − α

m

)

T

dm(x, y)

(9) where

α

m

=

1

meas(

B

r

)

Z

B

r

α(x, y)dm(x, y).

(10)

The proposition made in [7℄ is to extra t the surfa e prin ipal urvatures and

their orrespondingdire tionsat

p

fromthis ovarian ematrix,asitseigenvalues and eigenve tors. The next theorem he ks if this works asymptoti ally in the

ontinuousmodel.

Theorem1 Theeigenve torsofthese ondfundamentalformmatrix

Σ

givethe prin ipaldire tionswitherror

o(r

8

₎

.Buttheeigenvaluesof

Σ

arenottheprin ipal urvaturesasthey satisfy

λ

1

=

πr

8

256

(5k

2

1

+ 2k

1

k

2

+ k

2

2

) + o(r

8

)

and

λ

2

=

πr

8

256

(k

2

1

+ 2k

1

k

2

+ 5k

2

2

) + o(r

8

)

Proof Inthe ontinuousmodel

α

m

thereforeis losetozerobe ausetheintegrated fun tion is odd on a symmetri domain. More pre isely, using Lemma 1 in (10)

andwriting

α

m

= (α

mx

, α

my

),

α

mx

=

1

2πr

2

Z

x

2

+y

2

<r

2

(k

1

x

3

+ k

2

y

2

x

i

+ o(r

5

))dxdy = o(r

3

)

andsimilarly

α

my

= o(r

3

).

By Lemma 1 again, the ovarian e matrix (9) satises

Σ =

R

x

2

_+y

2

_<r

2

α(x, y)

·

α(x, y)

T

dxdy + o(r

8

),

and, using (8), we an al ulate its four omponents as follows.

Σ

11

=

1

4

Z

x

2

_+y

2

_<r

2

(k

1

x

3

+ k

2

y

2

x)

2

dxdy + o(r

8

) =

πr

8

256

(5k

2

1

+ 2k

1

k

2

+ k

2

2

) + o(r

8

)

Byex hangingtheroles of

k

1

,

k

2

,and

x

,

y

respe tively,we get

Σ

22

=

πr

8

256

(k

2

1

+ 2k

1

k

2

+ 5k

2

2

) + o(r

8

).

Σ

being a symmetri matrix,

Σ

12

= Σ

21

and the integrated fun tion being odd,

Σ

12

=

1

4

Z

x

2

_+y

2

_<r

2

(k

1

x

3

+ k

2

y

2

x)(k

1

x

2

y + k

2

y

3

)dxdy + o(r

8

) = o(r

8

).

Thus,

Σ

isequivalenttoa diagonalmatrix whose prin ipaldire tionsare

t

1

and

t

2

, whi h validates the theoreti al requirements,

t

1

and

t

2

being the prin ipal dire tionsatpoint

p

.However,the orrespondingeigenvaluesare

λ

1

=

πr

8

256

(5k

2

1

+ 2k

1

k

2

+ k

2

2

) + o(r

8

)

λ

2

=

πr

8

256

(k

2

1

+ 2k

1

k

2

+ 5k

2

2

) + o(r

8

)

whi haredenitelydierentfrom

λ

1

= k

1

and

λ

2

= k

2

.Onlytheabsolutevalues of

k

1

and

k

2

ana tuallybe dedu edfrom

Σ

.

3.2 Anotherdis retese ondfundamentalform

Another method was also introdu ed in [7℄ whi h, in a nutshell, omputes the

ovarian e matrix of the unit normal ve tors proje tions onto the lo al tangent

plane.Byapplyingagainthe ontinuousasymptoti analysisofse tion3.1,weshall

seeinTheorem2that thismethod a tually omputesdis reteapproximationsof

the squaresofthe prin ipal urvatures.The dis rete algorithm isas follows. Let

M

be a

C

2

dened asthe eigenvaluesof the ovarian e matrix ofthe ve tors

v

i

. The ve tor

v

i

beingtheproje tion of

n

i

ontothetangentplane,wehave

v

i

=

n

i

· t

1

n

i

· t

2



.

Set

v

m

=

1

N

P

N

i=1

v

i

. Then this new dis rete ovarian e matrix writes

Σ

d

=

P

N

i=1

(v

i

− v

m

)

· (v

i

− v

m

)

T

. In the ontinuous framework, the lo al points on the surfa e have oordinates

m(x, y) = (x, y,

1

2

(k

1

x

2

_{+ k}

2

y

2

) + o(r

2

))

and the normal ve tortothissurfa e is

∂m

∂x

(x, y)

∧

∂m

∂y

(x, y) = (

−k

1

x,

−k

2

y, 1) + o(r).

It followsthat

v(x, y) =

1

p1 + k

2

1

x

2

+ k

2

2

y

2

−k

1

x

−k

2

y



+ o(r),

v

m

=

1

meas(

_B

r

)

Z

v(x, y)dm(x, y),

andthe ontinuous ovarian ematrixis

Σ :=

Z

B

r

(v(x, y)

_{− v}

m

)

· (v(x, y) − v

m

)

T

.

Theorem2 The eigenvaluesofthe ovarian ematrix

Σ

ofthe ve tors

v(x, y)

in thespheri al neighorhood

B

r

are

k

2

1

r

4

π

4

+ o(r

4

₎

and

k

2

2

r

4

π

4

+ o(r

4

_).

Proof Letus omputethemean

v

m

of

v(x, y)

onthespheri al neighborhood. By Lemma1theintegralonaspheri alneighborhoodisasymptoti allyequivalentto

theintegralona ylindri alneighborhoodandmorepre isely,

(πr

2

_)v

m

· t

1

= o(r

3

)

andsimilarly

(πr

2

)v

m

· t

2

=

Z

x

2

_+y

2

_<r

2

−k

2

y

p1 + k

2

1

x

2

+ k

2

2

y

2

dxdy + o(r

4

) = o(r

3

).

Thusthe oe ientsof

Σ

satisfy,againbyLemma1,

Σ

11

=

Z

x

2

_+y

2

_<r

2

k

1

2

x

2

1 + k

2

1

x

2

+ k

2

2

y

2

dxdy + o(r

4

) =

Z

x

2

_+y

2

_<r

2

k

1

2

x

2

+ o(r

4

)

=

k

2

1

r

4

π

4

+ o(r

4

)

Similarly,

Σ

22

=

k

2

2

r

4

π

4

+ o(r

4

)

and

Σ

12

= Σ

21

= o(r

4

_).

urva-3.3 Athirddis retefundamentalform

The methods analyzed in se tions 3.1 and 3.2 are akin to the original method

introdu ed in [33℄. Indeed, in [33℄ it was proposed to ompute the ovarian e

matrixofthenormalve torsoftheneighborhood(withoutproje tingtheminthe

lo alregressionplane)and thereforegeta

3

× 3

matrixinsteadofa

2

× 2

matrix. This isa tuallythe simplest imaginablemethod and we shall seethat itgivesa

resultsimilartose tion3.2.

Theorem3 Let

M

bea

C

2

surfa e, let

p

beapointof

M

.Thenthethree eigen-valuesofthe ovarian ematrix

C

oftheunit normalsina neighborhood ofradius

r

around

p

are asymptoti ally respe tively equal to 1 and to the squares of the prin ipal urvaturesat

p

.

Proof Anormalve torwrites

N =

1

p1 + k

2

1

x

2

+ k

2

2

y

2





−k

1

x

−k

2

y

1





+ o(r).

Asinthepreviousse tions,weeasilyobtainbyLemma1,

N

mx

= o(r), N

my

=

o(r), N

mz

= 1 + o(r)

.ThusagainbyLemma1,

C=

Z

x

2

+y

2

_≤r

2

1

1 + k

2

1

x

2

+ k

2

2

y

2





k

1

2

x

2

k

1

k

2

xy

k

1

x(1

− N

mz

)

k

1

k

2

xy

k

2

2

y

2

k

2

y(1

− N

mz

)

k

1

x(1

− N

mz

) k

2

y(1

− N

mz

) (1

− N

mz

)

2



dxdy + o(r

4

)

and,by al ulationsexa tlyanalogous toSe tion3.2,

C

11

=

k

2

1

r

4

π

4

+ o(r

4

)

,

C

22

=

k

2

2

r

4

π

4

+ o(r

4

₎

,

C

12

= C

21

= k

1

k

2

R

x,y

xydxdy = o(r

4

₎

,

C

13

= C

31

= C

23

= C

32

=

C

33

= o(r

4

)

.Thus the eigenvaluesare asymptoti allyequal to

k

2

1

r

4

π

4

and

k

2

2

r

4

π

4

, whi halso gives ba kthe squaresof the prin ipal urvaturesof the surfa e, but

againnottheirsign.

3.4 Afourth dis retefundamental form:thesurfa evariation

Weshallnowanalyzealastvariantintrodu edin[40℄,theso alledsurfa e

varia-tion.Itisagainbasedonalo al ovarian eanalysis.Unlikethepreviousmethods,

thesurfa evariationwasnot laimedtobea urvatureestimate,buttobea

mea-sureoftheneighborhood shape. This subse tionestablishes againalinkbetween

thisdis retequantityand theprin ipal urvaturesofthe surfa e.

Let

p

bea pointwith givenneighborhood

B

r

.Let

o

be thebary enterofthe neighborhood.In

R

3

,the oordinatesarewrittenwithsupers riptse.g.the

oordi-natesofapoint

u

are

(u

1

_{, u}

2

_{, u}

3

₎

.Thus,for

i = 1, 2, 3

,

o

i

₌

1

card B

r

P

p

k

∈B

r

p

i

k

.The entered ovarian ematrix

Σ = (m

ij

)

i,j=1,··· ,3

isdenedas

m

ij

=

P

p

k

∈B

r

(p

i

k

−

o

i

)

_{· (p}

j

_k

_{− o}

j

₎

for

i, j = 1, 2, 3

. Let

λ

0

≤ λ

1

≤ λ

2

be the eigenvalues of

Σ

with orrespondingeigenve tors

v

0

, v

1

, v

2

.For

k = 0,

· · · , 2

,

λ

k

=

X

p

i

∈B

r

Ea h eigenvaluegives the varian e ofthe pointset inthe dire tion ofthe

orre-spondingeigenve tor.Sin e

v

1

and

v

2

aretheve torsthat apturemostvariations, theydenethePCAregressionplane.Thenormal

v

0

tothisplaneisthedire tion

v

minimizing

P

p

i

∈B

r

h(p

i

− o), vi

2

.[40℄ denesthesurfa evariationby

σ =

λ

0

λ

0

+ λ

1

+ λ

2

.

(12)

Thisquantitymeasurestheratioofvarian ealongthenormaltothetotalvarian e.

Iftheneighborhoodishighly urved,its surfa evariation willbe highand ifthe

neighborhood is at the surfa e variation will be small. This quantity has the

propertytobeboundedbetween

0

(at ase)and

1/3

(isotropi distribution ase). Lemma 2 [43℄Inthelo alintrinsi oordinatesystem,thebary enterofa

neigh-borhood

B

r

ofpoint

p

has oordinates

x

o

= o(r

2

₎

,

y

o

= o(r

2

₎

and

z

o

=

Hr

2

4

+o(r

2

)

, where

H =

k

1

+k

2

2

is themean urvature at

p

.

Proof Wegivetheproofforthesakeof ompleteness.ByLemma1appliedtothe

numeratoranddenominator ofthefollowingfra tion,wehave

z

o

=

R

B

r

zdm

R

B

r

dm

=

R

x

2

_+y

2

_<r

2

z(x, y)dxdy + O(r

5

₎

R

x

2

_+y

2

_<r

2

dxdy + O(r

3

)

=

R

x

2

_+y

2

_<r

2



₁

2

(k

1

x

2

_{+ k}

2

y

2

) + o(x

2

+ y

2

) dxdy

R

x

2

_+y

2

_<r

2

dxdy

+ O(r

3

)

=

1

2πr

2

Z

r

ρ=0

Z

2π

θ=0

ρ

2

(k

1

cos

2

θ + k

2

sin

2

θ)ρdρdθ + o(r

2

)

=

r

2

8π

(k

1

π + k

2

π) + o(r

2

) =

Hr

2

4

+ o(r

2

).

Asimilarbutsimpler omputationyieldstheestimatesof

x

o

and

y

o

. Theorem4 Inthe lo al oordinatesystem thesurfa evariation

σ

satises

σ =

r

2

16

 k

2

1

+ k

2

2

2

−

1

3

k

1

k

2



+ o(r

2

)

(13)

Proof We needtoexplainwhatthe ovarian e eigenvaluesstandfor. Ea h

eigen-ve tor

v

i

and asso iated eigenvalue

λ

i

represent a prin ipal dire tion and the variationalongthisprin ipaldire tion,

λ

i

=

Z

m∈B

r

hom, v

i

i

2

dm.

Sin e we have

λ

0

≤ λ

1

≤ λ

2

, we an see that

λ

0

is asso iated to the dire tion withthe leastvariation namelythenormal dire tiontothe surfa e

oz

.Sin ethe eigenve tors formanorthonormal basis,wehave

λ

0

+ λ

1

+ λ

2

=

Z

m∈C

r

hom, v

0

i

2

+

hom, v

1

i

2

+

hom, v

2

i

2

dm =

Z

m∈C

r

Thisyields

λ

0

+ λ

1

+ λ

2

=

R

x

2

_+y

2

_<r

2

x

2

_{+ y}

2

_{+ (z}

_{− z}

o

)

2

dxdy

and

λ

0

+ λ

1

+ λ

2

=

πr

4

2

+ λ

0

+ o(r

6

)

(14)

Werst ompute

λ

0

,applyingagainLemma1togetba ktotheeasy ylindri al neighborhood.

λ

0

=

Z

x

2

+y

2

_≤r

2

(z

_{− z}

o

)

2

dxdy

=

Z

x

2

_+y

2

_≤r

2

z

2

dxdy + z

2

o

Z

x

2

_+y

2

_≤r

2

dxdy

− 2z

o

Z

x

2

_+y

2

_≤r

2

zdxdy

=

Z

x

2

+y

2

_≤r

2

z

2

dxdy +

H

2

_r

4

16

∗ πr

2

− 2

Hr

2

4

r

4

4

πH + o(r

6

₎

=

1

4

(k

2

1

Z

x

2

_+y

2

_≤r

2

x

4

dxdy + k

2

2

Z

x

2

_+y

2

_≤r

2

y

4

dxdy + 2k

1

k

2

Z

x

2

_+y

2

_≤r

2

x

2

y

2

dxdy

−

H

2

_r

6

16

π + o(r

6

₎

=

1

4

r

6

6

(

3π

4

(k

2

1

+ k

2

2

) + k

1

k

2

π

2

)

−

H

2

_r

6

16

π + o(r

6

)

where

H =

k

1

+k

2

2

isthemean urvature.Thus

λ

0

=

πr

6

32

(

k

2

1

+ k

2

2

2

−

1

3

k

1

k

2

) + o(r

6

₎

(15)

Using(14)and(15)weget

σ =

r

2

16

(

k

2

1

+k

2

2

2

−

1

3

k

1

k

2

) + o(r

2

₎

1 +

r

2

16

(

k

2

1

+k

2

2

2

−

1

3

k

1

k

2

) + o(r

2

)

whi hnallyyields:

σ =

r

2

16

 k

2

1

+ k

2

2

2

−

1

3

k

1

k

2



+ o(r

2

)

Theformulaofthe surfa evariation givenbyTheorem4 indeed measuresasort

of urvature.Tointerpretitwe annoti ethat

 thesurfa evariationissymmetri in

k

1

,

k

2

;

 inthe aseofapointlyingon asphere,

k

1

= k

2

= k

so

σ

sphere

=

r

6

24

k

2

;  inthe aseofasaddlepoint

k

1

= k =

−k

2

,

σ

saddle

=

r

6

12

k

2

so

σ

sphere

< σ

saddle

;  inthe aseofa ylinder

k

1

= k

,

k

2

= 0

,

σ

cylinder

=

r

6

32

k

2

.

Itfollowsfromthat thesurfa evariation isnota dis riminatingenough

4Asymptoti behaviorofMLS1 andMLS2

Thesimpleststatisti sthat anbe omputedinaspheri alneighborhoodarethe

bary enterandtheregressionplane.Lemma2statedthatsendingea hpointonto

thebary enterofitsneighborhoodapproximatesthemean urvaturemotion.The

next simplest statisti is the regression plane, and the se ond next is the lo al

degree2 regression surfa e. The maintool of the s ale spa eproposedin [17℄ is

the proje tion of ea h surfa e point

p

on the lo al regression plane. This PCA regressionplaneisdenedastheplane orthogonaltotheleast eigenve torofthe

enteredlo al ovarian ematrix,andpassingthroughthe entroidofthe

neighbor-hood.Theproje tionof

p

onthisplanewillbe alled

p

′

_{= M LS1(p)}

whereMLS

n

standsfor moving least squareof degree

n

. Indeed, thisproje tionmethod isthe simplestinstan eofthemovingleastsquaremethodbywhi hea hpointofa

sur-fa eisproje tedtoalo aldegree

n

polynomialregression.(Thelo albary enter ana tuallybe onsideredasanMLSoforder0,MLS0.)Thereissomeparti ular

interest in MLS1, be ause a re ent meshing method uses it as the simplest

re-versiblesmoothingtoolforpoint louds[17℄.Ontheotherhandmany loudpoint

pro essingmethodsinvolvesomevariantoftheMLS2methodtosmooth,

interpo-late, orsub-samplea point loud.MLS1and MLS2aresmoothing operatorsand

therefore ouldbe used ass alespa es,that is,asiterativesmoothing operators.

But,following[17℄MLS1indeedisas alespa e.MLS2isnot,asillustratedinthe

experiments ofSe tion5.Thetheorems ofthisse tion larifywhathappenswith

theselo alpolynomialregressionsbyrstre allingbrieywhyMLS1implements

amean urvaturemotion,andse ondbyshowingthatMLS2isinsensitivetorst,

se ond,and thirdorderintrinsi derivatives, andhasan order4 dieren etothe

originalsurfa e. Thestudyrevealsthefourth orderintrinsi dierential operator

asso iatedwith MLS2.

4.1 Theasymptoti behaviorofMLS1

Thenextlemma omparesthenormaltothePCAregressionplanewiththenormal

tothesurfa e,

n

at

p

.

Lemma 3 The normal

v

tothePCAregressionplaneinaspheri alneighborhood

B

r

at

p

∈ M

is equal to thesurfa enormal at point

p

,up to a negligible fa tor:

v

= n + O(r)

.

Proof The lo al PCA regression plane of point

p

is hara terized as the plane passingthrough thebary enteroftheneighborhood

B

r

andwith normal

v

mini-mizing:

I(v) =

Z

B

r

|hv, pp

′

i|

2

dp

′

s.t.

kvk = 1

Denotingby

(v

x

, v

y

, v

z

)

the oordinates of

v

,

I(v) =

Z

B

r

(v

x

x + v

y

y + v

z

1

2

(k

1

x

2

_{+ k}

2

y

2

) + o(r

2

))

2

dxdy.

Considering theparti ular value

v

= (0, 0, 1)

shows that theminimalvalue

I

min

of

I(v)

satises

I

min

≤ O(r

6

_).

v

x

≤ O(r)

and

v

y

≤ O(r)

. Thus, sin e

||v|| = 1

,

v

z

≥ 1 − O(r)

and therefore

v

= n + O(r)

.

Theorem5 Let

T

r

bethe operator dened on the surfa e

M

transforming ea h point

p

into itsproje tion

p

′

= T

r

(p)

on thelo al regressionplane. Then

T

r

(p)

− p =

Hr

2

4

n

+ o(r

2

_).

(16)

Proof ByLemma2thebary enter

o

of

B

r

haslo al oordinates

−→

po

= (o(r

2

_{), o(r}

2

_),

Hr

2

4

+

o(r

2

))

.Ontheotherhand

−−→

pp

′

isproportionaltothenormaltotheregressionplane,

v

. ThusbyLemma 3,

−−→

pp

′

= λ(O(r), O(r), 1

_{− O(r)).}

To ompute

λ

, we use the fa tthat

p

′

istheproje tion on theregressionplaneof

p

,andthat

o

belongs to thisplanebydenition.This impliesthat

−−→

pp

′

_⊥

−→

op

′

and therefore

λ

2

O(r

2

) + λ(1

_{− O(r))(H}

r

2

4

+ o(r

2

) + λ(1

_{− O(r))) = 0,}

whi hyields

λ =

Hr

2

4

+ o(r

2

₎

andtherefore

−−→

pp

′

= (O(r

3

), O(r

3

),

Hr

2

4

+ o(r

2

_{)) =}

Hr

2

4

n

+ o(r

2

_).

4.2 Theasymptoti behaviorofMLS2

Amongthe manyversionsofMLS2 proposedin the literature,we shallpi kone

whi h isa ommon denominator, and proneto a simpleasymptoti analysis. In

MLS2 a rst intrinsi referen e frame is rst al ulated, and the mean square

approximation byorder 2polynomialsismade inthisreferen eframe.The most

naturalframeisfoundbyapplyingMLS1,andthe oordinates

(x, y)

aretherefore the oordinatesintheregressionplaneinaspheri alneighborhood

B

r

.These ond stepistondthe losestorder2polynomialinthespheri alneighborhoodforthe

quadrati distan e.Be auseofLemma1we anspe ify,withoutlossofgenerality

orpre ision,thatthisminimizationismadeinthe ylindri alneighborhood

C

r

.In thatway,allintegrals omputedintheapproximationpro essareintegralsonthe

disk

x

2

_{+ y}

2

_{≤ r}

2

,whi h isnumeri allyandformally onvenient.Thusthe MLS2

algorithmwhi hwe shallanalyzeworks inthetwosteps:

1. ompute the regression plane of the manifold in the spheri al neighborhood

B

r

= B

r

(p)

∩ M

;

2. all

(x, y)

the referen e oordinates in the regression plane.Consider the re-stri tionofthe smooth manifoldtothe disk

D

r

:= x

2

_{+ y}

2

≤ r

2

,

z = f (x, y)

. Then nd the order 2 polynomial

g(x, y)

that best approximates

f

for the

L

2

(D

r

)

distan e;

3. set(inthereferen eframe)

M LS2(p) := (0, 0, g(0, 0))

.

ThenexttheoremshowsthatunlikeMLS1,whi hrevealsthemean urvature,the

dieren ebetweenapointsmoothedbyMLS2anditsoriginalpositionisverysmall

(oforder 4)and a tuallyrevealsafourth orderintrinsi operator ofbi-Lapla ian