Journal�

of ComputationalPhysics173,�

730–764(2001)

doi:10.1006/jcph.2001.6910,�

availableonlineathttp://www.idealibrary.comon

F�

acundoMemoli� � andGuillermoSapiro��Instituto

�deIngenierıa Electrica,Universidaddela Republica,Montevideo,Uruguay; Electrical

�and� ComputerEngineering, University of Minnesota,Minneapolis,Minnesota55455

E-mail:

memoli@iie.edu.uy;guille@ece.umn.edu

Received March26,2001;revisedAugust6, 2001

An�

algorithmfor thecomputationallyoptimalconstructionof intrinsic weighteddistance

�functionson implicit hyper-surfacesis introducedin this paper. Thebasic

idea�

is to approximatethe intrinsic weighteddistanceby the Euclideanweighteddistance

�computedin abandsurroundingtheimplicit hyper-surfacein theembedding

space,� therebyperformingall the computationsin a Cartesiangrid with classicaland� efficientnumerics.Basedonwork ongeodesicsonRiemannianmanifoldswithboundaries,

�we boundtheerrorbetweenthe two distancefunctions.We show that

this�

error is of thesameorderasthetheoreticalnumericalerror in computationallyoptimal,� Hamilton–Jacobi-based,algorithmsfor computingdistancefunctions inCartesian

�grids.Therefore,wecanusethesealgorithms,modifiedtodealwith spaces

with� boundaries,andobtainalsofor thecaseof intrinsicdistancefunctionsonimplicithyper

�-surfacesacomputationallyefficient technique.Theapproachcanbeextended

to�

solve a moregeneralclassof Hamilton–Jacobiequationsdefinedon theimplicitsurf� ace,following thesameideaof approximatingtheirsolutionsby thesolutionsinthe

�embeddingEuclideanspace.Theframework hereintroducedtherebyallows for

the�

computationsto beperformedonaCartesiangrid with computationallyoptimalalgorithms,� in spiteof thefact that thedistanceandHamilton–Jacobiequationsareintrinsic

�to theimplicit hyper-surface. c��

2001AcademicPress

Key Words: implicit hyper-surfaces;distancefunctions; geodesics;Hamilton–Jacobi

�equations;fastcomputations.

1. INTRODUCTION

Computing�

distancefunctionshasmany applicationsin numerousareasincludingmath-ematical� physics,imageprocessing,computervision, robotics,computergraphics,com-putational� geometry, optimalcontrol,andbrain research.In addition,having thedistance

730�

0021-9991/01$35.00Copyright c

�2001by AcademicPress

All rightsof reproductionin any form reserved.

DIST

ANCE FUNCTIONSAND GEODESICS 731!

function"

from a seedto a targetpoint, it is straightforward to computethecorrespondinggeodesic# path,sincethis is given by the gradientdirectionof the distancefunction,backpropagating� from thetargettowardtheseed(seefor example[18]). Geodesicsareusedfore� xamplefor pathplanningin robotics[35]; brainflatteningandbrainwarpingin compu-tational

$neuroscience[56, 57, 60, 61, 68]; crests,valleys, andsilhouettescomputationsin

computer% graphicsandbrainresearch[7, 30,62]; meshgeneration[64]; andmany applica-tions

$in mathematicalphysics.Distancefunctionsarealsoveryimportantin optimalcontrol

[59] andcomputationalgeometryfor computationssuchasVoronoidiagramsandskeletons[48]. It is thenof extremeimportancetodevelopefficienttechniquesfor theaccurateandfastcomputations% of distancefunctions.It is thegoalof thispaperto presentacomputationallyoptimal& techniquefor thecomputationof intrinsic weighteddistancefunctionson implicithyper-surfaces.1 It is well known already, andit will be furtherdetailedbelow, that theseweighted' distancescanbeobtainedasthesolutionof simpleHamilton–Jacobiequations.W

(e will alsoshow that the framework herepresentedcanbe appliedto a larger classof

Hamilton–Jacobi)

equationsdefinedonimplicit surfaces.Wealsodiscusstheapplicationofour& proposedframework to other, nonimplicit,surfacerepresentations.

1.1. DistanceFunction Computation and Its Hamilton–JacobiFormulation

Beforeproceeding,let usfirst formally definetheconceptof intrinsic weighted' distanceson& implicit hyper

*-surfaces.Let + be

,a (codimension1) hyper-surfacein IR

- d.

defined/

asthezero level setof a function 0 : IRd

. 1IR. That is, 2 is given by 3 x4 5 IRd

.: 6 7 x4 8 9 0

: ;.

W(

e assumefrom now on that < is=

a signeddistancefunction to the surface > . (This isnot a limitation, sinceaswe will discusslater, both explicit andimplicit representationscan% be transformedinto this form.) Our goal is, for a given point p? @ A , to computetheintrinsic

=gB -weighteddistancefunctiond

C gDE F p? G x4 H for"

all desiredpointsx4 I J K 2 NoteL

thatwearereferringto the intrinsic gB -distance,that is, the geodesicdistanceon the RiemannianmanifoldM N O P gB 2II

- Q(II

-standsfor the R dC S

1T U V dC W1X identity

=matrix) and not on the

embedding� Euclideanspace.For a given positive weightgB defined/

on thesurface(we areconsidering% only isotropicmetricsfor now), the gB -distanceon Y (that coincideswith thegeodesic# distanceof theRiemannianmanifold Z [ \ gB 2II ] ) i

^sgiven by

dC gD_ ` p? a x4 b cd inf

=epxf [ g ]

hL

igD j k l m n (1)

where'

L gD o p q rs b

agB t u v l w x y z{ |

l } ~ dlC

(2)

is=

the weightedlength functional defined/

for piecewise C1 curv% es � : [a� � b�

] � � , andCpx� [ � ] denotesthesetof curvespiecewiseC1 joining

�p? to

$x4 , traveling on � . In general,

1Although�

all the examplesin this paperaregoing to be reportedfor two-dimensionalsurfacesin 3D (heredenoted�

as3D surfaces),thetheoryis valid for generaldimensionhyper-surfaces,andit will bepresentedin thisgenerality� . A numberof applicationsdealwith higherdimensions.For example,for thegeneraltheoryof harmonicmaps,� in orderto dealwith mapsontogeneralopensurfaces,it is necessaryto havethisnotionof intrinsicdistance[41]. In addition,higherdimensionsmight appearin motionplanning,whenexplicitly assumingthattherobotisnotmodeledby apoint, therebyaddingadditionalconstraintsto its movements.

2�

Thiscancertainlybeextendedto any subsetof � .�

732!

MEMOLI AND SAPIRO

we' will considerthedefinitionto bevalid for any gB defined/

over thedomainthatthecurvemaytravel through.

W(

e needto computethis distancewhenall the concerningobjectsare representedindiscrete

/form in thecomputer. Computingminimal weighteddistancesandpathsin graph

representations� is anold problemthathasbeenoptimallysolvedby Dijkstra [21]. Dijkstrashowedanalgorithmfor computingthepathin O � n� log

�n� � operations,& wheren� is

=thenumber

of& nodesin thegraph.Theweightsaregiven on theedgesconnectingbetweenthegraphnodes,� andthealgorithmiscomputationallyoptimal.In theory,wecouldusethisalgorithmtocompute% theweighteddistanceandcorrespondingpathonpolygonal(notimplicit) surfaces,with' theverticesasthegraphnodesandtheedgestheconnectionsbetweenthem(see[33]).Theproblemis that theoptimalpathscomputedby this algorithmarelimited to travel onthe

$graphedges,giving only afirst approximationof thetruedistance.Moreover, Dijkstra’s

algorithmis notaconsistentone:it will notconverge to thetruedesireddistancewhenthegraph# andgrid is refined[42, 43]. Thesolutionto this problem,limited to Cartesiangrids,w' asdevelopedin [27, 51,52,59] (andrecentlyextendedby OsherandHelmsen,see[45]).Tsitsiklis [59] first describedan optimal-controltype of approach,while independentlySethian[51, 52] andHelmsen[27] bothdevelopedtechniquesbasedonupwindnumericalschemes.The solutionpresentedby theseauthorsis consistentandconvergesto the truedistance

/[49, 59], while keepingthe sameoptimal complexity of O � n� logn� � . Later this

w' ork was extendedin [32] for triangulatedsurfaces(seealso[7, 36] for relatedworksonnumericson non-Cartesiangrids).We shouldnotethat thealgorithmdevelopedin [32] iscurrently% developedonly for triangulatedsurfaceswith acutetriangles.Therefore,beforethe

$algorithmcanbeapplied,asaninitializationstepthesurfaceshavetobepreprocessedto

removeall obtusetrianglesor otherpolygonspresentin therepresentation[31]. Following[52], wecall thesefast

�marchingalgorithms.

Thebasicideabehindthecomputationallyoptimaltechniquesfor finding weighteddis-tances,

$meaningthesefastmarchingalgorithms,is tonotethatthedistancefunctionsatisfies

a Hamilton–Jacobipartial differentialequation(PDE) in the viscositysense;see,for ex-ample,[38, 50] for thegeneraltopicof distancefunctionsonRiemannianmanifolds(andanice� mathematicaltreatment),and[12,23,31,44,46,52] for theplanar(andmoreintuitive)case.% ThisHamilton–Jacobiequationis given by

� �d

C gD� � gB � (3)

where' � � is=

thegradientintrinsic to thesurface,anddC gD� is

=thegB -distancefrom agivenseed

point� to therestof themanifold.3

Thatis, wecantransformtheproblemof optimaldistancecomputationinto theproblemof& solvingaHamilton–Jacobiequation(recallthatgB is

=known, it is thegiven weight),also

knownastheEikonalequation.In ordertosolvethisequation,thecurrentstateof knowledgepermits� usto accuratelyandoptimally(in acomputationalsense)find (weighted)distanceson& Cartesiangridsaswell asonparticulartriangulatedsurfaces(aftersomepreprocessing,namelytheeliminationof obtusetriangles,see[6, 32]). Thegoalof thispaperis to extendthis

$to implicit hyper-surfaces.In otherwords,wewill show how to solvetheaboveEikonal

equation� for implicit hyper-surfaces  .

3¡

Note¢

that £ ¤ anddg¥¦ becometheclassicalgradientanddistancerespectively for Euclideanspaces.

DIST

ANCE FUNCTIONSAND GEODESICS 733!

Recall§

thatalthoughall theapplicationsin this paperwill bepresentedfor 3D surfaces,the

$theoryis valid for any d

C-dimensionalhyper-surfaces,andwill thenbepresentedin this

generality# .

1.2. DistanceFunction and Geodesicson Implicit Surfaces

Themotivationsbehindextendingthedistancemapcalculationto implicit surfacesarenumerous:� (a) In many applications,surfacesarealreadygiven in implicit form, e.g.,[10,13,16,25,45,46,53,66,62], andthereis thenaneedto extendto this importantrepresen-tation

$thepreviouslymentionedfasttechniques.Wecouldof coursetriangulatetheimplicit

surface,eliminateobtusetriangles,andthenusetheimportantalgorithmproposedin [32].This is notadesirableprocessin generalwhentheoriginaldatais in implicit form, sinceitaffectsthedistancecomputationaccuracy becauseof errorsfrom thetriangulation,andalsoaddsthecomputationalcostof thetriangulationitself,triangulationthatmightnotbeneededby

,the specificapplication.If, for example,what it is neededis to computethe distance

between,

aseriesof pointsonthesurface,thecomputationalcostaddedby thetriangulationis

=unnecessary. Notethatfinding a triangulatedrepresentationof theimplicit surfaceis of

course% dimensionalitydependent,andaddstheerrorsof the triangulationprocess.More-o& ver, accuratetriangulationsthat ensurecorrectnessin the topologyarecomputationallye� xpensive, andonceagainthereis noreasonto performafull triangulationwhenwemightbe

,interestedjust in theintrinsicdistancebetweenafew pointsontheimplicit surface.(b) It

is=

ageneralagreementthatthework onimplicit representationsandCartesiangridsis morerobustwhendealingwith differentialcharacteristicsof thesurfaceandpartialdifferentialequations� on thissurface.NumericalanalysisonCartesiangridsis muchmorestudiedandsupportedby fundamentalresultsthantheworkonpolygonalsurfaces.It isevenrecognizedthat

$thereisnoconsensusonhow tocomputebasicdifferentialquantitiesover atriangulated

surface(see,for example,[22]), althoughthereis quiteanagreementfor implicit surfaces.Moreover, representinganhyper-surfacewith structuredelementssuchastrianglesis cer-tainly

$difficult for dimensionsother than 2 or 3. (c) If the computationof the distance

function"

is just a partof a generalalgorithmfor solvinga givenproblem,it is not elegant,accurate,norcomputationallyefficienttogobackandforth fromdifferentrepresentationsofthe

$surface.

Beforeproceeding,we shouldnote that althoughthe whole framework and theory ishere

*developedfor implicit surfaces,it is valid for othersurfacerepresentationsaswell

after preprocessing.This will be explainedanddiscussedlater in the paper(Section5).Moreover, we will later assumethat the embeddingis a distancefunction. This is not alimitation,

�sincemany algorithmsexist to transforma genericembeddingfunction into a

distance/

one;seealsoSection5. Therefore,the framework herepresentedcanbeappliedboth

,to implicit (naturally)andothersurfacerepresentationssuchastriangulatedones.

In order to computeintrinsic distanceson surfaces,a small but importantnumberoftechniques

$have beenreportedin theliterature.As mentionedbefore,in a very interesting

w' ork Kimmel andSethian[32] extendedthefastmarchingalgorithmto work on triangu-latedsurfaces.In its currentversion,thisapproachcanonly beusedwhendealingwith 3Dtriangulated

$surfacesandits extensionto dealwith higherdimensionsseemsveryinvolved.

Moreover, it canonly correctlyhandleacutetriangulations(therebyrequiringapreprocess-ing

=step).And of course,it doesn’t applyto implicit surfaceswithout somepreprocessing

(a triangulation).

734!

MEMOLI AND SAPIRO

Another¨

very interestingapproachto computingintrinsic distances,this time workingwith' implicit surfaces,wasintroducedin [16]. Thiswill befurtherdescribedbelow, but firstlet’

�s make somecommentson it. First, this is an evolutionary/iterative approach,whose

steadystategives thesolutionto thecorrespondingHamilton–Jacobiequation.Therefore,this

$approachis not computationallyoptimal for the classof Hamilton–Jacobiequations

discussed/

in thispaper. (Althoughwhenproperlyimplemented,thecomputationalcomplex-ity of this iterative schemeis thesameasin thefastmarchingmethodhereproposed,theinner

=loop is morecomplex, makingtheiterative algorithmslower.)4 Second,very careful

discretization/

mustbe doneto the equationproposedin [16] becauseof the presenceofintrinsic

=jumpfunctionsthatmightchangethezerolevel-set(i.e.,thesurface).Ontheother

hand,thenumericalimplementationis notnecessarilydonevia theutilizationof monotone©scª hemes, asrequiredby ourapproachandall thefastmarchingtechniquespreviouslymen-tioned

$(therebyhaving a theoreticalerror « ¬ ­ ® x4 ¯ [19]), andbetteraccuracy might then

be,

obtained.In

°order to computethe intrinsic distanceon an implicit surface,we must thensolve

the$

correspondingHamilton–Jacobiequationpresentedbefore.In order to do this in acomputationally% efficient way, we needto extendthe fastmarchingideasin [27, 45, 51,52, 59], which assumea Cartesiangrid, to work in our case.Sincean implicit surfaceis representedin a Cartesiangrid, correspondingto theembeddingfunction, thefirst andmostM intuitive ideais thento attemptto solve the intrinsic Eikonal equationusing± the fast

�mar© ching technique.

$The first steptoward our goal is to expressall the quantitiesin the

intrinsic=

Eikonalequationby its implicit-extendedrepresentations.� Whatwe meanis thatthe

$intrinsic problem� (weconsidergB ² 1 for simplicity of exposition),

³ ´ µd

C ¶ ·x4 ¸ ¹ º 1 for p? » ¼

dC ½ ¾

p? ¿ À 0: Á (4)

with' p? Â Ã the$

seedpoint, is to beextendedto all IR- d

.(or at leastto abandsurroundingÄ ),

^andthederivativesareto betakentangentiallyto Å Æ Ç 0

: È. Consideringthentheprojection

of& theEuclideangradientontothetangentspaceof É to$

obtaintheintrinsicone,anddenotingby

,d

Cthe

$Euclideanextensionto theintrinsic distanced

C Ê, we have to numericallysolve, in

the$

embeddingCartesiangrid, theequation

Ë Ìd

C Íx4 Î Ï 2 Ð Ñ Ò d

C Óx4 Ô Õ Ö × Ø x4 Ù Ú 2 Û 1 for x4 Ü IRd

.d

C Ýl Þ p? ß ß à 0

: á (5)

where' l â p? ã is=

theray throughp? normal� to thelevel setsof ä .This is exactly the approachintroducedin [16], as discussedabove, to build-up the

e� volutionaryapproach,given by thePDE

åt æ sgª n ç è 0 é ê ë ì í î 2 ï ð ñ ò ó ô õ ö 2 ÷ 1ø ù 0

: ú(6)

where' û0 ü x4 ý þ ÿ � x4 � 0

: �is

=the initial valueof the evolving function,generallya step-like

4�

Thegeneralframework introducedin [16] is applicablebeyondtheHamilton–Jacobiequationsdiscussedinthis�

paper(seealso[8, 17]). Herewe limit thecomparisonbetweenthe techniquesto theequationswherebothapproaches� areapplicable.

DIST

ANCE FUNCTIONSAND GEODESICS 735!

function"

(convolved with the signum)that tells insidefrom outsideof the zerolevel-set.Onethenfindsd

C � � � � � � �.

Of course,in order to obtain a computationallyoptimal approach,we want to solvethe

$stationaryproblem(5), andnot its iterative counterpart(6). It turnsout that thebasic

requirements� for theconstructionof afastmarchingmethod,evenwith therecentextensionsin

=[45], do not hold for this equation.This can be explicitly shown, and hasalso been

hintedby Kimmel andSethianin their work on geodesicson surfacesgiven as graphsoffunctions.

" 5

To recap,the fast marchingapproachcannotbe directly applied to the computationof& intrinsic distanceson implicit surfacesdefinedon a Cartesiangrid (Eq. (5)), and thestateof the art in numericalanalysisfor this problemsaysthat in order to computein-trinsic

$distancesoneeitherhasto work with triangulatedsurfacesor hasto usethe iter-

ative approachmentionedabove. The problemswith both techniqueswere reportedbe-fore, and it is the goal of this paperto presenta third approachthat addressesall theseproblems.�

1.3. Our Contrib ution

Thebasicideaherepresentedis conceptuallyvery simple.We first considera small h�

of& fsetof � . That is, sincetheembeddingfunction � is=

a distancefunction,with � asitszerolevel set,we considerall pointsx4 in IR3 for which � � � x4 � � � h

�. This gives a region in

IRd.

with' boundaries.We thenmodify the (Cartesian)fastmarchingalgorithmmentionedabove for computingthe distancetransforminsidethis h

�-bandsurrounding� . Note that

hereall thecomputationsare,asin theworks in [27, 51, 52, 59], in a Cartesiangrid. Wethen

$usethis Euclideandistancefunction as an approximationof the intrinsic distance

on& � . In Section2 we show that theerrorbetweenthesetwo distances,underreasonableassumptionson the surface � , is of the sameorderasthe numericalerror introducedbythe

$fastmarchingalgorithmsin [27, 51, 52, 59].6 Therefore,

�whenadaptingthesealgo-

rithmsto work on Euclideanspaceswith boundaryadaptationdescribedin Section3, weobtain& an algorithm for the computationof intrinsic distanceson implicit surfaceswiththe

$samesimplicity, computationalcomplexity, andaccuracy as theoptimalfastmarching

techniques$

for computingEuclideandistancesonCartesiangrids.7 In°

Section3 wealsoex-plicitly� discussthenumericalerrorof our proposedtechnique.Examplesof thealgorithmhereproposedaregiven in Section4. SinceOsherandHelmsenhave recentlyshown thatthe

$fastmarchingalgorithmcanbe usedto solve additionalHamilton–Jacobiequations,

we' show that the framework hereproposedcanbe appliedto equationsfrom that classaswell; this is donein Section5. This sectionalsodiscussesthe useof the frameworkherepresentedfor nonimplicit surfaces.Finally, someconcludingremarksare given inSection6.

5�

Wehavealsobenefitedfrom privateconversationswith StanOsherandRonKimmel to confirmthisclaim.6

�In contrastwith works suchas[1, 47], wherean offset of this form is just usedto improve the complexity

of the level-setsmethod,in our casetheoffset is neededto obtaina smallerrorbetweenthecomputeddistancetransform�

andtherealintrinsicdistancefunction;seenext section.7

!Although in this paperwe dealwith the fastmarchingtechniques,othertechniquesfor computingdistance

functions"

on Cartesiangrids,e.g.,thefasttechniquereportedin [11] for uniform weights,couldbeusedaswell,since# thebasisof ourapproachis theapproximationof theintrinsicdistanceby anextrinsicone.

736!

MEMOLI AND SAPIRO

2.$

DISTANCE FUNCTIONS: INTRINSIC VS. EXTRINSIC

The�

goalof thissectionistopresenttheconnectionbetweentheintrinsicdistancefunctionandtheEuclideanfunctioncomputedinsideabandsurroundingthe(implicit) surface.Wewill' completelycharacterizethedifferencebetweenthesetwo functions,mainly basedonresults� on shortestpathson manifoldswith boundary. Theresultsherepresentedwill alsojustify

�the useof the Cartesianfastmarchingalgorithmsfor the computationof intrinsic

weighted' distanceson implicit surfaces.Recallthatwe aredealingwith a closedhyper-surface% in IRd

.represented� asthezero

level-setof adistancefunction & : IRd. '

IR. Thatis, ( ) * + , 0: -

. Ourgoalis to computea gB -weighteddistancemapon thissurfacefrom aseedpointq. / 0 .

Let

1h

23x4 5 6 B

7 8x4 9 h

� : ; <x4 = IR

- d.

: > ? @ x4 A B C h� D

be,

theh�

-ofE fsetof& F (hereB7 G

x4 H h� I

is=

theball centeredat x4 with' radiush�

).^

It is well knownthat

$for a smoothJ , K L h is also smoothif h

�is sufficiently small, seeAppendix A for

references.� Mh is

=thenamanifold© with smoothboundary.

Our computationalapproachis basedon approximatingthe solution of the intrinsicproblem� (d

C gDN O p? P is=

theintrinsic gB -weighteddistanceon Q )^

R Sd

C gDT U p? V W gB for"

p? X Yd

C gDZ [ q. \ ] 0: ^ (7)

by,

thatof theEuclidean_

(or extrinsic)^

one,

`d

C gDah

b c p? d e gB for p? f gh

dC gDh

hb i q. j k 0

: l (8)

where' gB is asmoothextensionof& gB in adomaincontainingm h, anddC gDn

hb o p? p is

=theEuclidean

gB -weighteddistancein q h.Ourgoalistobeabletocontrol,for pointson r , s dC gDt u d

C gDvh

b w Lx y z { |

with' h�

. Notethatwehavereplacedtheintrinsicgradient} ~ by,

theEuclideangradientandthe

$intrinsic distanced

C gD� � p? � on& the surfaceby the EuclideandistancedC gD�

hb � p? � in � h. We

ha*

ve thentransformedtheproblemof computinganintrinsic distanceinto theproblemofcomputing% adistancein anEuclideanmanifoldwith boundary.

W(

e will show thatundersuitable(andlikely) geometricconditionson � we' canindeedcontrol% � d

C gD� � dC gD�

hb � L � � � � with' h

�. In orderto materializethis,wefirst needto briefly discuss

the$

extensiongB andto review somebasicbackgroundmaterialon Riemannianmanifoldswith' boundary.

2.1.�

The Extensionof the Weight g�W

(e requirethat gB � � � gB , and that gB is

=smoothª andnonnegative within � h. Thereare

situationswhenonehasa readily availableextension,andothersin which the extensionhas

*to be“invented.” Wecall theformernatur� al extensionandthelattergB eneral extension.

Both�

cases,asarguedbelow, will providesmoothfunctionsgB .

DIST

ANCE FUNCTIONSAND GEODESICS 737!

In°

many applications,the weight gB : � � IR-

depends/

on the curvaturestructureof thehyper-surface. Denoting B� � � � : � � s Iª Rd

. �d

.the

$secondfundamentalform of   , and¡ ¢

B£ ¤ ¥

x4 ¦ § the$

setof its eigenvalues,thismeansthat

gB ¨ x4 © ª F« ¬ ­ ®

B ° x4 ± ² ³

where' F«

is=

agivenfunction.In thiscaseit is utterlynatur� al to$

takeadvantageof theimplicitrepresentationby notingthatB µ x4 ¶ · H ¸ ¹

Txº » ¼ x4 ½ for x4 ¾ ¿ , whereH À is theHessianof ÁandTx4  is thetangentspaceto à at x4 (see[37]). Thenatur� al e� xtensionthenbecomes

gB Ä x4 Å Æ F Ç H È ÉT

Êxº Ë Ì xº Í Î x4 Ï Ð x4 Ñ Ò

h Ó (9)

where' Ô Õ x4 Ö ×Ø Ù yÚ Û IRd.

: Ü Ý yÚ Þ ß à á x4 â ã äThisextensionis valid for å x4 æ IRd

.: ç è é x4 ê ë ì 1í î ï ð , whereñ ò absolutelyboundsall

principal� curvaturesof ó ; seeAppendixA.When

ôtheweightgõ cannotö bedirectlyextendedto bevalid for atubularneighborhoodof

the÷

hyper-surface,onehasto do that in a pedestrianway. Onesuchextensioncomesfrompropagatingø thevaluesof gõ alongthenormalsof ù in aconstantfashion;i.e.,

gõ ú xû ü ý gõ þ ÿ � � xû � � � xû � �h � (10)

where� � � � : IRd � �

standsfor the normalprojectiononto � . This extensionis welldefined

�andsmoothas long as thereis a uniquefoot

�in � for every xû in the domainof

the÷

desiredextension � . Takingh�

sufficiently smallwe canguaranteethat � � � h if� �

is�

smooth.SeeAppendixA for somedetails.In

�practice,thisextensioncanbeaccomplishedsolvingtheequation[14]

�t � sgn� � ! " # $ % & ' 0

(

with� initial conditionsgiven by any ) * + , 0( -

, suchthat . / 0 1 0( 2 3 4 5

gõ . Thengõ 6 7 8 9: ; < = > ? @ .

2.2.A

ShortestPathsand DistanceFunctions in Manif oldswith Boundary

Sincewe wantto approximatetheproblemof intrinsic distancefunctionsby a problemofB distancefunctionsin manifoldswith boundary, andto prove that the latter convergesto

÷the former, we needto review basicconceptson this subject.We will mainly include

resultsC from [2, 3, 63]. We areinterestedin theexistenceandsmoothnessof thegeodesiccurvö eson manifoldswith boundary, sinceour convergenceargumentsbelow dependonthese

÷properties.We will assumethroughoutthis sectionthat D E F mG H is a connectedand

completeRiemannianI

manifoldwith boundary(thiswill laterbecometheh�

-offset J h with�the

÷metric gõ 2II , whereII now standsfor thed

K Ld

Kidentitymatrix).

DEFINITION 2.1 Let pM N qO P Q , thenif dK R S T U V W

: X Y Z [ IR is thedistancefunctionin \ (with its metricmG ),

]a shortestpathbetweenpM andqO is apathjoining themsuchthat

its�

RiemannianlengthequalsdK ^ _

pM ` qO a .

Nob

w, sincec is complete, for every pair of points pM andqO there÷

existsa shortest� pathjoining

dthem;see[2]. Thefollowing resultsdealwith theregularityof this shortestpath.

738e

MEMOLI AND SAPIRO

FIG. 1. Thef

minimal pathis Cg 1,h but notC2

i.

THEOREM 2.1 Let j k l mG m ben

a C3 manifoldG with C1 boundaryn

B. ThenanyshortestpathM of o is C1.p

Whenô q r s

mG t is aflat manifold(i.e., u is acodimension0 subsetof IRd

andthemetricmG is

�isotropicandconstant),it is easyto seethatany shortest� pathmustv bea straightline

whene� ver it is in theinteriorof w , andashortestpathof theboundaryB when� it is there.This

xwill bethesituationfor usfrom now on.

It mightseemabit awkwardthatonecannotachieveahigherregularityclassthanC1 forthe

÷shortestpaths,evenby increasingtheregularityof y z B, butasimplecounterexample

will� convincethereader. Think of { as IR| 2 with� theopenunit discremoved (seeFig. 1),

andits Euclideanmetric.Theaccelerationin all theopensegment } AP~ is �0,(

andin all theopenB arc � P Q

� �is

� � �er ; that is, it pointsinwardandhasmodulus1. That is, even in most

simpleexamples,C2 regularity is not achievable.It is, however, very easyto checkthatinthis

÷case �� is

�actuallyLipsc

�hitz.

F�

or thegeneralsituation,in [3,39] theauthorsprovedthatshortestpathsdohaveLipsc�

hitzcontinuousfirst derivatives,whichmeansthatin factshortestpathsaretwicedifferentiablealmost� everywhereby

�Rademacher’sTheorem.Thisfactwill beof greatimportancebelow.

For a morecomprehensive understandingof the theoryof shortestpathsanddistancefunctions

�in Riemannianmanifoldswith boundary, see[2, 3,39,63] andreferencestherein.

2.3. ConvergenceResult for the Extrinsic DistanceFunction

Wô

enow show therelationbetweentheEuclideandistancein theband� h andtheintrinsicdistance

�in thesurface� . Below wewill denoted

K � �� dK 1� , andd

K �h

� �� dK 1�

h� .

Observation2.1. Sinceweassumetheimplicit surface� to÷

becompact,thecontinuousfunction

�d

K �: � � � � IR

|attainsits maximum.Therefore,we candefinethediameterof

DIST�

ANCE FUNCTIONSAND GEODESICS 739e

the÷

setas

diamK � �   ¡¢ sup

p£ ¤ q ¥ ¦ dK § ¨

pM © qO ª « ¬ ­

Observation2.2. Since ® ¯ ° h we� have that for every pair of points pM andqO in� ±

,d

K ²h

� ³ pM ´ qO µ ¶ dK · ¸

pM ¹ qO º , so in view of thepreviousobservationwehave

dK »

h� ¼ pM ½ qO ¾ ¿ diam

K À Á Â ÃpM Ä qO Å Æ Ç

Observation2.3. Sinceweareassuminggõ to÷

beasmoothextensionof gõ to÷

all È É Ê h

(we stressthe fact that the extensiondoesnot dependon h), gõ will� be Lipschitz in Ë ,andwe call K gÌ its associatedconstant.Further, we will denoteMgÌ ÍÎ maxÏ xÐ Ñ Ò Ó gõ Ô xû Õ andMgÌ Ö× supØ xÐ Ù Ú Û gõ Ü xû Ý .

Wô

eneedthefollowing Lemma�

whose� simpleproofweomit (see,for example,[18]).

LÞ

EMMA 2.1. Whena gõ -shortestpathtravelsthroughan interior region, its curvature isabsolutely� boundedby

Bß

gÌ àá supâxÐ ã ä å

æ çgõ è xû é ê

gõ ë xû ì íThe

xfollowing Lemmawill beneededin theproof of the theorembelow. Its proof canbe

foundin AppendixB.

LEMMA 2.2. Let f : [a� î bn

] ï IR bea C1 ð [a� ñ bn] ò function

�such that f ó is Lipschitz.Letô õ L

� ö ÷[a� ø b

n] ù denote

K(oneú of )

]f

� û’s� weakderivative(s� )

].p Then

b

af

� ü 2 ý xû þ dxK ÿ

f f� � b

a� b

af

� �xû � � � xû � dx

K �

Wô

e are now readyto presentone of the main resultsof this section.We boundtheerror� betweenthe intrinsic distanceon andthe Euclideanonein the offset h. As wewill� seebelow, in the mostgeneralcase,the error is of the orderh

� 1� 2 (h�

being�

half theofB fset width). We will later discussthat this is also the orderof the theoreticalerror forthe

÷numericalapproximationin fastmarchingmethods.Thatwill leadusto concludethat

ourB algorithmdoeskeeptheconvergenceratewithin thetheoreticallyproven orderfor fastmarchingv methods’numericalapproximation.However, for all practicalpurposes,theorderofB convergencein thenumericalschemesusedby fastmarchingmethodsis thatof h

�; see

[49]. We will alsoarguethat for all practicalpurposeswe canguaranteeno decayin theoB verallrateof convergence.WedeferthedetaileddiscussiononthistoafterthepresentationofB thegeneralboundbelow.

Tx

HEOREM 2.2. Let�

A and B be two pointson thesmoothhyper-surface� (seeFig. 2).

Let dgÌh � d

K g�h

� � A � B � and� dg� � dK g� � A � B � .p Then, for

�pointson thesurface� , we� havethat

for�

sufficientlysmallh

dK g� � d

K gÌh � h

� 12 C � h� �

diamK

S! " #

wher� e C $ h� %depends

Kon the global curvature structure of & and� on gõ , and� approachesa

constantwhenh ' 0 ((

it doesnotdependon A nor B, we� givea preciseformof C ( h� )in the

prM oof)].p

740e

MEMOLI AND SAPIRO

FIG. 2. Tf

ubularneighborhood.

Pr�

oof. LetÞ

dK

h * dK +

h� , A

- .B

ß / 0d

K 1 2d

K 3 4A

- 5B

ß 6; andlet 7 : [0 8 d

Kh] denotea 9 h gõ -distance

minimizing arc-lengthparameterizedpathbetweenA : ; < 0( =and B > ? @ dK h A , suchthatB CD E F 1. Let G H I J K L M N O P Q R S T U V W X Y be

�the orthogonalprojectionof Z ontoB [ .

This curve will be assmoothas \ for small enoughh�

; seeAppendixA. For sufficientlysmall h

�, the boundaryof ] h will� be smooth,since ^ is

�smoothand no shockswill be

generated_ (seenext sectionandAppendixA). Sowecanassumethat ` is�

C1 andthat ab is�

Lipschitz,sinceit is ashortestpathwithin asmoothRiemannianmanifoldwith boundary;seeSection2.2above.

It�

is evidentthat(this is asimplebut key observation)

L gÌ c d e f dK gÌ

h

g1hi d

K gÌj k 2lm L gÌ n o p qsince

(1) r s t h andgõ u v w gõ(2) x neednotbea gõ -shortestpathbetweenA andB onB y .

Wô

e thenhave

dK gÌz { d

K gÌh | } L~ gÌ � � � � L

~gÌ � � � � � � L~ gÌ � � � � L

~gÌ � � � �

� d

h�

0

�gõ � � � � �� � �

gõ � � � � ��   ¡ dtK

¢ d

h�

0

£gõ ¤ ¥ ¦ § ¨© ª «

gõ ¬ ­ ® ¯ °± ² ³ dtK ´ d

h

�0

µg¶ · ¸ ¹ º »¼ ½ ¾ g¶ ¿ À Á Â ÃÄ Å Æ dt

K

Ç d

h�

0g¶ È É Ê Ë Ì ÍÎ Ï Ð Ï ÑÒ Ó Ô dt

K Õ d

h�

0

Ög¶ × Ø Ù Ú g¶ Û Ü Ý Þ dt

K

ß Mà

gÌd

h

�0

á âã ä åæ çdt

K èK

égÌ

d

h�

0

ê ë ì í îdt

K

ï MgÌd

h

�0

ð ñ ò ó ô õ ö ÷ø ù ú û ü ý þ ÿ � � � H � � � � �� dtK

K gÌd

�h

�0

� � � � � � � � � �dt

K

� MgÌd

�h

�0

� � � � � � � � ! dtK "

h M#

gÌd

�h

�0

$H % & ' ( )* + dt

K ,K gÌ h d

#h -

W.

enow boundthefirst two termsat theendof theprecedingexpression.

DIST�

ANCE FUNCTIONSAND GEODESICS 741e

1. Wefirst boundthesecondtermin theprecedingexpression;thiswill beaningredientto

/theboundingof thefirst termaswell. Wehave

0H

1 2 3 4 5 67 8 9 sup: ;: < = > ? 1@ p£ :d

A Bp£ C D E F h G

HH

1 I JpM K L M N supO

p£ :dA P

p£ Q R S T hU maxV W X Y Z pM [ \ ] ^ _ ` pM a b c d

wheree f g pM h andi j pM k denotel

thelargestandthesmallesteigenvalueof H m n pM o , respectively.No

pw, aswe know from AppendixA, themaximumabsoluteeigenvalueof H

1 q rpM s , K

é tpM u ,

isv

boundedby

Ké w

pM x y z {1 | } ~ � pM � � � � �

wheree � � isv

themaximumabsoluteeigenvalueof H1 � � �

; thatis,

� � �sup�xÐ � � � max�

1� i � dA � � � i � H � � x� � � � �

wheree �i �   ¡ standsfor the i -th eigenvalueof asymmetricmatrix.

Then¢

dA

h�

0

£H

1 ¤ ¥ ¦ §s¨ © © ª« ¬ s¨ ­ ® ds

K ¯d

Kh

° ±1 ² h

# ³ ´ µ

2. Let usdefinethefunction f¶

: [0 · dK

h] ¸ IR, f¶ ¹

t º » ¼ ½ ¾ ¿ t À À . Thenformally Áf¶ Ât à ÄÅ Æ Ç È É

t Ê Ê Ë ÌÍ Î t Ï and f¶ Ð

t Ñ Ò H1 Ó Ô Õ Ö

t × × [ ØÙ Ú t Û Ü ÝÞ ß t à ] á â ã ä å æ t ç ç è ¨é ê t ë . Since ìí î ï ð isv

Lipschitz, and ñ is regular we can guaranteethat òf¶ ó ô õis also Lipschitz, so f

¶ ö ÷ øeù xists

almosteverywhere.Wewantto bound

dA

h�

0

ú ûf

¶ üt ý þ dt

K ÿ

W.

e note first that f¶ �

0� � �

f¶ �

dK

h � � 0,�

and � f¶ �

t � h#

, � f¶

t � � � � �1� h� � � Bß

gÌ for�

almosteù very t � [0 � d

Kh]. In fact,wehavethatfor thosesubintervalsof [0 � d

Kh] in whichtheshortest

path� travels through � � h, either f¶ �

t � h#

, or f¶ !

t " # $ h#

for the whole subinterval, andtherefore

/f

¶ %t & is

vconstantfor eachsubinterval, so f

¶ 't ( ) 0

�there.Ontheotherhand,when* is in the interior of + h, it is a g, -geodesic,so its accelerationis boundedby BgÌ , as we

ha-

veseenin Lemma2.1.Therefore,weconcludethat . f¶ /

t 0 1 2 1 H3 4 5 6 7t 8 8 [ 9: ; t < = >? @ t A ] B C B

ßgÌ .

CombiningD

all thiswehave thatfor almostevery t E [0 F dK

h],

Gf

¶ Ht I J K B

ßgÌ L supM N

: O P Q R 1S dA T

p£ U V W X hYZH

[ \ ]pM ^ [ _ ` a ] b c supd

p£ :dA e

p£ f g h i h j maxV k l m n pM o p q r s t pM u v w

andthegiven boundfollowsasbefore.

Applying Cauchy–Schwartzinequalityweobtain

dA

h�

0

x yf

¶ zt { | dt

K } ~d

Kh �

dA

h�

0

�f

¶2 � t � dt

K �

742e

MEMOLI AND SAPIRO

Nop

w usingLemma2.2:

d�

h�

0

�f

¶ 2 � t � dtK � �

f f¶ � d� h

�0 �

d�

h�

0f

¶f d

¶t � � d

�h

�0

f¶

f d¶

t

� d�

h�

0

�f

¶ � �f

¶ �dt

K � �d

Kh � h

� � �1 � h

� � � � BgÌ �

Finally�

,

d�

h�

0

� � �   ¡ ¢ £ ¤¥ ¦ dtK § ¨

dK

h © h� ª «

1 ¬ h� ­ ® ¯ BgÌ °

Using±

bothcomputedbounds,wefind that

dK g̲ ³ d

K gÌh ´ diam

K µ ¶ · ¸h

�MgÌ

¹ º1 » h

� ¼ ½ ¾ BgÌ ¿ MgÌ À h� Á Â

1 Ã h� Ä Å Æ K gÌ Ç h

� È É

(11)

From�

theprecedingLemmaweobtain:

CÊ

OROLLARY 2.1 FËor a givenpointq Ì Í

dK gÌÎ

h� Ï Ð qO Ñ Ò Ó Ô d

K gÌÕ Ö qO × Ø ÙL Ú Û Ü Ý hÞ 0ß à 0

á â

Remark.ã

Theä

rateof convergenceobtainedwith thetechniquesshown aboveis of orderåh

�. A quicklookover theproofof convergenceshowsthatthetermresponsiblefor theh

� 1æ 2

rateç is d�

h�

0 è éfê ët ì dt

K. All othertermshavethehigherorderof h

�. Supposewecanfind afinite

ícollectionî of (disjoint) intervals I

ïi ð ñ a� i ò b

ni ó suchthatsgnô õ öfê ÷

isø

constant( fê

isø

monotonic)withinù eachI i , ú N

ûi ü 1I i ý [0 þ d

Kh] ÿ whereù N

�is thecardinalityof thatcollectionof intervals,

and�f

ê �t � � 0

áfor t � [0 � d

Kh] � N

ûi � 1 I

ïi . Then,wecouldwrite

d�

h�

0

f

ê �t � dt

K � Nû

i � 1

sgnô � �fê � � �ai

� � bi� � bi

�ai

��f

ê �t � dt

K � Nû

i � 1

sgnô � �fê � !ai

� " bi� # $ f

ê %b

ni & ' f

ê (a� i ) *

+ Nû

i , 1

-f

ê .b

ni / 0 f

ê 1a� i 2 3 4

Nû

i 5 1

6 7f

ê 8b

ni 9 : ; < f

ê =a� i > ? @

A 2Nh�

since fê B

t C D E F G H t I I and J K L M traN

velsthrough O h PobtainingQ a higherrateof convergence,h

�. It is quiteconvincing thatcaseswhereN

� R ScanT be consideredpathological.We thenarguethat for all practicalpurposesthe rateofconT vergenceachieved is indeedh

�(at least).Moreover, for simplecaseslike a sphere(or

otherQ convex surfaces),it it veryeasytoshow explicitly thattheerroris (atleast)of orderh�

.8

Notwithstanding,U

we arecurrentlystudyingthespaceof surfacesandmetricsgV for whichweW canguaranteethat N

� X Y, andadvancesin this subjectwill bereportedelsewhere.

8Z

In[

this case,asin thecaseof convex surfaces,thegeodesicis composedof two straightlinesinsidetheband,tangent\

to its innerboundary, andageodesicon theinnerboundaryof theband;seeFig. 1.

DIST�

ANCE FUNCTIONSAND GEODESICS 743e

Thisä

showsthatwecanapproximatetheintrinsicdistancewith theEuclideanoneontheofQ fsetband ] h. Moreover, aswe will detailbelow, theapproximationerror is of thesameorderQ asthe theoreticalnumericalerror in fastmarchingalgorithms.Thereby, we canusefastalgorithmsin Cartesiangrids to computeintrinsic distances(on implicit/implicitizedsurfaces),enjoying their computationalcomplexity without affectingtheconvergencerategi^ ven by theunderlyingnumericalapproximationscheme.

3._

NUMERICAL IMPLEMENT ATION AND ITS THEORETICAL ERROR

In this sectionwe first discussthe simple modificationthat needsto be incorporatedinto

`the (Cartesian)fastmarchingalgorithmin orderto dealwith Euclideanspaceswith

manifoldswith boundary. We thenproposea way of estimatingthe (now discrete)offseth

�, andboundthetotalnumericalerrorof ouralgorithm,therebyshowing ourassertionthat

theN

errorwith ouralgorithmis of thesameorderastheoneobtainedwith thefastmarchingalgorithmfor Cartesiangrids(or triangulated3D surfaces).

Asa

statedbefore,we are dealingwith the numericalimplementationof the Eikonalequationb insidean open,bounded,andconnecteddomain c (this will later becometheofQ fset d h).

eThegeneralequation,whenP

� fxg h is

`theweight(it becomesgV for

iourparticular

case),T is given by

j kf

ê lxg m n o P

� pxg q r xg s t

fê u

r v w 0á x (12)

withW r theN

seedpoint. Note that following theresultsin theprevioussection,we arenowdealing

ywith theEikonalequationin Euclideanspace,andsotheEuclideangradientis used

above.Theupwindnumericalschemeto beusedfor thisequationis of theform z { xg 1 | } xg 2 ~� � � � � xg d

� � � xg � [49],

d�j

� �1 max� 2 � f

ê �pM � � mG j

� � 0á � � � �

xg � 2P� 2 � pM �

mG j� � min� � f

ê �pM � � xg �ej

� � � fê �

pM � � xg �ej�     ¡ (13)

whereW fê

is`

thenumericallycomputedvalueof fê

fori

everypoint pM in`

thediscretedomain

¢ £ ¤ ¥ ¦xg § ¨© ª « ¬ Z

­ ®xg ¯ °

Here, ±ej� withW j

² ³1 ´ 2 µ ¶ ¶ ¶ µ d

K, aretheelementsof thecanonicalbasisof IRd

�.

W·

e now describethe fastmarchingalgorithmfor solving the above equation.For thisweW follow thepresentationin [52]. For clarity wewrite down thealgorithmin pseudo-codeMform.

iDetailson theoriginal fastmarchingmethodon Cartesiangridscanbefoundin the

mentionedreferences.At all timestherearethreekindsof pointsunderconsideration:

¸ Narr¹

owBand. Thesepointshave to themassociatedanalreadyguessedvaluefor fê

,andareimmediateneighborsto thosepointswhosevaluehasalreadybeen“frozen.”º A

»live. Thesearethepointswhose f

¼v½ aluehasalreadybeenfrozen.

744e

MEMOLI AND SAPIRO

¾ F¿

ar Away. Thesearepoints that haven’t beenprocessedyet, so no tentative valuehasbeenassociatedto them.For that reason,they have f

¼ À Á, forcing themnot to be

consideredT aspartof theup-windingstencilin theGudunov’s Hamiltonian.Thestepsof thealgorithmincludethefollowing Initialization

Ã:

1. Set fÄ Å

0Æ

for every point belongingto theset[AÇ

live]. Thesearetheseedpoint(s)if they lie on the grid. If the seedis not a grid point, its correspondingNeighbors

¹ 9 aresetA

Çlive andaregiven an initial value f

Äsimply computedvia interpolation(taking into

accountthedistancefrom theneighborgrid pointsto theseedpoint).2. Find a tentative value of f

Äfor

ievery Neighbor

¹ofQ an A

Çlive pointÈ and tag each

Narr¹

owBand.3. Set f

Ä É Êfor all theremainingpointsin thediscretedomain.Ë Advance

-:

1. Beginningof loop:Let Ì pMminÍ Î be

Ïthepoint Ð [Narr

¹owBand] Ñ whichW takestheleast

v½ alueof fÄ

.2. Insertthepoint pM

minÍ toN

theset[Alive] andremove it from [Narr¹

owBand] Ò3. Tag as Neighbors

¹all thosepoints in the discretedomainthat canbe written in

theN

form pMminÍ Ó Ô xg Õej

� , andbelongto [Narr¹

owBand] Ö [F¿

arAway]. If a Neighbor¹

is`

in[FarAway], remove it from thatset([FarAway]) andinsertit to [Narr

¹owBand] ×

4.Ø

RecalculatefÄ

fori

all Neighbors¹

usingÙ Eq.(13)5. Set[Neighbor

¹] Ú emptyset.

6. Backto thebeginning(step1).

The boundaryconditionsare taken such that points beyond the discretedomain havef

Ä Û Ü.

Theconditionthatischeckedall thetime,andthatreallydefinesthedomainthealgorithmis

`working within, is the onethat determinesif a certainpoint qO is

`Neighbor

¹oQ fag iven

pointÈ pM thatN

belongsto the domain.The only thing onehasto do in order to make thealgorithmwork in thedomain Ý h specifiedby Þ xg ß IRd

�: à á â xg ã ä å h

� æis changetheway

theN

Neighbor¹

checkingT is done.Moreprecisely, weshouldcheck

qO ç Neighbor¹ è

pM é if`

f ê ë ì í î qO ï ð ñ h� ò

&& ó qO canT bewritten like pM ô õ xg öej� ÷ ø ù

theN

emphasisherebeingonthetest“ ú û ü qO ý þ ÿ h�

.” Wecouldalsoachievethesameeffectbygi� ving aninfinite weight to all pointsoutside � h; that is, we treattheoutsideof � h asanobstacle.� Therefore,with anextremelysimplemodificationto thefastmarchingalgorithm,we� make it work as well for distanceson manifoldswith boundary, and therefore,forintrinsic

�distanceson implicit surfaces.This is of coursesupportedby the convergence

resultsin theprevioussectionandtheanalysison thenumericalerrorpresentedbelow.

3.1.�

Bounding the Offset h�

W�

enow presentatechniquetoestimateh�

, thesizeof theoffsetof thehyper-surface that

actuallydefinesthecomputationaldomain � h. Theboundson h�

arevery simple.On onehand,weneedh

�to

belargeenoughsothattheupwindschemecanbeimplemented,meaning

that

h�

hasto belargeenoughto includethestencilusedin thenumericalimplementation.

9�

F or agrid point p� , any of its 2D-neighboringpointscanbewritten like p� � � x� i

� �e� i� .

DIST�

ANCE FUNCTIONSAND GEODESICS 745�

FIG. 3. W�

edepictthesituationthatleadsto thelowerboundfor h�

in the2D case.In red:thecurve.In black:the�

centersof B � x� � � � d1� 2 � x� . In green:thepointsof ! " # h� $ % x� & thatfall insideB ' x� ( d1) 2 * x� + for somex� , - ,.

and/ in bluethosethatdon’t.

Ontheotherhand,h�

hasto besmallenoughto guaranteethat 0 h remainssimplyconnectedwith� smoothboundariesandthat g1 remainssmoothinside 2 h.

Let3 4 5

be6

asbeforea boundfor theabsolutesectionalcurvatureof 7 , andlet 8 xg be6

the9

grid size.In addition,let W be6

themaximaloffsettingof thesurface: that9

guaranteesthat

9theresultingsetremainsconnectedanddifferentpartsof theboundaryof thatsetdo

not toucheachother. We show below thata suitableboundingof h�

is (recall thatd;

is thedimension

<of thespace),

=xg > d

; ?h

� @min

1A B C W D (14)

Let3

usintroducesomeadditionalnotation.Wedenotebycell the9

unit cell of thecomputa-tional

9grid.Letxg be

6apointin E h; wedenotebynF G xg H the

9numberof cellsC1 I xg J K L L L K CnM N xÐ O P xg Q

that9

containxg . It isclearthatif xg R S T Uh V W xg X (it is agrid point),thenxg is containedin 2d

�cellsY having xg asa vertex. It is alsoclearthatnF Z xg [ \ 2d

�. For a givencell C we] call ^ _ C `

the9

setof pointsof a b c h d e xg f that9

composeC (i.e., its vertices).We will denoteby C g xg hthe

9set nM i xÐ j

i k 1 Ci l xg m , andby n o xg p the9

set nM q xÐ ri s 1 t u Ci v xg w w .

Thelowerboundcomesfrom forcing that,for every xg x y , all pointsin C z xg { lie within|h (noteof coursethatwewanth

�to

9beassmallaspossible);seeFig. 3. Thatis,

xÐ } ~ C � xg � � �h �

Onceagain,thisconstraintcomesin orderto guaranteethatthereare“enough”pointstomakethediscretecalculations.Wetry tomakeC � xg � � C � xg � l � , whereC � xg � l � standsfor thehypercube

�centeredin xg , with sidelength2l , andsidesparallelto thegriddingdirections.

Theworstscenariois whenxg is a point in thediscretedomain,andwe musthave l � � xg .Finally, weobserve thatC � xg � l � � B � xg � l � d

; �. Theconditionthenbecomes

xÐ � � B� �

xg � � xg � d; �   ¡

h ¢x£ ¤ ¥ B

� ¦xg § h

� ¨ ©

which] providesthelowerbound,h� ª «

xg ¬ d;

.

746�

MEMOLI AND SAPIRO

The­

upperboundincludestwo parts.First, we shouldn’t go beyond W, sinceif we do,dif

®ferentpartsof theoffsetsurfacemight toucheachother, a situationthatcanevencreate

a nonsimplyconnectedband ¯ h. Thesecondpartof theupperboundcomesfrom seekingthat

9whentraveling on a characteristicline of ° at a point p± of² ³ , no shocksoccurinside´

h. It is a simplefact that this won’t happenif h� µ 1¶ · ; seeAppendixA. It is extremely

important¸

to guaranteethis bothto obtainsmoothboundariesfor ¹ h andto obtainsmootheº xtensionsof themetricg1 (g1 ).

»Note

¼of coursethatin general,h

�andalso ½ xg canY bepositiondependent.We canusean

adaptivegrid,andin placeswherethecurvatureof ¾ is high,or placeswherehighaccuracyis

¸desired,wecanmake ¿ xg small.

3.2.À

The Numerical Err or

ItÁ

is timenow to explicitly boundthenumericalerrorof ourproposedmethod.As statedabove, it isourgoaltoformallyshow thatwearewithin thesameorderasthecomputationallyoptimal² (fastmarching)algorithmsfor computingdistancefunctionson Cartesiangrids.Note

¼that thenumericalerror for thefastmarchingalgorithmon triangulatedsurfaceshas

notbeenreported,althoughit is of courseboundedby theCartesianone(sincethisprovidesaparticular“triangulation”).

3.2.1. NumericalÂ

Error Boundof theCartesianFastMarchingAlgorithm

The­

aim of this sectionis to bounda quantitythatmeasuresthedifferencebetweenthenumericallycomputedvalue d

; gÃÄ Å p± Æ Ç È andthe real valued; gÃÉ Ê p± Ë Ì Í . Any suchquantitywill

compareY bothfunctionson Î , but in principlethenumericallycomputedvaluewill not bedefined

®all over thehyper-surface.Sowe will bedealingwith an interpolationstage,that

we] commentfurtherbelow in Section3.2.2.Let

3usfix a point p± Ï Ð , andlet f

Ä Ñ Ò Óbe

6thenumericallycomputedsolution(according

to9

(13)),and fÄ Ô Õ Ö

the9

real viscosity× solutionof theproblem(12).Theapproximationerroris

¸thenboundedby (see[49])

maxpØ Ù Ú Û Ü Ý Þ x£ ß à f

Ä áp± â ã f

Ä äp± å æ ç CL è é xê ë 1

2 ì (15)

where] CL is aconstant.In practice,however, theauthorsof [49] observedfirst-orderaccu-rací y. As wehaveseen,wealsofind anerrorof orderh

î 1ï 2 forð

thegeneralapproximationofthe

9weightedintrinsicdistanceon ñ with] thedistancein theband ò h, andapracticalorder

of² hî

(seeRemarkandTheorem2.2).Beforeproceedingwith thepresentationof thewholenumericalerrorof our proposed

algorithm,weneedthefollowing simplelemmawhoseproofweomit.

L3

EMMAó 3.1. F

ôor a convex setD õ ö , and÷ y ø zù ú D

û, f

Äsatisfies

üf

Ä ýzù þ ÿ f

Ä �y� � � � � P

� �L � � � zù y� � �

Remark. Using�

the precedingLemmaand(15), it is easyto seethat for xê suchthatC � xê � � � ,

�f

Ä �p± � � f

Ä �q� � � � 2CL � � xê 1

2 ! " P� #

L $ % & ' (d

; )xê * + p± , q� - . / xê 0 (16)

a relationwewill shortlyuse.

DIST1

ANCE FUNCTIONSAND GEODESICS 747�

3.2.2. TheInterpolationError

Sincefollowing our approachwe arenow computingthedistancefunction in theband2h, in thecorrespondingdiscreteCartesiangrid, we have to interpolatethis to obtainthe

distance®

on the zerolevel-set3 . This interpolationproducesa numericalerror which weno4 w proceedto bound.

Given thefunction 5 : 6 7 8 9 : xê ; < IR ( = being6

agenericdomain,whichbecomestheband

6 >h for

ðourparticularcase),wedefinethefunction? @ A B : C D IR

Ethrough

9aninterpola-

tion9

scheme.Wewill assumethattheinterpolationerroris boundedin thefollowing way:10

supyF G H I x£ J K L M yN O P Q R S T U xê V W X max

zY Z [ \ x£ ] ^ _ z` a b minzY c d e x£ f g h z` i

for every xê j k .

3.2.3. TheTotal Error

Wl

e now presentthe completeerror (numericalplus interpolation)introducedby ouralgorithm, without consideringthe possibleerror in the computationof g1 (or in otherw] ords,weassumethattheweightwas alreadygiven in thewholeband m h).

nLet p± be

6apoint in o . Wedenoteby

p d; gÃq r p± s t u : v w IR

Ethe

9intrinsic g-distancefunctionfrom p± to

9any point in x .y d

; gÃh z p± { | } : ~ h � IR the

9g1 -distancefunctionfrom p± to

9any otherpoint in � h.

� d; gÃ

h � p± � � � : � � � h � � xê � � IR the9

numerically� computedv× alueof d; gÃ

h � p± � � � to9

any pointin

¸thediscretedomain.� � � d; gÃ

h � � p± � � � : � � IRE

the9

resultof interpolatingd; gÃ

h (that’sonly specifiedfor pointsin� � �h �   xê ¡ )

¢to pointsin IRd

£ ¤ ¥.

The¦

goalis thento bound § d; gè © p± ª « ¬ ­ ® ¯ d° gÃ

h ± ² p± ³ ´ µ ¶L · ¸ ¹ º , andweproceedto dosonow.

Let xê be»

in ¼ andyN in ½ ¾ xê ¿ , then

d° gÃÀ Á p± Â xê Ã Ä Å d

° gÃh Æ p± Ç xê È É d

° gÃÊ Ë p± Ì xê Í Î d° gÃ

h Ï p± Ð xê Ñ Ò d° gÃ

h Ó p± Ô xê Õ Ö d° gÃ

h × p± Ø yN ÙÚ

d° gÃ

h Û p± Ü yN Ý Þ d° gÃ

h ß p± à yN á â d° gÃ

h ã p± ä yN å æ ç è d° gÃh é ê p± ë xê ì í (17)

andusingProposition2.2, Lemma3.1 (15), andsimplemanipulations(in that order)weobtainî

d° gÃï ð p± ñ xê ò ó ô d

° gÃh õ p± ö xê ÷

ø C ù hî údiam

° û ü ýh

î 12

þ ÿ M�

gà � xê � yN � � CL � � xê � 12

þ � maxyF � � x£ d° gÃ

h � p± � yN � � minyF � � � x£ � d

° gÃh � p± � yN � �

Thelasttermcanbedealtwith using(16).Sincewewantbothhî �

0�

and h�x£ � � , in order

to�

have increasingfidelity in theapproximationof d° gÃ

h by»

its numericcounterpartd° gÃ

h ,11 we can! choose(for instance)h

î "Cx£ # $ xê % & for someconstantCx£ ' ( d

°and ) * + 0� ,

1- . We

10 Onemayimagineseveralinterpolationschemessatisfyingthisnot-stringent-at-allcondition.11 Thisway, wewill haveanincreasingnumberof pointsin . h

/ .

7480

MEMOLI AND SAPIRO

then�

obtain

d° gÃ1 2 p± 3 4 5 6 7 d

° gÃh 8 p± 9 : ;

L < = > ? @ A B xê C D2E F G H xê I J K L (18)

whereM N O P xê Q R S goesT to a constant(that dependson U ) a¢

s V xê W 0,�

andthis providesthedesired

Xbound.

T¦

o recap,wehaveobtainedthattheuseof anEuclideanapproximationin theband Y h to�

the�

intrinsicdistancefunctiononthelevel-setZ doesn’X

t (meaningfully)changetheorderofthe

�wholenumericalapproximation,in theworstcasescenario.While in themostgeneral

case! thetheoreticalboundfor theerrorof ourmethodis of orderhî 1[ 2 andthegeneralorder

ofî theerrorof theunderlyingnumericalschemeis \ ] xê ^ 1_ 2, for all practicalpurposestheapproximationerror(over ` )

¢betweenbothdistances(d

° gÃa andd° gÃ

h )¢

isof orderhî

(seeremarkafterCorollary2.1),andthepracticalnumericalerrorbetweend

° gÃh andd

° gÃh is

bof order c d xê e f

(for someg h [ 12 i 1j for ourfirstorderschemes).Then,thepracticalboundfor thetotalerror

becomes»

somethingof order k l xê m minn o p q r . Therefore,choosingabig enoughs t u 1) dispelsany concernsaboutworseningtheoverallerrorratewhendoingCartesiancomputationsonthe

�band.12

T¦

o conclude,let’s point out thatsincewe areworking within a narrow band( v h) o¢

ft hesurfacew , we areactuallynot increasingthedimensionalityof theproblem.We canthenwM ork with a Cartesiangrid while keepingthesamedimensionalityasif we wereworkingonî thesurface.13

4. EXPERIMENTS

Wx

enow presentanumberof 3D examplesof ouralgorithm.Recallthatalthoughall theey xamplesaregiven in 3D, thetheorypresentedabove is valid for any dimensiond

° z3.

T¦

wo classesof experimentalresultsarepresented.We show a numberof intrinsic dis-tance

�functionsfor implicit surfaces,aswell asgeodesicscomputedusingthesefunctions.

In{

orderto computeinterestinggeodesics,weusealsononuniformweights,permittingthecomputation! of crest/valleys,andoptimalpathswith obstacles.Wealsoexperimentallycom-pare| our resultswith thoseobtainedon triangulatedsurfacesusingthefastmarchingtech-niquedevelopedby Kimmel andSethian(for thisweusetheresultsandsoftwarereportedin [6]).

Figures}

4, 5, and6 show the intrinsic distancefunction for implicit surfacescomputedwithM themethodhereproposed(g~ � 1). An arbitraryseedpointon theimplicit surfacehasbeen

»chosen,andpseudocolorsareusedto improve the visualization.Redcorresponds

to�

low valuesof thedistanceandblueto thehigh ones.We observe that,asexpected,thedistance

X(colors)vary smoothly, andthat closepointshave similar colorsandfar points

ha�

veverydifferentcolors(closeandfar measuredon thesurfaceof course).In Fig.7wecomparetheresultof ourapproachwith thatof fastmarchingonatriangulated

surface (all triangulated-surfacecomputationswere donewith the packagereportedin[6]). We alsoshow absolutedifferences(error) betweendistancesobtainedthroughboth

12 Note�

thatthenumericalschemeusedby thefastmarchingalgorithmdecreasesits accuracy whennondiffer-entiable� pointsof thedistanceappear, thiscanhappenfor instancewhenthedomaincontainsthecut locusof theinitial�

set[15]. In any case,� � x� � 12 is theslowesterrorrateachievable.

13 Thenumberof pointsin thebandcanberoughlyestimatedby thequantity2 h a�

rea[ � ] when � x� � 1.

DIST1

ANCE FUNCTIONSAND GEODESICS 749�

FIG. 4. Distancemapsfromapointonthesphere,torus,andteapot(threeviewsarepresentedfor eachmodel).

approaches.The particularpatternof the error is the subjectof future research.In Fig. 8weM show level lines of the intrinsic distancefunction computedwith the techniquehereproposed.|

Before�

concludingthis part of the experiments,let’s give sometechnicaldetailsonthe

�implementation.The codefor the examplesin this paperwas written in C++. For

visualization× purposes,VTK wasused.Mostof the“hardcode”wasdonetakingadvantageofî Blitz++’s doubletemplatizedarraysandrelatedroutines,see[9]. The implicit modelsused� in this paperwereobtainedfrom [67] (othertechniques,e.g.,[40], couldbeusedaswell).M All thecodewascompiledandrun in a450Mhz PentiumIII, with 256Mb of RAM,wM orkingunderLinux (RedHat6.2).Thecompilerusedwas � � � � � � � � � � � � andthelevel ofoptimizationî was3. InTable1weshow runningtimesof theintrinsicdistancemapalgorithmfor

�someof theimplicit modelsweused,alongwith thecorrespondingof� fset-value(h

î)

¢and

sizeandnumberof grid pointsin � h for eachmodel.

4.1. Geodesicson Implicit Surfaces

T¦

o find geodesiccurveson the implicit surface,we backtrackstartingfrom a specifiedtar

�get point toward the seedpoint, while traveling on the surfacein the directiongiven

FIG. 5. Distancemapfrom apointonaportionof white/graymatterboundaryof thecortex.

FIG. 6. Distancemapfrom oneseedpoint on a knot. In this picturewe evidencethat thealgorithmworkswell� for quiteconvolutedgeometries(aslong ash is properlychosen).Notehow pointsclosein theEuclideansense� but faraway in theintrinsic sensereceiveverydifferentcolors,indicatingtheir large(intrinsic)distance.

750�

DIST�

ANCE FUNCTIONSAND GEODESICS 751 

FIG. 7. Top: Distancemapfrom a singleseedpoint (situatedat thenose)on animplicit bunny (g ¡ 1). Thefigureon thetop-left was obtainedwith theimplicit approach(dh

¢ )£

herepresented,while theoneon thetop-rightwas derived with thefastmarchingon triangulatedsurfaces(d

¤ ¥)£

technique.Bottom:Threeviews of theabsolutedifferencebetweenboth distancefunctions(d

¤h

¢ ¦ d§ ). The maximaldifference(error betweenthe distances)is4 ¨ 1439,being91© 5599themaximalcomputeddistancein theband.

byª

the (negative) intrinsic-distancegradient.This meansthatafterwe have computedtheintrinsicdistancefunctionasexplainedabove, wehaveto solve thefollowing ODE(whichobviouslykeepsthecurveon « ):

¬­® ¯ ° ± ² d

³ g´µh

¢ ¶ · ¸¹ º 0» ¼ ½

p¾ ¿ À ÁwhereÂ

à Äd

³ g´Åh

¢ Æ p¾ Ç ÈÉ Ê d³ g´Ë

h¢ Ì p¾ Í Î Ï d

³ g´Ðh

¢ Ñ p¾ Ò Ó Ô Õ Ö p¾ × Ø Ù Ú p¾ Û

TABLE 1

Model Size #Ü Ý Þ h¢ ß à xá â h RunningTime (secs)

Brain 122 ã 142 ä 124 168å 603 1 æ 75�

9 ç 4Bunnè

y 81 é 80 ê 65ë

38ì 107 1 í 75�

1 î 99Knot 80 ï 81 ð 44 16ñ 095 1 ò 0 0

ó ô76

Sphereõ

70 ö 70 ÷ 70�

11ø 800 1 ù 75 0 ú 65ë

Torus 64 û 64 ü 64ë

21ý 704�

1 þ 75 1 ÿ 16Teapot 80 � 55

� �46 24� 325

�1 � 75 1 � 22

�

752 

M�

EMOLI AND SAPIRO

FIG. 8. Top: Level lines for the intrinsic distancefunction depictedin Fig. 7 (left). Bottom: Level linesfor

the intrinsic distancefunction depictedin Fig. 4 (secondrow). In both rows, the (22) levels shown are0ó

03ó �

0 � 05 0 � 1 � � � � � 0 � 95� 0 � 97�

percentof the maximumvalueof the intrinsic distance,andthe coloring of thesurf� acecorrespondsto the intrinsic distancefunction. Threeviews are presented.Note the correctseparationbetween�

adjacentlevel lines.Notealsohow theselinesare“parallel.”

is�

thegradientof d³ g´�

h¢ at p¾ � � projected� ontothetangentspaceto � � � � � 0

» at p¾ . Since

we mustdiscretizetheabove equation,onecanno longerassumethatat every instantthegeodesic! path " will lie on the surface,so a projectionstepmustbe added.In addition,sinceall quantitiesareknown only at grid points,an interpolationschememustbe usedto

#performall evaluationsat positionsgiven by $ . We have useda simpleRunge–Kutta

inte�

grationprocedure,with adaptivestep,namelyanODE23procedure.Beforepresentingexamplesof geodesiccurves,weshouldnotethatweareassumingthat% &d

³ g´'h

¢ , theextrinsicgradientof thedistancein theband,is agoodapproximationof ( ) d³ g´* ,

the#

intrinsicgradientof theintrinsicdistance(andnot justd³ g´+

h¢ agoodapproximationof d

³ g´,aswehavepreviouslyproved).Boundingtheerrorbetweenthesetwo gradients,e.g.,usingthe

#framework of viscositysolutions(sinceintrinsicdistancesarenotnecessarilysmooth),

is thesubjectof currentwork (seealsonext sectionfor anumericalexperiment).The figuresdescribednext illustrate the computationof geodesiccurves on implicit

surfacesfor differentweightsg- . In all thefiguresthegeodesiccurve isdrawn ontopof thesurface,which is coloredasbefore,colorsindicatingtheintrinsicweighteddistance.

In.

Fig. 9 we presentboth the geodesiccurve computedwith our techniqueand theone computedwith the fast marchingalgorithm on triangulatedsurfacesfollowing theimplementationreportedin [6].

In.

Fig.10weshow thecomputationof sulci (valleys)onanimplicit surfacerepresentingthe

#boundarybetweenthe white andgray matterin a portion of the humancortex (data

obtainedfrom MRI). Here the (extended)weight g- is a function of the meancurvaturegi! ven by [6]

g- valley / x0 1 2 3 4 M 5 x0 6 7 miny8 9 :

h¢ M ; y< = p> ?

DIST�

ANCE FUNCTIONSAND GEODESICS 753 

where M@

standsfor themeancurvatureof thelevel setsof A , so it is computedsimply asM

@ Bx0 C D E F G x0 H . In theexamplepresentedwe usedI J 100and p¾ K 3. More detailson

the#

useof this approachfor detectingvalleys (andcreases)canbefoundin [6] andin thereferencesL therein.

In Fig.11weshow thecomputationof geodesiccurveswith obstaclesonimplicit surfaces.This is animportantcomputationfor topicssuchasmotionplanningonsurfaces.

4.2. SimpleNumerical Accuracy Validations

WM

e concludetheexampleswith somesimplenumericalvalidations.Sincefor a sphere,for instance,therealdistancescanbecomputed,wecomparethesewith thosenumericallycomputedN with ouralgorithm.Aspreviouslyexplained,for thiscasetheerrorof ourproposedband-based

ªapproximationof the intrinsic (continuous)distanceis of orderh

O(actually, it

canN beshown thattheorderis slightly superlinear).We have testedthecomputeddistancebetween

ªgiven seedpointsin thespherefor anumberof differentgrid sizes(resolutions)in

the#

cube[0 P 1]3, obtainingtheerrorsgiven in Table2. In obtainingthedatawe have usedh

O Q2

R S TxU V 0W 7. It canbeobservedanoverall error

Xd

³ Y Z [ \d

³h ] ^ L

_ ` a b crategiven approximatelyby d e xU f 0g 653.

Althougha thoroughstudyof theapproximationof h i d³ g´j by

ª k ld

³ g´mh

¢ will bethesubjectof futureresearch,we will presentbelow somenumericevidence.Noteof coursethat themainconcern,aspreviously explained,is at thecut locus(singularitieson thegradientofthe

#distancefunction).To thebestof our knowledge,completeanalysisof theaccuracy of

the#

gradientof theintrinsicdistancehasnot yetbeenperformedfor triangulatedsurfaces.

TABLE 2

Size h OverallNumericalError

100 0 n 079621 0 o 111035120 0 p 070081 0 q 101189140 0 r 062912 0 s 090596160 0 t 057298 0 u 084766180 0 v 052764 0 w 077357200 0 x 049012 0 y 072119220 0 z 045849 0 { 069262240 0 | 043140 0 } 064085260 0 ~ 040789 0 � 061003280 0 � 038727 0 � 057780300 0 � 036901 0 � 056024320 0 � 035271 0 � 053569340 0 � 033806 0 � 051469360 0 � 032480 0 � 048853380 0 � 031273 0 � 047292400 0 � 030170 0 � 046195420 0 � 029157 0 � 044747440 0 � 028223 0 � 043254460 0 � 027358 0 � 041999480 0 � 026555 0 � 040501500 0 � 025807 0 � 039396

754�

MEMOLI AND SAPIRO

FIG.�

9. Top: Distancemap(weight � 1) andgeodesiccurve betweentwo pointson an implicit bunny. Weshow two geodesicssuperimposed,theblackoneis theoneobtainedvia theimplicit backpropagationdescribedin thetext, while thewhiteoneis obtainedwhenperformingthebackpropagationcomputationin thetriangulatedsurface.It is importantto notethatin bothcasesthedistancefunctionusedis theonecomputedwith our implicitapproach;to feedthedatatothetriangulatedsurfacesback-propagationalgorithm,wefirst interpolatedtheintrinsicdistanceto pointsonto the triangulatedsurface.We canclearlyseethatbothgeodesicsoverlapalmostentirely,justifying

�theproposedimplicit backpropagationapproachwhencomparedto theoneonthetriangulatedsurface.

Bottom:We repeatthetop figure,but now for thewhite curve (� � ) thedistanceusedwas alsocomputedon thetriangulatedsurface.In otherwords,the black curve (� h

� ) correspondsto completeimplicit computations,bothdistanceandbackpropagation,while the white onecorrespondsto completecomputationson the triangulateddomain.For thisparticularexample,thegeodesicobtainedwith thecomputationsontheimplicit surfaceisactuallyshorterthantheoneobtainedwith computationson thetriangulatedrepresentation.

DIST�

ANCE FUNCTIONSAND GEODESICS 755 

FIG. 10. Thesefour figuresshow the detectionof valleys over implicit surfacesrepresentinga portion ofthehumancortex. We usea meancurvaturebasedweighteddistance.In theleft-uppercornerwe show themeancurvatureof thebrainsurface(clippedto improve visualization).It is quiteconvincing that this quantitycanbeof greathelp to detectvalleys. In the remainingfigures,we show two curvesover the surface,whosecoloringcorrespondto themeancurvature(not clipped,from red,yellow, greento blue,asthevalueincreases).Theredcurve correspondsto the natur� al geodesic  (g¡ ¢ 1), while the white curve is the weighted-geodesicthat shouldtravel through“nether” regions.Indeed,a very cleardifferenceexistsbetweenboth trajectories,sincethewhitecurve makesits way throughregionswherethe meancurvatureattainslow values.The figure in the right-lowquadrantis azoomedview of thesamesituation.

WM

emakeall ourcomputationsagainfor simplicity, over asphere,takingg£ ¤ 1 (wewilldiscard

¥thesuperscriptsg£ andg£ for

¦theremainsof thissection).As anindicatorof how well§ ¨

d³ ©

h¢ approximatesª « d

³ ¬(over ­ )

¬we look at how muchthe quantity ® ¯ ° d

³ ±h

¢ ² dif¥

fersfrom 1. In Fig.12weshow ninehistogramsof theaforementionedquantity, for 1000pointson the sphereandfor nine (increasing)valuesof h

O. It canbe observed that the valuesof³ ´ µ

d³ ¶

h¢ · spreadmoreandmoreash

Ogro! ws.

5.¸

EXTENSIONS

5.1.¹

GeneralMetrics: SolvingHamilton–JacobiEquationson Implicit Surfaces

Sincethe very beginning of our exposition we have restrictedourselves to isotropicmetrics.As statedin the introduction,this alreadyhasmany applications,andjust a few

756�

MEMOLI AND SAPIRO

FIG. 11. Distancemapandgeodesiccurvebetweentwo pointsonanimplicit bunny surfacewith anintrinsicobstacleº on it. Wenow useabinaryweight,g » ¼ 1½ ¾ ¿ ,À beinginfinity at theobstacle.Thispermits,asillustratedin thefigure,thecomputationof optimalpathswith obstacleson implicit surfaces.Thebluepathcorrespondstotheobstacle-weighteddistancefunction,andthewhiteoneto thenatural(gÁ Â 1)distancefunction.Bothgeodesicsareshown over thesurfaceof thebunny, thepseudocolorrepresentingtheweighteddistancefor thesurfacewithobstacle.Theobstacleis alsoshown in blue.Notethatthegeodesicis nottouchingtheobstacledueto thelow gridresolutionusedto defineit in thisexample(low resolutionwhichmakesit actuallynotabinarybut amultivaluedobstacle).

wereà shown in the previous section.Sincethe fastmarchingapproachhasbeenrecentlyeÄ xtendedto moregeneralHamilton–Jacobiequationsby OsherandHelmsen[45], we areimmediately

Åtemptedto extendour framework to theseequationsaswell. Theseequations

have applicationsin importantareassuchasadaptive meshgenerationon manifolds,[28],andsemiconductorsmanufacturing.

Then,Æ

we areled to investigatetheextensionof our algorithmto generalmetricsof theform, G : Ç È IRd

É Êd

É, that is, a positive definite2-tensor. Our new definitionof weighted

lengthË

becomes

LÌ

G Í Î ÏÐÑ b

aG Ò Ó Ô t Õ Õ [ Ö× Ø

t Ù Ú ÛÜ Ýt Þ ] dt

ß à

DISTá

ANCE FUNCTIONSAND GEODESICS 757â

FIG. 12. Histogramsof ã ä å dæ hç è for several(increasing)valuesof h, for 1000pointsuniformly distributed

oné asphere.Fromleft to right andtop to bottom,thehistogramsareplottedfor increasingvaluesof hê

.ëandtheproblemis to find for every xì í î (for afixed pï ð ñ ),

ò

dß Gó ô xì õ pï ö ÷ø inf

Åùpxú [ û ]

üL

ÌG ý þ ÿ � � (19)

As before,we attemptto solve theapproximateproblemin theband � h, with anextrinsicdistance

�

dß G�

hç � xì � pï � � inf

Åpxú [ � h

ç ]

�L

ÌG � � � � (20)

whereÃ

LÌ

G � � ��� b

aG � � � t � � [ �� �

t � � ! "t # ] dt

ß

for$

an adequateextensionG of% G. The solutionof the extrinsic problemsatisfies(in theviscosity& sense)theEikonalequation

'G ( 1 ) * xì + , d

ß G-h

ç . / dß G0

hç 1 1 2 (21)

The first issuenow is the numericalsolvability of the precedingequationusing a fastmarching3 typeof approach.OsherandHelmsen[45] have extendedthecapabilitiesof thefastmarchingto dealwith Hamilton–Jacobiequationsof theform

H 4 xì 5 6 f7 8 9

a: ; xì <

758â

MEMOLI AND SAPIRO

for$

geometricallybasedHamiltoniansH= >

xì ? @pï A : B C D IRE d

É F GIR

E dÉ H

IRE

thatI

satisfy

H= J

xì K Lpï M N 0 iO

f Ppï QR S0OH

= Txì U Vpï W is

Åhomogeneousof degree1 in Xpï

pïi H

=pY i

Z [ xì \ ]pï ^ _ 0O

for 1 ` i a dß b

xì c d e f gpï h(22)

It easilyfollows thattheseconditionshold for (21)considering

H= i

xì j kpï l mn o G p 1 q r xì s [ tpï u vpï ] wwhenà thematrixG x 1 y xì z is diagonal.Therefore,wecansolvethiskind of Hamilton–JacobiequationsÄ (theextrinsicproblem)with theextendedfastmarchingalgorithm.

In{

order to show that our framework is valid for theseequationsaswell, all what webasically

|needto dois to provethattheextrinsicdistance(20)ontheoffset } h con~ vergesto

theI

intrinsiconeontheimplicit surface� , i.e.,(19).Thiscanbedonerepeatingthestepsinthe

Iconvergenceproofpreviously reportedin Section2.3for isotropicmetrics.Combining

thisI

with the resultsof Osherand Helmsenwe then obtain that our framework can beappliedto alargerclassof Hamilton–Jacobiequations:generalintrinsicEikonalequations.Theextensionof theseideasto evenmoregeneralintrinsic Hamilton–Jacobiequationsofthe

Iform H

� �xì � � � u� � a� � xì � xì � � remains� to be studied,andeventualadvanceswill be

reportedelsewhere.

5.2.�

Nonimplicit Surfaces

Theframeworkwepresentedwasheredevelopedfor implicit surfaces,althoughit appliesto

Iothersurfacerepresentationsaswell. First, if thesurfaceis originally given in polygonal

or% triangulatedform,orevenasasetof unconnectedpointsandcurves,wecanuseanumberof% availabletechniques,e.g.,[34,40,47,55,58,65,67] (andsomeverynicepublicdomainsoftware[40]), to first implicitize thesurfaceandthenapplythetechniquehereproposed.14

Note�

that theimplicitation needsto bedoneonly oncepersurfaceasa preprocessingstepandwill remainvalid for all subsequentusesof thesurface.This is important,sincemanyapplicationshavebeenshown to benefitfrom animplicit surfacerepresentation.Moreover,as we have seen,all what we needis to have a Cartesiangrid in a small bandaroundthe

Isurface� . Therefore,thereis no explicit needto performanimplicitation of thegiven

surfacerepresentation.For example,if thesurfaceisgivenby acloudof unconnectedpoints,weà cancomputedistancesintrinsic to thesurfacedefinedby thiscloud,aswell asintrinsicgeodesic� curves,withoutexplicitly computingtheunderlyingsurface.All thatis neededisto

Iembedthiscloudof pointsin aCartesiangrid andconsideronly thosepointsin thegrid

atadistanceh�

or% lessfrom thepointsin thecloud.Thecomputationsarethendoneon thisband.

|

6.�

CONCLUDING REMARKS

In{

this article we have presenteda novel computationallyoptimal algorithm for thecomputation~ of intrinsic distancefunctionsandgeodesicson implicit hyper-surfaces.The

14 The�

sametechniquescanbeappliedto transformany givenimplicit functioninto adistanceone.

DISTá

ANCE FUNCTIONSAND GEODESICS 759â

underlying� idea is basedon using the classicalCartesianfastmarchingalgorithm in anof% fsetboundaroundthegiven surface.We have providedtheoreticalresultsjustifying thisapproachandpresenteda numberof experimentalexamples.The techniquecanalsobeappliedto 3D triangulatedsurfaces,or evensurfacesrepresentedby cloudsof unconnectedpoints,� afterthesehavebeenembeddedin aCartesiangrid with properboundaries.Wehavealsodiscussedthat the approachis valid for moregeneralHamilton–Jacobiequationsaswell.Ã

Man�

y questionsremainopen.Recently, T. Barth (and independentlyD. Chopp)haveshowntechniquesto improvetheorderof accuracy of fastmarchingmethods.It will beinter-estingÄ toseehow themethodproposedherecanbeextendedtomatchsuchaccuracy.Relatedto

Ithis,wearecurrentlyworkingontighterboundsfor theerrorbetweend

ß g��h

ç anddß g�� ,aswell as

bounds|

for theerrorbetweentheircorrespondingderivatives.Weareinterestedin extendingthe

Iframeworkpresentedhereto thecomputationof distancefunctionsonhighcodimension

surfacesandgeneralembeddings.Moregenerally, it remainsto beseenwhatclassof intrin-sicHamilton–Jacobi(or in general,whatclassof intrinsicPDEs)canbeapproximatedwithequationsÄ in theoffsetband � h. In an even moregeneralapproach,whatkind of intrinsicequationsÄ canbe approximatedby equationsin otherdomains,with offsetsbeing just aparticular� andimportantexample.Even if fastmarchingtechniquesdo not exist for theseequations,Ä it mightbesimplerandevenmoreaccurateto solvetheapproximatingequationsin

Åthesedomainsthanin theoriginalsurface� . Theframeworkherepresentedthennotonly

of% fersasolutiontoafundamentalproblem,but alsoopensthedoorstoanew areaof research.

APPENDIX�

A: DISTANCE MAPS IN EUCLIDEAN SPACE

W�

e now presenta few important resultson distancemaps.Thesehave beenmainlyadapted(andadopted)from [4, 5, 26,54].

Where�

ver � is smoothweknow thatit satisfiestheEikonalequationÄ�   ¡ ¢ £

1 ¤ (A.1)

TheÆ

distancefunctionsatisfiesthisPDEeverywherein theviscosity¥ sense[29,20]. It is alsowellà known thatwithin asufficiently smallneighborhoodof ¦ § ¨ © ª 0

O «, ¬ ­ ® ¯ is

Åsmooth° ,

or% at leastassmoothas ± . Theseassertionscanbe madeprecisethroughthe followingLemma

²from [24]:

L²

EMMA A.1.³

Let´ µ

be¶

a Ck·

(k¸ ¹

2) codimension1 closedhyper-surfaceof IRdÉ.º Then,

thesigneddistancefunctionto » is Ck· ¼

U ½ for¾

a certainneighborhoodU of ¿ .ºDif

Àferentiating Á Â Ã Ä 2 Å 1, weobtain

D Æ Ç È É Ê Ë Ì 0O Í

Therefore,

H Î Ï Ð Ñ 0O

(A.2)

meaningÒ that thenormalto Ó at pï isÔ

aneigenvectorof theHessian,associatedto thenulleigenÕ value.Differentiatingagainweobtain

D3 Ö × Ø Ù Ú D2 Û Ü 2 Ý 0O Þ

(A.3)

760â

MEMOLI AND SAPIRO

Theß

next Lemma,whosedetailedproof can be found in [4], is mainly basedin therelations(A.2) and(A.3), and it is usedto verify that the function à : á â ã ä å æ ç IRd

è éd

èdefined

êby ë ì t í î H

ï ð ñpï

0 ò t ó ô õ pï0 ö ö ÷ pï

0 isÔ

any point in themanifold ø ù ú 0O û ü

satisfiesthe

ýfollowing ODE:

þÿ � t � � � 2 � t � � 0O

t � � � � LEMMA A.2. Theeigenvectorsof H � ar� econstantalongthecharacteristiclinesx � s° � �

xì 0 � s° � � � xì � s° � � (ar� c lengthparametrized, xì 0 is a pointonto � )ò

of� � within� anyneighbor-hood

�where it is smooth, and� theeigenvaluesvaryaccording to

�i � s° � ! i " 0O #

s° $i % 0O & '

1 (W

�eusetheaboveformulato boundthemaximumoffset ) * + of, - . / 0

O 0that

ýkeeps1 2 3 4 5

smooth,we just take 6 7 8 9 maxÒ1: i ; d

è <1 = > i ? 0O @ A B C

1.W

�enow obtainboundson theeigenvaluesof theHessianof thedistancefunction:

CD

OROLLARY A.1. TheeigenvaluesE i F pG H of� H I J pG K (principalG curvaturesof L xM : N O xM P QR SpG T U )

Var� eabsolutelyboundedby

W Xi Y pG Z [ \ ] ^

1 _ ` a b pG c d e f gwher� e h i absolutely� boundsall eigenvaluesof H j k pG l m pG n o ; and� p q r pG s t is sufficientlysmall.°

Tß

o conclude,let’spresentsomeconceptsonprojectionsontotheimplicit surfaceu , zerolevel-setof thedistancefunction v . It isclearthattheprojectionof apoint pG w IRd

èonto, x

isÔ

given by

y z {pG | } pG ~ � � pG � � � � pG � �

This projectionis well definedaslong asthereis only onexM � � suchthat � � � pG � � xM .This canbe guaranteedwhenworking within a small tubular neighborhoodof a smoothsurface� . Moreover, thismapis smoothwithin acertaintubularneighborhoodof � [54]:

Tß

HEOREM� A.1.

�If

� � �IR

� dè

is a compactCk·

(k¸ �

2) codimension1 hyper�

-surface, thenthereisah � � � � 0

�suc° hthatthemap � � iswelldefinedandbelongstoCk

· �1 � � xM : d

  ¡xM ¢ £ ¤ ¥

h� ¦ §

IR� d

è ¨ ©

APPENDIXª

B: TECHNICAL LEMMA

LEMMA B.1. Let f : [a« ¬ b¶] ­ IR bea C1 ® [a« ¯ b

¶] ° function

¾such that f ± is Lipschitz.Let² ³ L

´ ´ µ[a« ¶ b

¶] · denote

  ¸one¹ of º f

¾ »’s° weakderivative. Thenonehas

b

af

¾ ¼ 2 ½ xM ¾ dx  ¿

f f¾ À b

a Áb

af

¾ ÂxM Ã Ä Å xM Æ dx

 

DISTÇ

ANCE FUNCTIONSAND GEODESICS 761È

Proof. Let ext É f¾ Ê Ë

denoteê

theLipschitzextensionof f¾ Ì

toý

all IR giÍ ven by

exÕ t Î f¾ Ï Ð Ñ

xM Ò Ó f¾ Ô Õ

a« Ö for×

xM Ø a«f

¾ Ù ÚxM Û for xM Ü [a« Ý b

¶]

f¾ Þ ß

b¶ à

for×

xM á b¶

Thenlet ext â f¾ ã

beä

given by any (bounded)primitiveof ext å f¾ æ ç

, thatis,ext è f¾ é ê

exÕ t ë f¾ ì í

.Let

²ˆî ï L

´ ðIR

�denote

êext ñ f

¾ ò ó’sweakderivative, andwehave that ˆô õ

[aö b] and ÷ coincideø asweakù derivativesof f

¾. Let ú û ü ý þ ÿ � � � � 0� be

äa family of boundedsupportmollifiers.Thenwe

defineê

thefunction

f¾ � �

exÕ t � f¾ � � �

It is clearthatwewill have ( � meansuniformconvergence)

(a)

f¾ � � � 0� �� � exÕ t � f

¾ �o, ver compactsetsof IR

(b)

f¾ ��

� � 0� �� � exÕ t f¾ ! "

o, ver compactsetsof IR

(c)

f¾ # #$ % & 0' ( ˆ) locally

*in L

´ 2 + IR� ,

Since f¾ -. / C 0 1 IR

� 2, wecanuseintegrationby partsto concludethat

b

af

¾ 34 2 5 xM 6 dx7 8

f¾ 9: f

¾ ; b

a <b

af

¾ = >xM ? f

¾ @ @A B xM C dx7 D

NoE

w the left-handside will converge to ba f

¾ F 2 G xM H dx7

inÔ

view of I b¶ J; the first term in

theý

right-handsidewill obviously converge to f f¾ K L b

a. For the remainingtermwe observethe

ýfollowing, using Cauchy–Schwartz inequality (let M N O : L2 P [a« Q b

¶] R S L2 T [a« U b

¶] V W IR

denoteê

L´ 2 X [a« Y b

¶] Z ’s internalproduct):

b

af

¾ [ \xM ] f

¾ ^ ^_ ` xM a dx7 b b

af

¾ cxM d e f xM g dx

7h i j f

¾ k kl m f¾ n o p q r s

f¾ t u v u w

f¾ x xy z f

¾ { |f

¾ } ~ �f

¾ �f

¾ � �� � � � �� � b¶ �

a« � max�x� � [a� b] �

�f

¾ � �� � x� � � � f¾ � �

f¾ �

L2 � [a� b] ��

max�x�   [a¡ b] ¢

£f

¾ ¤x� ¥ ¦ § f

¾ ¨ ¨© ª « ¬ L2­ ®

[a b] ° ±No

Ew, everythingis undercontrolsince

maxÒ²x� ³ [a b] µ

¶f

¾ · ·¸ ¹ xº » ¼ ½ ¾ ¿ ÀL Á  [aà b] Ä Å

762È

MEMOLI AND SAPIRO

Hence,Æ

wehaveproved

b

af

¾ Ç Èxº É f

¾ Ê ÊË Ì xº Í dx7 Î Ï 0Ð Ñ b

af

¾ Òxº Ó Ô Õ xº Ö dx

7

theý

laststepof theproof. ×

ACKNOWLEDGMENTS

Many colleaguesand friends helpedus during the courseof this work and they deserve our most sincereacknoØ wledgment.Prof. Miguel Paternain,in theearlystagesof this work, pointedout very relevant references.Prof.Ù

StephanieAlexander, Prof. David Berg, and Prof. RichardBishop helpedus with our questionsaboutgeodesicsÚ onmanifoldswith boundaries.Prof.Stanley Osherhelpeduswith issuesregardingthefastmarchingalgo-rithmin general,andhisrecentextensionsin particular, whileProf.RonKimmeldiscussedwith ustopicsrelatedtohisworkontriangulatedsurfacesandprovidedgeneralcommentsonthepaper. Prof.Osheralsohelpedwith thelit-eratureÛ onrateof convergenceof Hamilton–Jacobinumericalapproximations.Prof.OmarGil helpeduswith somedeepÜ

insightsinto technicalaspectsof thework. Thiswork benefitedfrom rich conversationswith AlbertoBarte-saghiÝ (whoalsoprovideduswith someof hiscodeto extractgeodesicsin triangulatedsurfaces),CesarPerciante,Luis Vasquez,FedericoLecumberry, andProf.GregoryRandall.Wealsoenjoyedimportantdiscussionwith Prof.VÞ

icentCasellesonthedensityof zerosof generalfunctions.WealsothankProf.Luigi Ambrosiofor hiscommentsonß theregularityof level-setsof distancefunctions.Weareindebtedto AlvaroPardofor hiscarefulreadingof themanuscriptandfor hissuggestions.WethankProf.L. T. ChengandRichardTsaifor theircomments.Prof.H. Zhaoproà videdsomeof theimplicit surfaces.FM in particularwantsto thankOmarGil, whohasbeenagreatandpatientteacherá

, andGregoryRandallfor hisconstantsupportandencouragement.FM performedpartof thiswork whilevisitingâ theUniversityof Minnesota.ThisworkwaspartiallysupportedbyagrantfromtheOfficeof Naval ResearchONR-N00014-97-1-0509,ã

theOffice of Naval ResearchYoungInvestigatorAward,thePresidentialEarly CareerAwardsfor ScientistsandEngineers(PECASE),aNationalScienceFoundationCAREERAward,by theNationalScienceä

FoundationLearningandIntelligentSystemsProgram(LIS), by theComisionSectorialdeInvestigacionßCientå

ıfica (CSIC),theComisionAcademicadePosgrado(CAP),andTecnocomthroughascholarshipto FM.

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