Rendering for Rapid Prototyping and CAM
Appli ations
Al este S alas and Piero Pili
Laboratory for Advan ed Planningand Simulations
GEMMS Area
Energy and IndustrialPro esses
CRS4
(Center for Advan ed Studies, Resear h
and Developmentin Sardinia)
January2002
Abstra t
This do ument des ribes somepreliminary results of theappli ation of dire t
volumevisualizationandrealisti imagesynthesiste hniquesinComputerAided
Manufa toring(CAM) eld.
The nal goalis to analyze theuse volume data representation anddire t
volume visualization te hniques respe tively as basi geometri element and
renderingmethodfornumeri al ontrol(NC)pro essplanningandsimulation.
Inparti ularwewanttostudytheirappli ationinthepro essofhumanorgans
repli ation by rapidprototypingte hniques. Volume renderingte hniques will
be used to evaluate the geometri re onstru tion of volumetri data (e. g.
parts of human body extra ted from a CT s an as the arotid arteries), in
order toobtainthebest polygonalapproximationof adataset. Thepolygonal
model resulting from this evaluation stage will be used to drive its physi al
re onstru tionviarapidprototyping.
Avolumerenderingprogram, alledVolCastIA,ispresented. VolCastIAisa
softwareenviromentbuilttoevaluatetheuseofstateoftheartimagesynthesis
methods inthevolumerendering ontext.
1 Introdu tion
The Volume Rendering eld gained high popularity in thelast years. It o ers
a way to inspe t volumetri datasets withouttheneed to prepro ess them by
geometri samplingte hniques(forexample,themar hing ubeste hnique)that
an auselossofinformationandevidentre onstru tionartifa ts.
The use of volumetri datasets representation as entral element in rapid
prototyping industrial appli ations is be oming an interesting resear h prob-
industrial millingpro ess eld.
1.1 Volume rendering and rapid prototyping
Rapidprototypingte niquesgivethepossibilitytoobtainaphysi alprototype
ofapolygonalmodel. Theyareoftenusedintheindustrial eld,forexampleto
evaluatethe hara teristi sofanewobje t,beforestartingitsmassprodu tion
anditslaun honthemarket.
Furthermore,rapidprototyping analsobeusedtore onstru tpartsofthe
human body extra ted from a CT s an,for example in order to evaluate the
possibilityto makeaprothesisusingthepatient'sdatasetasadire t referen e
model.
Sin e the rapid prototyping pro ess is a tually driven by geometri data,
it is important to build arobustsystemwhi h is ableto produ ean a urate
polygonalmesh from lassi edorsegmentedvoxels. Forthisreasonthehybrid
photo-realisti visualization of polygonaland volumtri data isuseful to eval-
uate andmeasure thedi eren esbetweentheoriginal datasetandthederived
polygonalmodeltobeprodu ed byrapidprototypingdevi e.
1.2 Volume rendering and industrial milling pro ess
1.2.1 Current te hniques for millingpro ess simulation and visual-
ization
The urrent te hniques for the visualization and simulation of the industrial
millingpro essarebasedonthe\ lassi al"approa hforeverythree-dimensional
visualization problem: using a polygonal model | that, in this eld, o ersa
representationoftheobje ttobema hined.
Themainproblemofthisapproa histhatthemillingpro essis on eptually
based onvolume manipulations (i. e. theremoval ofparts from a solidblo k
of material), while the polygonal representation is based on surfa es (i. e.
everyobje tisapproximatedwithameshofpolygonsthat representsitsouter
surfa e).
Thisdis repan y ausesaseriesofproblem| rstofall,duringthemilling
pro essplanningandsimulation: thetransformationsthatthemillingtoolper-
formsonasolidobje tmustbesimulatedusinggeometri omputationsbased
on planes, rather than onvolumes. This is a ompli ated task, expe ially re-
gardingthe he kofpossible ollisions betweenthemilling tool(that ouldbe
omposed byup to5arti ulatedaxis) andthe polygonalstru turesof theob-
je ttobemilled. Furthermore,everyalterationontheinitialpolygonalmodel
( ausedbythemillingpro essitself)impliesthe al ulationofanewpolygonal
mesh, withvariousdegreesofapproximationandlossofpre ision.
The same problems an be found during the simple visualization of the
millingpro ess. Infa t,thisvisualizationmusto erawaytounderstandsome
fundamental information about the milling phases: for example, the amount
and onsisten yoftheresidualmaterial,andtheregionsoftheobje tsinwhi h
it is a umulated. This kind of information an be hardly represented using
lassi al polygonal visualization te hiques, and the results are not satisfying,
bothfrom theaesteti andinformativepointofview(see gure1);
simulation,byFidiaS.p.A.. The olouredspotsonthema hinedobje trepre-
sentthe volume of residualmaterial. This displa ement ofthe spots does not
representthereal onsisten yoftheresidualmaterial,but itisdue toapprox-
imationsrelatedto thepolygonalvisualizationte hniques. Morein detail,the
volume of the residual material is sampled in some zones of the obje t, and
everysampleisrenderedwithanappropriate olour(basedonthevolume on-
sisten y). Thereisnowaytoknow(atleastintuitively) theamountofresidual
materialin thezonesoftheobje tthat havenotbeensampled.
niques
The volumetri representation of theobje tto be ma hined ansolveanum-
berofproblems, sin ethevolumemanipulations impliedinthemillingpro ess
aredire tlyrepresentedbyvolumetri information(onwhi hvolumerendering
operates).
During the milling pro ess planning and simulation, the alteration of the
initialvolumetri modeloftheobje ttobemilleddoesnotneedapproximations
orlossofpre ision:thesameresolutionispreservedduringallthemillingphases.
Thereareno ompli atedpolygonalstru turesthat anintera twiththemilling
tool|thema hinedobje tisrepresentedwithadis retegridofvalues,andit
makeseasiertodete tpossiblepathsforthetoolandavoid ollisions.
Furthermore, during the milling pro ess visualization, volume rendering
te hniques makeit easy to lassify di erent materials (e. g. nal obje t and
residualmaterialinthemillingpro ess)withdi erentvisualproperties. There
is no need to arbitrarily sample the ma hined obje t, in order to dete t the
residual material information: this information is arried by the volumetri
dataitself.
Ontheothehand,volumemanipulationandrenderingisa umputationally
intensivetask. Inthepastyears,itsappli ation eldshavebeenrestri tedbyits
hardwarerequirements: fastandexpensivepro essors(ordedi atedhardware),
andhighamountsofexpensivememorywererequiredtoperformalltheneeded
omputationsin areasonabletime. [17℄[16℄[1℄[6℄[15℄
Today,thelowpri esof omputerhardware(bothforpro essorsandmem-
ory), and the resear h progress in volume manipulation alghorithms, make it
possibleto performvolumemanipulationson
Andoneofthe newappli ation elds anbetheindustrial milling pro ess
simulationandvisualization.
2 Resear h goals
The primary goal of this study is the realisti visualization of the volumetri
dataset,bothforgeometri alre onstru tionandmillingpro essevaluation. In
this ontextweareinterestedonhighqualityrenderingofvolumetri dataand
its omparisonwiththegeometri alrepresentationintheformatSTLobtained
bythethegeometri re onstru tionalgorithm.
In this kind of appli ation, volumetri representation and photo realisti
image synthesiste hniquesplay afundamental role: ana urate volume visu-
alization aneasetheevaluationofageometri ally-re onstru teddatasettobe
usedforrapidprototyping. Furthermore,duringtherapidprotototing,andalso
themillingpro ess,visualization,theoperatorneedstohavearealisti preview
oftheobje tunder manufa toringin ordertoevaluatethe nal results.
However,studying theuse ofvolumevisualizationin therapidprototyping
andinindustrialmillingprodu tionpro essesisamore ompli atedtask: photo-
realisti image rendering is not usually onsidered to be a primary goal for
volumevisualization. Infa titsappli ations,mainlyderivevedfromthemedi al
eld, tend to enhan e the dataset understanding, rather than the rendering
appearan e,orthe delitytothere onstru teobje t[14℄.
iedanddevelopedduringtheresear hwillneed,in ase ond phase,tobe om-
pared with the state-of-the-art polygonal geometri representations and ren-
deringte hniques urrentlyusedinlayer-by-layermanufa toringalgorithm ur-
rently used in rapid prototyping setup and also in milling ontrol simulation
programs.
As an advan ed obje tiveof this reasea h will bethe development of new
rapid prototyping and milling pro ess planning strategies, that an be made
possibletheuseofvolumetri representationforrapidprototypingpro essitself.
3 The volume rendering pipeline
Volumevisualizationen ompassesbothmultidimensionalimagesynthesys and
multidimensionalimagepro essing. Itprovidesfortheanalysisandunderstand-
ingof3Dvolumetri s annedorexperimentaldata. TheVolume Visualization
pipelinetakesininputthegeometri modelandprodu esasoutputa2Dshaded
proje tionpi ture.
In order to identify the volume visualization te hniques usable for photo
realisti volumerenderingappli ation,itisne essaryto hara terizeea hstage
ofthevolumevisualizationpipeline.
Anintrodu tionofthepipeline anbefoundin[7℄and[14℄. Inthisse tionwe
introdu etheKaufmannomen latureandwegiveonoverviewofthemainstages
of the pipeline. We are interested a photo realisti hybrid volume rendering
system i.e. a framework whi h is ableto produ e2D pi tures whi h ombine
together3Dvolumetri dataandsurfa esofgeometri obje ts.
3.1 Voxelization stage
The rst stage of the pipeline is the voxelizationof the geometri model (for
examplethe3Dsolid tobemilled)wherewe omputethe3Dvoxelvolume(a
digitalanddis reterepresentationoftheobje t). Theprimarygoalofthevox-
elizationofgeometri models(3Ds an- onverting)istodeviseaframeworkfor
dis retespa e thatisa loseanalogof ontinuousspa e. Thedigitalrepresen-
tation hasto on rm the delityand onne tivityrequirements. Forexample,
adigitalsurfa eformed by voxelizationhasa\thi kness" hara terizedbythe
absen eoftunnelsorholesofa ertain onne tivity.
3.2 Manipulation stage
On e the 3D voxelimage has been re ontru ted (voxelized) or a quired by a
s annerapparataasaCTandMRIdisposal(inthis asethevoxelizationstage
is substitutesby thea quisition,enhan ementandre onstru tionstages), itis
then3D enhan edby3Dimage-pro essingte hniques (su hasmedian lter-
ingornonlinearrankordertoredu ethestair aseartifa t)inthemanipulation
ofvolumetri datastage. Themanipulationstage alsoin lude geometri trans-
formation.
Then we meet the lassi ationstage. The stage is either athresholdingstep
or amaterial lassi ationstep. Tresholding is used primarly in the surfa e-
renderingapproa h, in whi h adensityvalueoftheinterfa ebetweentwoma-
terials issele ted. All valuebelowit areasso iateda\0"(transparent,white)
value, while allthose aboveare set to \1". Forvolumerendering, non binary
material lassi ation is performed at the lassi ation stage of the pipeline.
In generalthe problem of lassifyng voxelin a datasetthat ontainsmultiple
materialisdiÆ ult. Classi ationforvolumerenderingmayassignsu hopti al
propertiesas lightattenuation (e.g. opa ity level) and brightness(e.g. olor,
emission) to ea h voxelin the dataset. The attenuation and brightnessvalue
are assigned onthe basisof thevoxel value(e.g. density),the per entuage of
material prensent in thevoxel,theposition ofthe voxel, theknowledgeabout
thematerial (e.g. bone,tissue,orfatin CTdata). Di erent lassi ersrequire
di erentkindofremderingalgorithmsandresultin di erentkindofimages.
3.4 Mapping stage
At the and of the lassi ation stage, the data rea hes the Mapping stage.
Itmapsthe3Dvoxeldataintogeometri ordisplayprimitives. Regardlessthe
originofthedataset,it anbestored,manipulated,anddisplayedusingavariaty
ofdisplay primitives. Although volumevisualizationfo useson3D primitives,
thevolumevisualizermay apitalizeontheadvantegesofthemoretraditional,
lowerdimensional primitives, to representand visualize thevariour aspe ts of
the same dataset. The hoi e of the display primitive depends on whether
volume rendering, surfa e rendering or hybrid rendering is used. There are
severalte hniquesforextra tingthedisplayprimitivefromthevolumedataset.
These in lude: the uberille model, a surfa e tra king algorithm that olle t
theexteriorvoxelfa es,avery nepolygonalmesh generatedbythemar hing
ubesalgorithm,a loudofpoints. Therearealsomanydatastru tures, losely
relatedto thedisplayprimitives,that anbeusedto representdataset. These
in lude spatial enumeration using voxel matrix, a sparse voxel matrix, or a
atvoxelmatrix, ellde ompositionasino trees,run-lenhten oding,surfa es
oftheobje ts,primitiveinstan ing, onstru tivesolid geometryandboundary
models.
3.5 Viewing stage
On ethedisplayprimitivearegeneratedtheyareproje tedtoforma2Ds reen
image by the volume viewing stage. The proje tion algorithm depends heav-
ily on the display primitive generated at the mapping stage and on whether
volumerenderingorsurfa erenderingisused. Conventionalviewing algorithm
andgeometryengines anbeexploidedtodisplay onventionalgeometri prim-
itive. However,whenvolumeprimitivesaredisplayeddire tly,aspe ialvolume
renderingalgorithmisused. Thisalgorithm apturesthe ontentsofthevoxels
outside aswell as inside the volume beingvisualized. There are various algo-
rithms used to reate a 2D proje tionfrom the 3D volumetri dataset using
volume rendering. Between them we are interested to forward (iamge-order)
proje tion algorithms, where for ea h pixel the vopxel ontribution for that
resentedbybinary(obje t/ba kground)fun tion orbyagradued(multivalued,
per entuage)fun tion. Inthelatter ategory,themultivaluedfun tion maybe
lassi ed as representing apartial overage,variable densities, and/or graded
opa ities. Binary lassi ationmanifestsitselfasvisualartifa tinthedisplayed
image. In ontrast,tehgradedfun tionapproa husesa lassi ationin partial
opa ity, olor,insteadof binary lassi atio. Theresulting oored translu emt
gelisthenproje tedby arefullyresamplingand ompositingallthevoxelsalong
aproje tionraytoformthe orrespondingpixel. Thisapproa hgenerateshigh
qualityrenderinganditisthefo usofthisreasera h.
3.6 Shading stage
The nal stage ofthevolume visualizationpipelineis theshadingstage ofthe
2D proje tion. Conventional shading te hniques an be used when geometri
primitives are the display primitives. Otherwise, a spe ial volume (dis rete)
shading te hnique, in whi h the displayed surfa e gradient is re overedfrom
thevolume dataitself, is used. Theultimategoalof thevolumeshading isto
providea2Dimagethatispra tiv allyindistiagshiblefromaphotographofthe
real obje tbut retains all its minute details whi h may be of great s ienti
ormedi al value. Unlikeshading of geometri obje ts, where the normalsare
al ulated dire tly from the geometri information, in volume shading these
normals are notavailable and have to be re overedfrom the volume data are
on ernedin thisissue.
4 VolCastIA framework
Thehighestunderstandingofthevisualizedvolumeisherea hievedbyo ering
arealisti representationofthevolumeitself|andhere omesthene essityto
explore various volume renderingte hniques,in order to rea hthe best photo
realisti results. Inthisse tionwe on entrateoure ortsinthefollwingstages:
1. lassi ationandmappingstages:
(a) gradient al ulation;
(b) opa ities lassi ation;
( ) materials lassi ation;
2. viewingandshadingstages:
(a) gradientinterpolation;
(b) opa itiesinterpolation;
( ) olors/materialsinterpolation;
(d) rayproje tion(parallel,perspe tive);
(e) raypro essing(interpolation methods, depth ueing, maximumin-
tensityproje tion,et .).
Main algorithms whi h hara terize ea h stage have been implemented in
VolCastIAin orderto evaluatetheirin uen e inthegeneration of nalimage.
Themostimportantalgorithmsarepresentedin thenextse tions.
Thegradient al ulationmethoddeterminestheshadingappearen e(see3.33.6)
ofthe nalmodel(whi hhasgotagreatin uen eintherenderingappearen e),
and,in some ases,itin uen estheopa ities lassi ationphase.
Todes ribinggradientestimationalgorithms that wehaveimplementedin
VolCastIAweusethefollowingnotation:
f(x;y;z)isthevoxelvalueatposition(x;y;z);
rf(x;y;z)isthegradientve torforthevoxel(x;y;z);
rf(x;y;z)
x
,rf(x;y;z)
y
andrf(x;y;z)
z
arethe omponentsofthegra-
dientforthevoxel(x;y;z);
krf(x;y;z)kis thegradienteu lideannormafor thevoxel(x;y;z),nor-
malized to be 0 krf(x;y;z)k 1. Thus, given rf the maximum
gradientlenghtfoundin avolumetri dataset,wehave:
krf(x;y;z)k= q
rf(x;y;z) 2
x
+rf(x;y;z) 2
y
+rf(x;y;z) 2
z
rf
Givenpreviousnotation,thegradient omponentsmaybe al ulatedby:
Centraldi eren e;
Intermediatedi eren e;
Sobelestimator.
Nextparagrafsdis uss ea hte hnique.
4.1.1 Central di eren e
The entral di eren eis oneofthe most ommongradientoperatorsand it is
usedbyMar Levoyin[11℄,anditisalsoimplementedin VolCastIA.
rf(x;y;z)
x
= 1
2
(f(x 1;y;z) f(x+1;y;z))
rf(x;y;z)
y
= 1
2
(f(x;y 1;z) f(x;y+1;z))
rf(x;y;z)
z
= 1
2
(f(x;y;z 1) f(x;y;z+1))
Hen e,its onvolutionkernelis:
1
2 0
1
2
Figure2reportsarenderingthat usesthis method.
agerenderedwiththe entraldi eren e
gradient al ulationmethod.
Figure 3: Zoom on a detail of an im-
agerenderedwiththeintermediatedif-
feren e gradient al ulationmethod. It
is possible toobservethat theimage is
moredetailedthan gure2,butitisalso
lesssmooth.
4.1.2 Intermediate di eren e
Thisisasimplegradientoperator,similartothe entraldi eren emethod. The
voxelreferen ed, wherethegradientis al ulated,isusedinthe omputation.
rf(x;y;z)
x
=f(x;y;z) f(x+1;y;z)
rf(x;y;z)
y
=f(x;y;z) f(x;y+1;z)
rf(x;y;z)
z
=f(x;y;z) f(x;y;z+1)
Its onvolutionkernelis:
1 1
The main advantages of the intermediate di eren e operator are its sim-
pli ity (it requires only a few al ulations), and its sensibility (this gradient
estimator anper eiveveryrapid voxel values hanges, and so it anshowa
numberof thinydetails in the nal image). Ontheother hand,therendering
result is aften very rough, sin e the gradient ve tors an show very frequent
(andsometimesunpleasant) hangesintheirdire tionandintensity.
Figure3reportsanimagerenderedusingtheintermediatedi eren ekonvo-
lutionkernel.
renderedwith thesobelgradient al u-
lation method. This image is morevi-
sually pleasantthantheones in gures
2 and3, at thepri e ofa lossof detail
dueto thehighinterpolation.
4.1.3 Sobelestimator
TheSobeloperator onvolutionkernel(forthez dire tion)is:
2
4
2 0 2
3 0 3
2 0 2 3
5
(forx= 1)
2
4
3 0 3
6 0 6
3 0 3 3
5
(forx=0)
2
4
2 0 2
3 0 3
3 0 3 3
5
(forx=1)
Inorderto al ulatethegradientforagivenvoxel,thisoperatorusesallthe
26voxelsinitsneighborhood. Thisdeterminesaverygoodgradientestimation,
butalsoaheavyloadduetothelargenumberofmathemati aloperationsneeded
to omplete theoperation, andalsoan e e t of\blurring"onthe nalimage,
dueto thehigh levelofinterpolationonthegradientvalues.
Figure4reportsarenderingthat usesthis method.
4.2 Transfer fun tions
In order to assign a olour or an opa ity to one or more voxel values, it is
ne essaryto de neatransferfun tion that, usingthatvalue,determinesother
hara teristi s(e. g. olour,opa ity,re e tan e,roughness...).
For this task, a tually VolCastIA relies on the program user, that reates
a on guration le ontaining the proper asso iations between isovalues and
material/opa ities.
Aninterestingoptionwouldbethepossibilitytohelptheuserinthisphase
|i. e. givingsomesortofautomationthat,aftertheanalysisofthevolumetri
dataset, de nes one or more transfer fun tions suitable for viewing the most
interestinginformationofthatdataset.
Variousexamplesofthiste hnique,withdi erentapproa hes, anbefound
in [4℄,[5℄and[8℄.
The variationof the valueopa ityin therendered model is amethod to hide
unessentialinformationonthe nal image,orto enhan eitsimportantdetails
(see3.3).
InVolCastIAopa ities lassi ationstageis omputedbyusinganexternal
data lewhi hstore,intables,theopa ityvalue
v
asso iatedwiththeisovalue
f
v
,bothde ned bytheuser.
Di erent opa ities lassi ation alghoritms are available in VolCastIA to
ompute the opa ity value of ea h voxel. Next paragraphsdes ribe methods
andparametersthatdeterminethequalityofthe nal image.
4.3.1 Linearopa ities lassi ation
Thismethodisquitesimple,andithasbeeninspiredbyDrebin,Carpenterand
Hanrahanworks[3℄. Essentially,everyvoxelisassignedwiththeopa ityofthe
isovalue itbelongs to. The voxels betweentwoisovaluesare assigned with an
opa ityderivedfrom thelinearinterpolationoftheopa itiesoftheisovaluesat
bounds|andthisinterpolationisin uen edbyaparameter0p
v
1.
Given two isovaluesf
v
n and f
v
n+1
, with theirrespe tiveopa ities
v
n and
v
n+1
,and given a\transitionvalue" t
v
n
=f
v
n +p
v
n (f
v
n+1 f
v
n
) (whi h de-
termines the beginning of the linearinterpolation between the isovalues), the
opa ity (x;y;z)forthevoxelatposition(x,y, z)isgivenby:
(x;y;z)= 8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
vn
iff
vn
f(x;y;z)<t
vn
vn+1
f(x;y;z) t
vn
fv
n+1 fv
n
+
vn
fv
n+1
f(x;y;z)
fv
n+1 fv
n
ift
vn
f(x;y;z)<f
vn+1
0 otherwise
This lassi ationmethod is quite dire t, and learly it shows thevolume
onsisten iesofthevariousregionsinsideavolumedataset. Itworkswellwhen
every voxel in the dataset is adja ent to other voxels belonging to the same
isovalue,orto the adja entones (i. e. if f
vn
f(x;y;z)<t
vn
, then t
vn
1
f(x1;y1;z1)<f
vn+1
). Anyway,itsbehaviourmustbe ompletelydriven
\byhand"(via thef
v and p
v
parameters),and there arenooptimizations for
theenhan ementof ertaintypesofinformation(apartthevolume onsisten y
itself).
Animagerenderedusingthismethodisshownin gure5.
Thenext opa ities lassi ationmethods o er some ways to extra t more
detailed stru turalinformationsfromthevolumedata.
4.3.2 Isovalues ontour surfa es lassi ation
Thismethodhasbeendes ribedbyMar Levoyin[11℄. Ithasbeenimplemented
(slightlymodi ed) in VolCastIA.Givenan isovalue f
v
and its relatedopa ity
v
,theopa ityforthevoxelatposition(x;y;z)willbe
v
whenf(x;y;z)=f
v
and rf(x;y;z) = 0. For the voxels whose values are di erent from f
v , the
ear opa ities lassi ation method. It
ispossibleto observethevolumeofthe
tissue(thathasbeenmademoretrans-
parentthanthebone).
Figure 6: Image rendered using the
region boundary opa ities lassi ation
method. Thevolumeofthetissueisal-
mostdisappeared,whilethesourfa esof
the boundary regions between twodif-
ferentmaterialsareenhan ed.
v
andf(x;y;z),andinverselyproportionaltokrf(x;y;z)k.
Therelatedexpressionis:
(x;y;z)=
v 8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
1 ifkrf(x;y;z)k=0^f(x;y;z)=f
v
1 1
rv
f
v
f(x;y;z)
krf(x;y;z)k
ifkrf(x;y;z)k>0^
f(x;y;z) r
v
krf(x;y;z)kf
v
f(x;y;z)+r
v
krf(x;y;z)k
0 otherwise
The r
v
parameter represents the thi kness of the transition from the f
v
isovalue,andin uen estherateoftheopa ityfallingo .
Whenmore than oneisovalue has to berendered, the above expression is
applied separately to ea h element,and theresults arethen ombined. Given
variousisovaluesf
v
n
;1nN;n2,withtheirrespe tiveopa ities
v
n and
transition thi knesses r
v
n
, the totalopa ity for a voxelat position (x;y;z) is
givenby:
tot
(x;y;z)=1 N
Y
n=1 (1
n
(x;y;z))
Where
n
(x;y;z)hasbeen al ulatedwiththepreviousexpression,forevery
isovalue.
This lassi ationmethodisparti ularlyusefulwhentherenderingisfo al-
izedtothestru tureofoneormoreomogeneousregionsofthevolumedataset.
Thezonesin whi hthevoxelvaluesare onstant(i. e. krf(x;y;z)k=0) and
equaltoanisovaluearerenderedwiththeirfullopa ity,whiletransitionregions
(i. e. krf(x;y;z)k>0) arerenderedwithanopa ityinverselyproportionalto
thespeedofthetransitionitself. This lassi ationenhan estheunderstanding
ofsometypesofdataset,su hastheonesderivedbyele trondensitymaps[11℄,
or,aswewillseelater,derivedfromthedis retizationofageometri fun tion.
4.3.3 Regionboundarysourfa es lassi ation
Mar Levoydes ribed this lassi ation algorithm in [11℄. It has been imple-
mented(slightlymodi ed) in VolCastIA.This method hasbeendeveloped for
therenderingofhumanbody CTdatasets.
Humantissuesare quitehomogeneous, and, forsimpli ity, it is possibleto
assume that,in agiven volume, everyoneofthem tou hes atmost twoother
types,whi hare expressedin theCTdatasetwith di erentvalues. Given this
assumption,itispossibletoimplementanopa ities lassi ationmethodthat,
insteadoftheregionsofadatasetbelongingtooneormoreisovalues,enhan es
thesurfa esattheboundaryoftheisovaluesthemselves. Ito erstophysi ians
abetterunderstandingofaCTdataset,sin etheyareoftenmoreinterestedin
thetissuesboundaryregions,ratherthanintheinternal tissues(that ouldbe
anobsta lefora learvisualization).
patientwitha heekbonefra ture.
Figure8:RenderingofaCTs anofthe
head ofa adaver. Thebeamsproje t-
ing from the mouth are an artifa t in
the dataset, and are due to dental ll-
ingss atteringX-rays.
Inordertoremoveinternaltissue lassi ation,itispossibletomakethevox-
elsopa ities (x;y;z)linearlyproportionaltothegradientmagnitudekrf(x;y;z)k.
Thisway,theinternaltissue(wherekrf(x;y;z)k!0)arerenderedas(almost)
ompletelytransparent.
Given two isovaluesf
vn and f
vn+1
, with theirrespe tiveopa ities
vn and
v
n+1
,theopa ityforavoxelatposition (x;y;z)isgivenby:
(x;y;z)=
krf(x;y;z)k
h
8
>
>
>
>
>
<
>
>
>
>
>
:
vn+1
f(x;y;z) f
vn
fv
n+1 fv
n
+
vn
fv
n+1
f(x;y;z)
fv
n+1 fv
n
iff
vn
f(x;y;z)f
vn+1
0 otherwise
In this expression, h 2 <
+
is a parameter used to make the
krf(x;y;z)k
h
fra tiontendmorerapidlyto1(or0), thusmodifyingthegradientmagnitude
in uen e on the algorithm \expanding" and highlighting (or \shrinking" and
attenuating) the surfa es dete ted at the isovalues boundary regions. After
applyingtheaboveexpression,thevaluesof (x;y;z)greaterthan1willbeset
to1.
In gures 7 and 8 it is possible to observe the renderings of two medi al
datasets, using the region boundary sourfa es lassi ation. Furthermore, in
tionmethod.
4.4 Materials lassi ation
Inthisphaseofthevolumerenderingpipelineeveryvoxelvalueisassignedwith
aper entuageofoneormorematerials,thatwillbeusedtodeterminethe nal
olourofthevoxelwhenitwillbeusedforthesamplingof viewrays.
Currently, onlyone materials lassi ation method isimplemented in Vol-
CastIA
Linear materials lassi ation
Like the linear opa ities lassi ation method seen above, this algorithm has
beeninspiredbyDrebin, Carpenter andHanrahan'sworks[3℄works,and basi-
allyworksthesameway(itisin luen edbythesamep
v
parameter).
Given twoisovalues f
vn and f
vn+1
, with theirassigned materialsm
vn and
m
vn+1
(with their olours
vn and
vn+1
),and givena\transitionvalue"t
vn
=
f
vn +p
vn (f
vn+1 f
vn
)(thesameseenbefore),the olour (x;y;z)forthevoxel
atposition(x,y,z)is givenby:
(x;y;z)= 8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
v
n
iff
v
n
f(x;y;z)<t
v
n
vn+1
f(x;y;z) tv
n
f
v
n+1 f
vn
+
v
n
fv
n+1
f(x;y;z)
f
v
n+1 f
vn
ift
v
n
f(x;y;z)<f
v
n+1
0 otherwise
4.5 Ray pro essing
Variouskindofpro essingofthedataa quiredduringtherayvolumetraversal
o erthepossibilitytoobtaindi erentresultsonthe nalimage.
4.6 Depth ueing
Thehumanper eptionofdepth inthree-dimensionals enesismostlybasedon
thevisualfo us|thatenhan estheforegroundobje ts ontours,whileblurring
theba kgroundones.
A similar help to the understanding of the depth of a three-dimensional
dataset(and,moreingeneral,ofevery omputer-synthesizedthree-dimensional
obje t) anbea hievedbyapplyingthedepth- ueinge e t:thatmeans,making
theba kgroundobje tsdarkerthantheforegroundones,simulatingthepresen e
ofauniformlight-absorbingfog.
4.7 Interpolation methods
Variousinterpolationmethodshavebeenimplementedinordertoassignavalue
toeveryraysample,basedonthevaluesoftheadja entvoxels. Thesemethods
areexplainedinthe\Interpolationmethods"se tion.
This volumerenderingte hniqueis basedon visualizingonly thesample with
the greater value found by therayduring volume traversal [10℄. Even if this
te hnique is notstri tly relatedto the realisti representation of a volume, it
ouldbeusefulto givemoreinformationabouttherendereddataset.
4.9 Sum proje tion
Thismethod isbasedonthevisualizationofthesumofallthevaluessampled
during ray traversal, with a visual e e t similar to the one given by a X-ray
image.
4.10 Interpolation methods
Everysampleofeveryrayshotinthevolumetri datasethastobe omputedby
interpolatingvariousinformation(i. e. opa itiesandmaterials/ olours) arried
by the adja ent voxels. This interpolation may be performed using various
te hniques,that usuallygiveamorerealisti resultatthepri eofanin reased
omputational omplexity.
4.11 Parallel and perspe tive rendering
The di eren ebetweenparallel and perspe tivevolume renderingis usually a
matter of performan e vs. realism: perspe tive volume rendering o ers the
maximum imageunderstanding, while theparallel rendering,at the pri eofa
slightlyper eptible lossof proportionin therepresentedobje t,o erse various
methods to enhan e the rendering speed, by using data a ess patterns that
exploit a he oheren y[13℄.
InVolCastIA,noparallelrenderingoptimizationalghoritmshavebeenimple-
mented|andthepossibilitytoobtainparallelrenderinghasbeenimplemented
in ordertoevaluatetheper eptibledi eren esbetweenthese twomethods.
4.12 Colour spa es
Inordertorea hthemaximumrealisminvolumevisualization,theVolCastIA
renderingpipelinehasbeenextended withthepossibilityto omputethe nal
image olours using various olor spa es: ommon RGB, CIE-xyz, CIE-luv,
and dire t olorspe trumanalysis andinterpolation. Thedi eren esin terms
of rendering realism and speed have been evaluated, in order to obtain the
maximumimage delity(obtainedbyusinga olourspa ethatfollowsthehuman
visualsistem responseto oloursvariations[2℄),but alsoto hooseanoptimal
trade-o betweenthesetwoaspe ts.
4.13 Hybrid rendering of volumetri and geometri data
ThestudyofvolumerenderingCAD/CAMandrapidprototypingappli ations
requires to evaluatethe di eren eswithgeometri rendering,bothin termsof
visualappearen e,andobje tsrepresentationpre ision.
various olumetri datasets: the s alar
elds of a sphere, a oni and a
torus. Thes alar eldshavebeen las-
si ed with the isosurfa es lassi ation
method, in order to enhan e the vox-
els near a spe i ed isovalue, thus ap-
proximating the original urve (e. g.
x 2
+y 2
+z 2
=0:5forthesphere).
Inthisphase,itisne essarytovisualize geometri and volumetri data,in
ordertoexaminethepossibilitytorepresentbothinaCAD/CAMorrapidpro-
totypingplanningprogram(sin egeometri obje ts ouldbeusefulto enhan e
the graphi al feedba k), and to evaluate the rendering pre ision (e. g. mea-
suring the di eren es betweena geometri ally generated sphere and its voxel
representation)( gure10).
Forthispourpose,ithasbeenimplementedamethodinspiredby[12℄,adding
toVolCastIAsomeofthe hara teristi sofaCSGprogramdevelopedinternally
byCRS4(Ray astCSG). In gure9itispossibletoobservetherenderingofa
CSG onerenderedwithavolumetri dataset,andin gure10isshownadetail
ofthe onevertex.
Referen es
[1℄ ArieE.Kaufman|ReuvenBakalash.Memoryandpro essingar hite ture
for 3d voxel-based imagery. IEEE Computer Graphi s and Appli ations,
8(11),November1988.
[2℄ AndrewS.Glassner. Prin iplesof DigitalImage Synthesis. MorganKauf-
man,1995. ISBN1-55860-276-3.
CSG one, to evaluate the di eren e
in pre ision with the oni volumetri
datasetin gure 9. TheCSG onehas
aradiusandheightof1,andis entered
intheoriginofthegeometri alreferen e
system. The volumetri dataset repre-
sentsthex 2
+y 2
(z 5) 2
=0 urve(so,
the oneand urveslopesaredi erent),
sampled in a 256x256x256grid, in the
range[ 1;1℄forea hdimension. Every
voxelhasasidelenghtof0.0078125.
Graphi s (ACMSiggraph Pro eedings),22(4):65{74,1988.
[4℄ TaosongHe,Li hanHong,ArieKaufman, andHanspeterP ster. Genera-
tionoftransferfun tionswithsto hasti sear hte hniques. InRoniYagel
and GregoryM. Nielson, editors, IEEE Visualization '96, pages227{234,
1996.
[5℄ JirHladuvka,AndreasKonig,andEduardGroller.Curvature-basedtrans-
fer fun tions for dire t volume rendering. In Bian a Fal idieno, editor,
SpringConferen e onComputer Graphi s 2000 (SCCG 2000), volume16,
pages58{65,may2000.
[6℄ D. Ja kel. Re onstru ting solids from tomographi s ans: the par um ii
system. Advan es inComputer Graphi s Hardware II,1988.
[7℄ A.Kaufman. Volume visualization. InVolumeVisualization. IEEECom-
puterSo ietyPressTutorial,1990.
[8℄ Gordon Kindlmannand JamesW. Durkin. Semi-automati generation of
transfer fun tions for dire t volume rendering. In IEEE Symposium on
VolumeVisualization,pages79{86,1998.
[9℄ GunterKnittel.Theultravissystem.Hewlett-Pa kardLaboratories, Visual
ComputingDepartment,2000.
[10℄ L. Mroz | E. Groller | A. Konig. Real-time maximum intensity pro-
je tion. In E. Groller, H. Lo elmann, and W. Ribarsky, editors, Data
Visualization '99,pages135{144.Springer-VerlagWien,1999.
[11℄ M.Levoy. Displayofsurfa esfromvolumedata.IEEEComputerGraphi s
andAppli ations,pages29{37,May1988.
[12℄ M. Levoy. A hybrid ray-tra er for rendering polygon and volume data.
IEEEComputer Graphi s andAppli ations,10(2):33{40,Mar h1990.
[13℄ P. La route| M. Levoy. Fast volume rendering using ashear-warpfa -
torization oftheviewing transformation. Pro eedings of SIGGRAPH '94,
pages451{458,July1994.
[14℄ BertholdLi htenbelt|RandyCrane|ShazNaqvi. Introdu tiontoVol-
ume Rendering. Hewlett Pa kard- Prenti eHall PTR,1998. ISBN 0-13-
861683-3.
[15℄ ClaudioT.Silva|ArieE.Kaufman|ConstantinePavlakos.Pvr: High-
performan evolumerendering. IEEE Computatonal S ien e& Engineer-
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[16℄ ToshiakiOhashi|TetsuyaU hiki|MarioTokoro. Athree-dimensional
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ROGRAPHICS1985, September1985.
[17℄ Samuel M. Goldwasser | R. Anthony Reynolds| Ted Bapty | David
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