Journal of Computational Physics 173, 730–764 (2001) doi:10.1006/jcph.2001.6910, available online at http://www.idealibrary.com on Facundo M´ emoli and Guillermo Sapiro Instituto de Ingenier´ ıa El´ ectrica, Universidad de la Rep ´ ublica, Montevideo, Uruguay; Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455 E-mail: [email protected]; [email protected]Received March 26, 2001; revised August 6, 2001 An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and efficient numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton–Jacobi-based, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hyper-surfaces a computationally efficient technique. The approach can be extended to solve a more general class of Hamilton–Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows for the computations to be performed on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton–Jacobi equations are intrinsic to the implicit hyper-surface. c 2001 Academic Press Key Words: implicit hyper-surfaces; distance functions; geodesics; Hamilton– Jacobi equations; fast computations. 1. INTRODUCTION Computing distance functions has many applications in numerous areas including math- ematical physics, image processing, computer vision, robotics, computer graphics, com- putational geometry, optimal control, and brain research. In addition, having the distance 730 0021-9991/01 $35.00 Copyright c 2001 by Academic Press All rights of reproduction in any form reserved.
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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
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
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
[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
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
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
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
$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
$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
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
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
$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
$(therebyhaving a theoreticalerror « ¬ ® x4 ¯ [19]), andbetteraccuracy might then
be,
obtained.In
°order to computethe intrinsic distanceon an implicit surface,we must thensolve
the$
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
(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
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°
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�
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
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�
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
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
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
`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
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
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
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 ¡ )
(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
4, 5, and6 show the intrinsic distancefunction for implicit surfacescomputedwithM themethodhereproposed(g~ � 1). An arbitraryseedpointon theimplicit surfacehasbeen
»chosen,andpseudocolorsareusedto improve the visualization.Redcorresponds
surface (all triangulated-surfacecomputationswere donewith the packagereportedin[6]). We alsoshow absolutedifferences(error) betweendistancesobtainedthroughboth
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
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
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�
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
#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].
#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
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§ ¨
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
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
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
this article we have presenteda novel computationallyoptimal algorithm for thecomputation~ of intrinsic distancefunctionsandgeodesicson implicit hyper-surfaces.The
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
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
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
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]:
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
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