Smooth Surface Etractio from Ustructured Poit-based Volume Data Usig PDEs Paul Rosethal ad Lars Lise Abstract Smooth surface etractio usig partial differetial equatios PDEs is a well-kow ad widely used techique for visualizig volume data Eistig approaches operate o gridded data ad maily o regular structured grids Whe cosiderig ustructured poit-based volume data where sample poits do ot form regular patters or are they coected i ay form, oe would typically resample the data over a grid prior to applyig the kow PDE-based methods We propose a approach that directly etracts smooth surfaces from ustructured poit-based volume data without prior resamplig or mesh geeratio Whe operatig o ustructured data oe eeds to quickly derive eighborhood iformatio The respective iformatio is retrieved by partitioig the 3D domai ito cells usig a kd-tree ad operatig o its cells We eploit eighborhood iformatio to estimate gradiets ad mea curvature at every sample poit usig a four-dimesioal least-squares fittig approach Gradiets ad mea curvature are required for applyig the chose PDE-based method that combies hyperbolic advectio to a isovalue of a give scalar field ad mea curvature flow Sice we are usig a eplicit time-itegratio scheme, time steps ad eighbor locatios are bouded to esure covergece of the process To avoid small global time steps, oe ca use asychroous local itegratio We etract a smooth surface by successively fittig a smooth auiliary fuctio to the data set This auiliary fuctio is iitialized as a siged distace fuctio For each sample ad for every time step we compute the respective gradiet, the mea curvature, ad a stable time step With these iformatios the auiliary fuctio is maipulated usig a eplicit Euler time itegratio The process successively cotiues with the et sample poit i time If the orm of the auiliary fuctio gradiet i a sample eceeds a give threshold at some time, the auiliary fuctio is reiitialized to a siged distace fuctio After covergece of the evolvutio, the resultig smooth surface is obtaied by etractig the zero isosurface from the auiliary fuctio usig direct isosurface etractio from ustructured poit-based volume data ad rederig the etracted surface usig poit-based rederig methods Ide Terms PDEs, surface etractio, level sets, poit-based visualizatio 1 INTRODUCTION With the permaetly improvig digital techologies, the amout of geerated, collected, ad stored data icreases steadily While costs for computatio power, data storage, ad data collectio declie, more ad more data has to be evaluated Whe operatig o cluster or shared-memory machies, umerical simulatios of physical pheomea ca produce huge data sets For fleibility, some of these simulatios are preferred to be carried out o a ustructured poit-based data structure rather tha a grid For eample, i astrophysics particle simulatios are quite frequetly used with the umber of particles ragig up to several millios, each storig multiple scalar values Aother eample of large ustructured poit-based volume data geeratio are the deploymet of sesor systems with large umbers of sesors A additioal attractive property of ustructured poit-based data approaches is that they aturally iclude all grid-based cofiguratios A ustructured poit-based data visualizatio approach ca always be applied to a gridded data set by eglectig the grid coectivity Surface etractio methods based o partial differetial equatios PDEs have a large variety of applicatios ad, i particular, are a well-kow techique for segmetatio of scalar volume data May algorithms ad approaches with differet modificatios of the mai idea eist Most of them address a specific problem or a specific type of gridded data sets Typically the algorithms operate o heahedral cells ad a give iitial surface is modified to eplicitly or implicitly miimize a give eergy fuctioal Eve though all these kow algorithms cover a wide area of prob- Paul Rosethal is with Jacobs Uiversity Breme, E-mail: prosethal@jacobs-uiversityde Lars Lise is with Jacobs Uiversity Breme, E-mail: llise@jacobs-uiversityde Mauscript received 31 March 2008; accepted 1 August 2008; posted olie 19 October 2008; mailed o 13 October 2008 For iformatio o obtaiig reprits of this article, please sed e-mailto:tvcg@computerorg lems, to our kowledge o algorithm eists that directly operates o ustructured poit-based volume data, where scalar fuctio values are give at poits i a three-dimesioal domai that have a arbitrary distributio ad o grid coectivity If this type of data has to be processed, it is typically resampled over a regular structured grid usig scattered data iterpolatio techiques Such a iterpolatio techique itroduces resamplig iaccuracies that icrease the ucertaity or error i the resultig visualizatio We propose a PDE-based surface etractio method that directly operates o a large ustructured poit-based volume data set, i e we are either resamplig the data over a structured grid or geeratig a global or local polyhedrizatio Istead, we determie some eighborhood iformatio for the sample poits, iitialize a auiliary fuctio o the sample poits, ad process this fuctio accordig to its approimated ad iterpolated properties like gradiet or mea curvature For our surface etractio method we use hyperbolic advectio to a isovalue of a give scalar field ad mea curvature flow I the cotet of level-set methods, the auiliary fuctio is typically referred to as level-set fuctio To esure umerical stability of the chose eplicit time itegratio scheme ad thus covergece of the overall evolutio process we choose appropriate eighbors for the derivative calculatio ad small eough time steps The drawback of small global time steps ca be circumveted by usig asychroous local itegratio After covergece of the process, the zero isosurface to the auiliary fuctio is etracted i form of a poit cloud surface represetatio ad visualized via poit-based rederig techiques usig splats The mai ideas of our smooth surface etractio method ad the structure of the etire pipelie are itroduced i Sectio 3 For storig the sample poits we use a spatial decompositio based o a threedimesioal kd-tree, as described i Sectio 4 The approimatio of gradiet ad mea curvature requires the computatio of earest eighbor iformatios for each sample poit These are computed i a preprocessig step ad stored durig the whole evolutio process I every step ad for every sample poit the gradiet of the auiliary fuctio ad the mea curvature is approimated usig a fourdimesioal least-squares method, as described i Sectio 5
While the reiitializatio of the auiliary fuctio is described i Sectio 6, the evolutio approach is eplaied i detail i Sectio 7 Cosideratios about stability ad time itegratio are preseted i Sectio 8 Sectio 9 eplais the last step i our pipelie, the zero isosurface etractio Theoretical ad practical results, icludig aalysis of error ad computatio times, are show i Sectio 10 2 RELATED WORK Commo ways to deal with ustructured poit-based volume data are to resample the data usig scattered data iterpolatio techiques or to compute a polyhedral grid that coects the ustructured data poits Afterwards segmetatio methods like isosurface etractio, regiogrowig methods, ad also PDE-based methods like level-sets or fiite elemets ca be applied to the gridded data to geerate the desired visualizatios Level-set methods, i particular, ted to operate o regular heahedral grids what facilitates discrete derivative computatios Scattered data iterpolatio is a well-studied field We refer to oe of may surveys o this topic for further details [10] Recetly, Park et al [25, 26] have show that scattered data recostructio for large data sets ca be achieved at iteractive or ear-iteractive rates whe discretizig the approach ad resamplig over a regular grid Ufortuately, such resamplig steps always itroduce iaccuracies I our case, whe observig data sets with highly varyig poit desity, the iterpolatio error would be eormous whe usig regular grids that fit today s memory costraits Adaptive grids ca reduce the error, but the more adaptive it gets the more complicated the processig becomes, which raises the desire to directly operate o the ustructured poit-based volume data Level-set methods go back to Sethia ad Osher [22, 23, 30], who first described the evolutio of a closed hypersurface As oe ca see, the geeral approach is youger tha classical surface etractio ad scalar volume data processig techiques Nevertheless, due to their fleibility already a large commuity is usig ad developig levelset methods May differet approaches eist ad the rage of applicatio areas is wide Still, to our kowledge, all eistig level-set algorithms require some kid of uderlyig grid ad coectivity iformatio amog the sample poits Bree et al [2] preseted a geeral framework for level-set segmetatio of a large variety of regular data sets Museth et al [20] use a level-set method to segmet o-uiform data sets, ad Eright et al [5] apply a level-set approach to a octree-based adaptive mesh Beside these, may other approaches, eg [18, 31, 33], eist that target a huge variety of level-set segmetatio tasks o differet structured data sets We wat to eplicitly metio the particle level-set methods [13], as they use free particles durig the level-set computatios However, i cotrast to our approach, particle level-set methods still require a uderlyig grid or mesh We preset a approach that operates directly o ustructured poit-based volume data Some level-set approaches such as active cotours [19] require a eplicit represetatio of the iitial surface, which is deformed util it coverges to the desired surface Our approach, istead, chages the scalar values i the sample poits over time such that the surfaces are givig implicitly I particular, we do ot eed to recostruct the scalar field at ay positios other tha the sample poits 3 GENERAL APPROACH Give a set of ustructured sample poits i D R 3 i space ad a volumetric scalar field f : D R with bouded domai D that is give at these sample locatios, we wat to etract a smooth isosurface Γ iso D with respect to a give isovalue f iso We first eed to build a data structure that stores ad hadles eighborhood iformatio This is doe usig a three-dimesioal kd-tree I a preprocessig step, we apply a stadard algorithm [11] o kd-trees for computig the earest eighbors for each sample We chose = 26 ispired by the regular case o a structured equidistat heahedral grid, where every sample has 26 earest eighbors i the L -metric Afterwards a auiliary fuctio ϕ is iitialized as a siged distace fuctio for every sample We choose a radial fuctio as iitial fuctio such that the iitial isosurfaces are spheres The auiliary fuctio is adapted i a iterative process steered by the PDE If the orms of the auiliary fuctio gradiets eceed a give threshold at ay time i the process, a reiitializatio step is applied to this fuctio This keeps the auiliary fuctio close to a siged distace fuctio, where distaces are measured with respect to the isosurface, ad assures good umerical behavior of the fuctio durig the PDE-based process Durig the adaptig process, the gradiet ϕ of the auiliary fuctio ϕ ad the mea curvature κ ϕ are approimated oly for the sample poits The approimatio is computed usig a four-dimesioal least-squares approach Havig all ecessary iformatios collected, a time step is performed with respect to the equatio = a f f iso ϕ+bκ ϕ ϕ, 1 which models hyperbolic ormal advectio [22], weighted with factor a > 0, ad mea curvature flow, weighted with factor b > 0 Sice we are usig a eplicit Euler time discretizatio for updatig the fuctio at the sample poits, the time steps are bouded by the Courat-Friedrichs-Lewy coditio [3] to permit umerical stability Oe ca follow a global or a local strategy for updatig the level-set fuctio values If oe chooses oe global time step for all sample poits, the step is bouded by the most restrictive stability coditio of the sample poits The required fuctio properties are fast to compute, but for most samples the time steps are smaller tha required Istead, oe ca use adaptive time itegratio [21, 24] Here, oe has a local time step for every sample Thus, the stability coditio of a sample does ot affect the time steps of other sample poits Sice calculatios per sample poit are more complicated ad time cosumig this method oly pays off for data sets with highly varyig poit desity I both cases, the auiliary fuctio is deformed at the sample poits over time util it reaches steady state, i e, util the fuctio values do ot chage more tha a give threshold from oe time step to the subsequet oe After covergece the isosurface to the isovalue zero is etracted from the geerated scalar field over the sample poits This surface has the the desired properties of Γ iso Note that the surface does ot eed to be etracted durig the PDE computatios It is give implicitly 4 SCATTERED DATA STORAGE For fast calculatios ad access to the sample poits some preprocessig steps are required For each sample we calculate the 26 earest eighbors with respect to the Euclidea distace This eighbor iformatio is used later o to approimate geometric properties of the auiliary fuctio To store the scattered data poits icludig space coordiates, data fuctio value, ad auiliary fuctio value we use a threedimesioal kd-tree T This data structure helps savig computatio time I order to save storage space the samples are ot directly stored i the kd-tree but i a vector of sample poit locatios P ad i a vector of values V Vector V stores all the fuctio values The odes of the kd-tree oly lik to P ad V Usig vector P, the kd-tree T is build recursively We start with depth 0 ad T = /0 ad correspodig vector P The, for every depth i i the kd-tree ad vector P, which is a subvector of P, we sort P i imod3 -directio, where 0, 1 ad 2 deote the three dimesios A ode is iserted ito T, P is split i two half-sized vectors P 1 ad P 2 ad the pivot-value of the splittig is stored i the iserted ode If P is odd-sized, a lik to the midpoit is also stored i the ode While proceedig recursively with the subvectors P 1 ad P 2, we get two childre for the just iserted kot If a subvector is empty the recursio stops 5 CALCULATING FUNCTION PROPERTIES To process the auiliary fuctio followig hyperbolic ormal advectio ad mea curvature flow, as modeled i Equatio 1, we have to
calculate the gradiet ad mea curvature i each sample poit ad for every time step Because of the ustructured distributio of the samples, we are ot able to use ay grid-based approach for calculatig fuctio gradiets or mea curvature Istead we use a four-dimesioal least-squares approach to approimate the gradiet ϕ of the auiliary fuctio ϕ ad the mea curvature κ ϕ Durig the calculatios oly a earesteighbors computatio of the sample poits is eeded Furthermore, o iformatio about the scalar field other tha at the sample poits is eeded 51 Gradiet Calculatio Let ϕ : R 3 D R be a differetiable fuctio, the the graph of ϕ is the submaifold graphϕ R 4 defied as graphϕ := {,ϕ : D} The gradiet ϕ i every poit ca be derived by projectig the ormal, ϕ of the taget hyperplae to graphϕ i to the volumetric domai We obtai that ϕ = pr R 3,ϕ, where pr R 3 : R 4 R 3 deotes the orthogoal projectio to the three first coordiates Oe- ad two-dimesioal illustratios of this geometrical cosideratio are preseted i Figures 1 ad 2 ϕ pr R,ϕ,ϕ graphϕ ϕ Fig 1 Relatio betwee the ormal to the graph ad gradiet of a oedimesioal scalar fuctio ϕ ϕ pr R 2,ϕ,ϕ ϕ T,ϕ graphϕ graphϕ Fig 2 Graph of a two-dimesioal fuctio ϕ with taget plae T,ϕ graphϕ, ormal,ϕ to the graph, ad gradiet ϕ Cosiderig the earest eighbors of i R 3, the taget hyperplae T,ϕ graphϕ R 4 is approimated usig a fourdimesioal least-squares fittig through the eighborig samples with associated fuctio values The ormal of the resultig hyperplae is projected to R 3 to get the fuctio gradiet This procedure fially results i a closed formula for the gradiet approimatio For a oe-dimesioal fuctio ϕ, represeted through the poits 1,ϕ 1,,,ϕ, we get i ϕ i i dϕ ϕ i d = 2 2 i 2 i This formula is a geeralizatio of several well-kow gradiet approimatio methods For eample, usig poits,ϕ 1 ad + h,ϕ 2 leads to the stadard forward differecig dϕ d = ϕ 2 ϕ 1 h Also cetral differecig is a special case of Equatio 2 poits,ϕ 1, +h,ϕ 2, ad h,ϕ 0 leads to dϕ d = ϕ 2 ϕ 0 2h Usig Such a closed formula ca also be derived for approimatios of gradiets i higher dimesios It turs out, that the chose least-squares approach also geeralizes the kow forward ad cetral differecig schemes o grids i higher dimesios The formulae with the closed forms for our applicatio i 3D are give i the appedi This least-squares approach results i a cosistet gradiet approimatio: If the distace to the used eighbors coverges to 0, the computed hyperplae coverges to the taget hyperplae to the graph of the fuctio Hece, the approimated gradiet coverges towards the eact gradiet 52 Mea Curvature Calculatio The calculatio of the mea curvature κ ϕ requires some more cosideratios We follow the ideas described by Osher ad Fedkiw [22] First, we ote that κ ϕ = ϕ ϕ More precisely, we have κ ϕ = ϕ 1 ϕ with + 1 ϕ 2 + ϕ 3 ϕ 2 ϕ 3 ϕ ϕ = ϕ1 ϕ, ϕ 2 ϕ, ϕ 3 ϕ Thus, we ca reduce the problem of mea curvature calculatio to the problem of three gradiet calculatios for the three dimesios of the ormalized gradiet of ϕ The ormalized gradiet of ϕ ca be approimated for every sample, as described i Sectio 51 The, the gradiet to every fuctio ϕ i ϕ : R 3 R, i = 1,2,3, is approimated usig agai a four-dimesioal least-squares approach Afterwards the ith compoets of gradiets ϕ i ϕ, i = 1,2,3, are take ad summed up to get a approimatio to the mea curvature κ ϕ Due to usig oly multiple cosistet gradiet calculatios, the cosistecy of this approach for mea curvature approimatio is obvious 6 REINITIALIZATION As described by Peg et al [28], the quality of the PDE-process sigificatly degrades if the auiliary fuctio ϕ is ot close to a siged distace fuctio of the isosurface I our approach the fuctio is iitialized as a siged distace fuctio to the isosurface Ufortuately the PDE process caot maitai this property
Hece, we have to reiitialize the auiliary fuctio to a siged distace fuctio to the ew isosurface if the orms of the gradiets of the fuctio eceed a certai threshold For this reiitializatio we choose a PDE approach solvig the special Eikoal equatio = sigϕ1 ϕ 3 This approach is much faster tha eact calculatio of the siged distace fuctio As we use a eplicit Euler time itegratio for the reiitializatio process, Equatio 3 leads to the time developmet equatio ϕ i+1 = ϕ i + t sigϕ i 1 ϕ i 7 SMOOTH ISOSURFACE EXTRACTION As described i Sectio 3, our goal is to etract a smooth isosurface Γ iso D R 3 with respect to a isovalue f iso of f : R 3 D R For solvig this problem we choose a PDE formulatio which is a combiatio of two well-kow approaches The hyperbolic advectio [22] + α ϕ = 0 models the trasport of the iterface i ormal directio with speed α We wat to etract a isosurface from the give data set, so our goal is to miimize E = ϕ f f iso d D This leads to the evolutio equatio +ϕ f f iso ϕ = 0 4 A secod property we wat to cosider is the smoothess of the resultig surface Γ iso Thus, our secod goal should be the miimizatio of the surface area of Γ, i e, we wat to miimize Γ = δϕ ϕ d, D where δ deotes the oe-dimesioal Dirac δ-fuctio with δ = d d H ad H beig the oe-dimesioal Heaviside fuctio, defied as { 1 for 0 H = 0 for < 0 These cosideratios lead to the model of mea curvature flow [6, 7, 8, 9], characterized by = κ ϕ ϕ, 5 where κ ϕ deotes the mea curvature to the level set Combiig hyperbolic ormal advectio 4 ad mea curvature flow 5, we get the evolutio equatio = a f f iso ϕ+bκ ϕ ϕ with scalig parameters a,b > 0, where gradiet ϕ ad mea curvature κ ϕ are computed as described i Sectio 5 This equatio is solved to steady state usig ϕ i+1 = ϕ i + t i i e a eplicit time discretizatio of order oe, 6 8 STABILITY AND TIME INTEGRATION 81 Stability Accordig to the La-Richtmyer equivalece theorem [32] covergece of a fiite differece scheme is equivalet to cosistecy ad stability As metioed i Sectio 51 our gradiet ad mea curvature approimatios are cosistet The stability of our differetial schemes ad followig costraits to the time step sizes will be ow observed For simplicity, we describe our aalysis for the two-dimesioal hyperbolic ormal advectio case, i e = a ϕ, a > 0 Applyig our fiite differece scheme, derived i Sectio 51, ad eplicit Euler time itegratio we get a evolutio equatio ϕ i+1 = E tϕ i, 7 for each sample R 2 with discrete solutio operator E We use the vo Neuma stability aalysis [14, 34], which makes use of spatial Fourier trasforms [1, 4], to verify stability of the time evolutio Equatio 7 Vo Neuma s theorem states that the evolutio process is stable with respect to the maimum orm, iff Ẽξ 1, for all ξ R, where Ẽ deotes the Fourier trasform of E For observig two eighbors per sample poit, y for the gradiet approimatio we ca assume the situatio,y, f 1, +1,y, f 2, +,y+ y, f 3 without loss of geerality I this case, vo Neuma stability aalysis leads to the followig coditios assurig stability λ y y λ 1 t 1 y λ + λ y 1 t 8 9 y λ λ y 10 λ 0, 11 where λ = a f ad λ y = a f y Oe easily sees, that Equatios 8 ad 9 ca be maitaied by choosig a small eough time step t Cotrary to this, the last two coditios affect the relatio betwee partial derivatives of ϕ ad, y From a descriptive poit of view, they assure the upwidig of the process ad describe two sectors i which the secod eighbor is allowed to be to assure stability I our approach, where the domai is R 3, the same cosideratios lead to similar coditios ad restrictios for the time step ad sample locatios Therefore, we are able to process the auiliary fuctio assurig stability by choosig appropriate eighbors ad a small eough time step I practice, we observed that whe choosig the 26 earest eighbors ad small eough time steps t i Equatio 6, we ever ra ito stability problems To derive a estimate for a time step t we trasferred the cosideratios by Osher ad Fedkiw [22] to our case The coditio for the hyperbolic ormal advectio becomes t a f f iso ϕ d mi ϕ + 1 + 2 < 1, 3 where d mi deotes the Euclidia distace to the earest eighbor, i e the radius of the miimal domai of depedece The coditio for the whole evolutio becomes t a f f iso ϕ d mi ϕ + 1 + 2 3 + 6b dmi 2 < 1
Although there is o evidece that this criterio esures stability, it worked out well for all the practical cases we cosidered While from a theoretical poit of view, we ca esure stability by choosig appropriate eighbors ad time steps with respect to Equatios 8 to 11, for practical purposes it is beeficial to use a larger umber of eighbors, as larger umber of eighbors 26 i our eamples esure better gradiet approimatios 82 Sychroous Time Itegratio Time steps have to be small eough to esure stability, i e, the physical domai of depedece is required to lie i the domai of depedece of the fiite differece scheme All these cosideratios are doe i a fied sample poit at a certai poit i time If we wat to apply a global time step for all sample poits, it is bouded by the most restrictive stability coditio whe cosiderig all poits Thus, if the sample poits have a highly varyig distributio or if the uderlyig scalar field has big local variatios, time steps for all poits may be bouded by the rather restrictive stability coditio of a few poits 83 Asychroous Time Itegratio To alleviate this potetial drawback, oe ca use asychroous time itegratio Here, like i global time itegratio, all sample poits start at the same poit i time The for each sample poit oe time step is computed, oly bouded by the local stability coditio t Fig 3 Sample poit time positios before the time itegratio marked i red ad after the first time step marked i gree As show i Figure 3 most of the sample poits are ow asychroous To compute the et step for ay of the sample poits, oe would have to evaluate the fuctio values at other sample poits at the same poit i time Fortuately, usig the liear Euler time itegratio we ca circumvet this time cosumig step If we save for every sample poit ot oly its curret fuctio value ad poit i time but also the previous fuctio value ad poit i time, we are able to recostruct all fuctio values betwee the previous two poits i time by liear iterpolatio t Fig 4 Progress of time lie through the sample poits The earliest sample poit is processed further usig the liearly iterpolated fuctio values marked i blue of its eighbors I Figure 4, the asychroous time itegratio process is visualized After havig computed the first time step, the time lie is set to the earliest subsequet poit i time Afterwards the fuctio values of the eighbors are liearly iterpolated at this poit i time This iterpolatio ca be performed for all eighbors, sice we are always updatig the earest poit i time, i e, we are processig the earliest sample poit Hece, for all other sample poits we ca access stored values for a future poit i time ad a past poit i time with respect to the curret time lie With the iterpolated properties, the auiliary fuctio at the curret sample poit is itegrated i time with the local time-step restrictio ad the time lie proceeds to the et poit i time If at ay time the orm of the auiliary fuctio gradiet eceeds the give threshold, the time lie is stopped, all sample poits are iterpolated to the curret poit i time, ad the reiitializatio process is started Durig reiitializatio the time steps are also bouded by a stability coditio ad asychroous time itegratio is applied i the same way as described before The whole evolutio process stops, if the fuctio values of the sample poits chage o more with respect to a certai tolerace, i e whe the process reaches steady state I this case, the fuctio values of the sample poits are iterpolated to the curret time lie ad we get a sychroous auiliary fuctio 9 ISOSURFACE EXTRACTION AND RENDERING To obtai the smooth isosurface to the iitial data set, the zero isosurface of the auiliary fuctio after covergece is etracted This is doe usig a isosurface etractio approach for ustructured poitbased volume data preseted by Rosethal ad Lise [29] A appropriate eighborhood is geerated for each sample out of the kd-tree structure For fast access to the odes of the kd-tree ad efficiet calculatio of the eighborhood, a efficiet ideig scheme has bee itroduced Eploitig the ideig scheme for fast eighbor computatio, isopoits are liearly iterpolated betwee eighborig samples with differet auiliary fuctio sigs, where the eighborhood approimates a atural eighborhood The surface ormals to the isopoits are iterpolated usig a four-dimesioal least-squares approach As a result we get the isosurface represeted as a poit cloud icludig surface ormals Sice o coectivity of the geerated poits of the poit cloud is kow, a poit-based rederig is favorable The oly iformatio we have, are the iterpolated fuctio values ad ormals of the surface poits as well as the earest eighbors From this iformatio we ca geerate a splat-based represetatio of the isosurface usig a leastsquares approach The resultig splats are redered usig a poitbased raytracig techique [16] 10 RESULTS AND DISCUSSION We applied the preseted approach to a variety of data sets to verify our method For performace aalysis we applied it to a resampled ustructured poit-based data set of 4M radomly distributed samples, geerated from the regular Hydroge data set of size 128 128 128 A illustratio of the evolutio process for this data set is show i Figure 5 To compare the computatio times, we dowsampled the data set to differet sizes These ad the followig computatio times were measured o a 266 GHZ XEON processor The rutime aalysis for the preprocessig is preseted i Table 1 # samples kd-tree geeratio NN-calculatio 500k 1 sec 23 sec 1M 2 sec 50 sec 2M 4 sec 111 sec 4M 9 sec 239 sec Table 1 Computatio times for the preprocessig of the Hydroge data set with differet sample quatities The preprocessig icludes the geeratio of the kd-tree ad earest eighbor calculatio The rutimes for the isopoit etractio were also aalyzed regardig the Hydroge data set A summary of the results icludig the
umber of samples, umber of etracted zero-level-set poits, calculatio time for the eighborhood iformatio, ad calculatio time for the poit etractio is show i Table 2 Because of some tuig i the implemetatio, we achieved sigificatly faster results tha those stated by Rosethal ad Lise [29] Step 0 Step 1 Step 2 Step 3 # samples # poits eighborhood poit etr 500k 7k 08 sec 05 sec 1M 12k 17 sec 11 sec 2M 18k 41 sec 19 sec 4M 29k 101 sec 26 sec Table 2 Isopoit etractio times for the hydroge data set The umber of etracted surface poits ad etractio times refer to the zero isosurface to the auiliary fuctio after covergece of the evolutio process The whole method applied to the 4M hydroge data set, icludig preprocessig, evolutio of the auiliary fuctio with sychroous time itegratio, ad isopoit etractio lasted 22 miutes A detailed aalysis of the computatio times of the evolutio process ad the reiitializatio is give i Table 3 We observed that a asychroous update step is sigificatly slower tha a sychroous oe Thus, usig a adaptive time itegratio scheme oly pays off i case of heavily varyig sample desity au fuctio evolutio reiitializatio sychr it 59k samp/sec 88k samp/sec asychr it 2k samp/sec 46k samp/sec Step 4 Step 5 Step 6 Step 7 Step 8 Fig 5 Poit-based raytracig of the smooth isosurface etractio process o the Hydroge data set with 4M radomly distributed sample poits Data set courtesy of Peter Fassbider ad Wolfgag Schweizer, SFB 382 Uiversity Tübige Table 3 Computatio times compariso for the evolutio ad the reiitializatio process with sychroous ad asychroous time itegratio The times are specified i thousad processed sample poits per secod Fially we applied our method to a real-world eample of a ustructured poit-based volume data sets, provided by astrophysical particle simulatios of Stepha Rosswog, Jacobs Uiversity, Breme, Germay I the simulatio, a set of particles represetig a White Dwarf passes a black hole ad is tor apart by the strog gravity The data sets represet a sapshot of this simulatio at a certai poit i time, where several physical properties are give Two smooth isosurfaces to differet isovalues were etracted from a desity data set with 500k sample poits They are show i Figure 6 The whole process with asychroous time itegratio lasted 68 miutes To compare our PDE-based surface etractio approach to direct isosurface etractio i terms of quality of the etracted surface, we etracted a isosurface from the 500k White Dwarf data set with both algorithms A visualizatio of the etracted poit clouds is preseted i Figure 7 The output geerated by the PDE-based approach does ot ehibit ay outliers ad results i a much smoother surface 11 CONCLUSION AND FUTURE WORK We have preseted a smooth surface etractio method that ca directly be applied to ustructured poit-based volume data sets No global or local three-dimesioal mesh geeratio or recostructio over a regular grid is applied The preseted approach is able to etract surfaces with respect to ormal advectio to a scalar field ad mea curvature flow I a preprocessig step, the samples are stored i a kd-tree ad for every sample the earest eighbors are calculated Afterwards, a auiliary fuctio is iitialized as a siged distace fuctio Subsequetly, the evolutio of this fuctio begis The eeded properties of the auiliary fuctio, like gradiet or mea curvature, are approimated usig four-dimesioal leastsquares approaches ad the fuctio is processed accordig to hyperbolic ormal advectio ad mea curvature flow If the auiliary fuctio departs from a siged-distace fuctio, a reiitializatio is performed For time itegratio of the evolutios, a sychroous or
Fig 7 Compariso betwee direct isosurface etractio o the left side ad smooth isosurface etractio o the right side for the 500K White Dwarf simulatio data set To illustrate the sigificat advatage of the PDE-based approach over direct isosurface etractio, we iclude a close-up view o the surface ad show a rederig of the surface poits with very small splats Fig 6 Isosurfaces of the 500k White Dwarf simulatio data set The uderlyig scalar field represets the desity i space We segmeted the data set regardig 40g/cm 3 surface o the left side ad 100g/cm 3 surface o the right side Data set courtesy of Stepha Rosswog, Jacobs Uiversity, Breme, Germay a asychroous approach are applied assurig stable time steps The auiliary fuctio is processed util it reaches a steady state To visualize the computed surface, a poit cloud represetig of the zero isosurface to the auiliary fuctio is etracted A poit-based rederig techique usig splats is eecuted o this poit cloud To aalyze our method, we applied it to several data sets I terms of future work a localizatio of the algorithm, similar to the ideas by Peg et al [28], as well as a itegratio of the reiitializatio step ito the processig of the auiliary fuctio, as proposed by Lefoh et al [15], could be cosidered to speed up the calculatios Also, the earest eighbor calculatio ca be improved usig faster algorithms [17] A geeral questio would be, if it is reasoable to use higher-order time discretizatio schemes for the evolutio process like the oes proposed by Gottlieb et al [12] Fially, the rederig egie should be improved, sice the splattig approach used has problems with hadlig sharp edges [27] A GRADIENT CALCULATION IN R 3 For the partial derivative of the fuctio ϕ : R 3 R, represeted through the poits i,y i,z i,ϕ i, i = 1,,, i y-directio we get X 1 i ϕ i + X 2 y i ϕ i + X 3 z i ϕ i + X 4 i y = ϕ Y, where 2 X 1 = i y i z 2 i z i + i z i i z i y i z i y + i i z i z i i z y y 2 i 2 X 2 = i 2 z i z 2 i + i z i i z i i z i + z i i i z 2 i i z i X 3 = i 2 y i z i i z i y + i y i i z i i z i X 4 = + i i z i y i i i 2 y i z 2 i y i z i z i + + i y i i z i i z i i y i z i z i i z 2 i y i z i i z i y i
ad Y = y i z i 2 i 2 2 i 2 2 + i z i y 2 i y i 2 + i y i i 2 y 2 i z 2 i + i z y 2 i 2 i y i i i y i + i z 2 i + 2 + 2 + 2 i y 2 i i y i y i i i z i i y i y i i z i i y i i i y i y z y i z i y 2 i z i i z i i 2 i z i z i y i z i 2 z i Aalogously, we obtai the partial derivatives i - ad z-directio ACKNOWLEDGEMENTS This work was supported by the Deutsche Forschugsgemeischaft DFG uder project grat LI-1530/6-1 REFERENCES [1] R Bracewell The Fourier Trasform ad Its Applicatios McGraw-Hill Sciece Egieerig, 3 editio, 1999 [2] D Bree, R Whitaker, K Museth, ad L Zhukov Level set segmetatio of biological volume data sets I Hadbook of Medical Image Aalysis, Volume I: Segmetatio Part A, pages 415 478, New York, 2005 Kluwer [3] R Courat, K Friedrichs, ad H Lewy O partial differece equatios of mathematical physics IBM J, 11:215 222, 1967 [4] H Dym ad H McKea Fourier Series ad Itegrals Academic Press Ic, 1972 [5] D Eright, F Losasso, ad R Fedkiw A fast ad accurate semilagragia particle level set method Computers ad Structures, 836-7:479 490, 2004 [6] J Escher ad G Simoett The volume preservig mea curvature flow ear spheres I Proceedigs of the America Mathematical Society, volume 126, pages 2789 2796, 1998 [7] L C Evas ad J Spruck Motio of level sets by mea curvature I J Diff Geometry, 33:635 681, 1991 [8] L C Evas ad J Spruck Motio of level sets by mea curvature II Tras America Math Soc, 3301, 1992 [9] L C Evas ad J Spruck Motio of level sets by mea curvature III J Geometric Aalysis, 22, 1992 [10] R Frake ad G M Nielso Geometric Modelig: Methods ad Applicatios, chapter Scattered Data Iterpolatio: A Tutorial ad Survey, pages 131 160 Spriger Verlag, New York, 1991 [11] J H Friedma, J L Betley, ad R A Fikel A algorithm for fidig best matches i logarithmic epected time ACM Tras Math Softw, 33:209 226, 1977 [12] S Gottlieb, C-W Shu, ad E Tadmor Strog stability-preservig highorder time discretizatio methods SIAM Rev, 431:89 112, 2001 [13] S E Hieber ad P Koumoutsakos A lagragia particle level set method J Comput Phys, 2101:342 367, 2005 [14] S Larsso ad V Thome Partial differetial equatios with umerical methods Spriger, 2005 [15] A E Lefoh, J M Kiss, C D Hase, ad R T Whitaker Iteractive deformatio ad visualizatio of level set surfaces usig graphics hardware I VIS 03: Proceedigs of the 14th IEEE Visualizatio 2003 VIS 03, page 11, Washigto, DC, USA, 2003 IEEE Computer Society [16] L Lise, K Müller, ad P Rosethal Splat-based ray tracig of poit clouds Joural of WSCG, 151 3, 2007 [17] J McNames A fast earest-eighbor algorithm based o a pricipal ais search tree IEEE Trasactios o Patter Aalysis ad Machie Itelligece, 239:964 976, 2001 [18] R B Mile A Adaptive Level-Set Method PhD thesis, Uiversity of Califoria, Berkeley, 1995 [19] B S Morse, W Liu, T S Yoo, ad K Subramaia Active cotours usig a costrait-based implicit represetatio I CVPR 05: Proceedigs of the 2005 IEEE Computer Society Coferece o Computer Visio ad Patter Recogitio CVPR 05 - 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