Indirect Volume Rendering

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1 Indrect Volume Renderng Balázs Csébalv Deartment o Control Engneerng and Inormaton Technology Budaest Unversty o Technology and Economcs Classcaton o vsualzaton algorthms Drect Volume Renderng DVR: The renderng engne can drectly rocess the volumetrc data Indrect Volume Renderng IVR : An ntermedate reresentaton s requred by the renderng engne: 3D Fourer transorm Fourer Volume Renderng FVR Random ont cloud Monte Carlo Volume Renderng MCVR Trangular mesh o an sosurace Marchng Cubes algorthm 2 / 44 Fourer Volume Renderng Tomograhc reconstructon The nut data s a set o roectons The 3D sgnal s reconstructed rom the roecton data on the onts o a samlng lattce Fourer Volume Renderng The nverse o the tomograhc reconstructon The nut volumetrc data s the result o the tomograhc reconstructon A roecton mage or a secc vewng drecton s reroduced rom the volume data Smulated X-ray renderng 3 / 44

2 Tomograhc reconstructon 4 / 44 Radon transorm t y δ cos + y sn t ddy 5 / 44 The Fourer roecton-slce theorem The Fourer transorm o the roecton 2πωt P ω t e Fourer roecton-slce theorem: P ω F ω cos ω sn F u v y P ω e 2π u+ vy dt dudv 6 / 44 2

3 2D Fourer-transorm The 2D Fourer transorm o y: F u v 2π u + vy y e ddy The Fourer transorm o the roecton: P ω t s e 2πωt dt 7 / 44 Coordnate transormaton t cos s sn sn cos y The roecton n the transormed t s sace: t t s ds 8 / 44 The Fourer roecton-slce theorem P ω 2πωt 2πωt t s e dt t s ds e dt P ω In the orgnal y sace: 2πωt y e ddy t cos + y sn 9 / 44 3

4 The Fourer roecton-slce theorem Fourer transorm o the roecton P ω 2πω cos + ysn y e ddy Fourer roecton-slce theorem: y 2D Fourer-transorm o a slce P ω F ω cos ω sn F u v P ω e 2π u+ vy dudv 0 / 44 Fltered Bac Proecton FBP Integraton by substtuton: y u ω cos v ω sn 2π 0 0 F ω e 2πω cos + ysn ωdωd ω s the Jacoban determnant o the transormaton / 44 Fltered Bac Proecton FBP y π 0 0 π 0 0 F ω + π e F ω e 2πω cos + ysn 2πω cos + π + ysn + π ωdωd + ωdωd From the denton o the Fourer transorm we obtan: F ω + π F ω 2 / 44 4

5 Fltered Bac Proecton FBP y π 0 π 0 P ω ω e F ω ω e 2πω cos + ysn 2πω cos + ysn dωd dωd convoluton lterng where the transer uncton s ω bac roecton 3 / 44 The transer uncton o the lter 4 / 44 Fltered Bac Proecton FBP. For each angle comute the Fourer transorm P o the corresondng roecton 2. Flterng n the requency doman P s multled by ω equvalent to a convoluton n the satal doman: 2πωt q t P ω ω e dω 3. Bac roecton: π q cos + y y sn d 0 5 / 44 5

6 Dscrete aromaton. For each angle comute the dscrete Fourer transorm P o the corresondng roecton 2. Flterng n the requency doman P s multled by ω 3. Inverse dscrete Fourer transorm o the result o ste Bac roecton as a nte sum: N y q cos + y sn 6 / 44 Fourer Volume Renderng 3D Fourer transorm Slcng Along a lane erendcular to the vewng drecton the 3D Fourer transorm s resamled 2D nverse Fourer transorm Based on the Fourer roecton slce-theorem the obtaned mage s the roecton o the orgnal 3D data 7 / 44 Comlety o FVR Assume that the volume resoluton s N 3 The comlety o the 3D DFT s ON 3 logn erormed only once n a rerocessng The comlety o the 2D slcng s ON 2 a comact resamlng lter s used The comlety o the 2D nverse Fourer transorm s ON 2 logn erormed or each rame The comlety o the renderng s ractcally roortonal to the number o the els The tradtonal methods ray castng slattng vst all the voels; thereore ther comlety s ON 3 8 / 44 6

7 Practcal roblems Due to the smoothng eect o the ractcal resamlng lters the central art o the volume s overemhaszed Premultlcaton: The 3D DFT s remultled by the recrocal nverse Fourer transorm o the resamlng lter 9 / 44 Drawbacs Alha comostng s not suorted Only smulated X-ray mages Only arallel roecton Lmted ractcal alcaton 20 / 44 Deth cueng Those voels that are arther rom the mage lane are rendered wth lower ntensty The deth erceton can be mroved Deth cueng can be mlemented n the requency doman by a 2D oeraton The 3D DFT does not have to be recalculated { d } H ν F ν D ν P ν H ν F ν P ν H ν D ν F ν P ν H ' ν FT 2 / 44 7

8 Hemshercal llumnaton Classcal hemshercal llumnaton: I + N L 2 Hemshercal llumnaton or volume data: L I + + L 2 2 Evaluaton n the requency doman: 2 FT{ } + FT { } ν L H ν 22 / 44 Hemshercal llumnaton hemshercal llumnaton hemshercal llumnaton wth deth cueng 23 / 44 Monte Carlo Volume Renderng The densty uncton s ntegrated n the regon that s roected onto the gven el The Monte Carlo ntegraton s aled 24 / 44 8

9 9 25 / 44 Monte Carlo Volume Renderng The contnuous reresentaton o the densty uncton: Monte Carlo ntegraton: s the th samle o a random varable where s the robablty densty uncton o the averagng gves an unbased estmaton or the ntegral the more roortonal to the ntegrand g the lower s the varance o the estmaton mortance samlng Z h M g M g E d g d g I 26 / 44 Monte Carlo Volume Renderng The calculaton o el I v s the vsblty uncton: Partal mortance samlng s roortonal only to the densty uncton : v E d v d I V V otherwse V v 0 v V d d [ ] M V M M v M E v d v E I 27 / 44 MCVR algorthm Random ont samles are generated wth a robablty roortonal to the contnuous densty uncton rst a random voel oston s selected then a random translaton s added where the robablty densty o the translaton s roortonal to a reconstructon ernel The ont samles are roected onto the screen The densty o a el s roortonal to the number o samles roected onto the gven el

10 Comlety The varance o the estmaton can be reduced by ncreasng the number o samles The normalzed ntenstes are quantzed The standard devaton can be reduced below the level o the quantzaton error It s easy to see that the number o the necessary samles s roortonal to the number o the els rather than the number o voels In ths sense the comlety o MCVR s ON 2 whch s better than the comlety ON 2 logn o Fourer Volume Renderng 28 / 44 Convergence 29 / 44 Images rendered by MCVR M 6M 30 / 44 0

11 Shadng based on gradents Not the orgnal data values are used as a robablty densty uncton The robablty s roortonal to the gradent magntude The well-dened sosuraces are enhanced The ont samles are shaded based on ther gradents beore the roecton The comostng s a smle ntegraton The results are smlar to that o the shaded Fourer Volume Renderng 3 / 44 Shadng based on gradents M 6M 32 / 44 Marchng Cubes The ntermedate reresentaton s a trangular mesh aromatng a gven sosurace For each cubc cell t s checed whether t s ntersected by the sosurace Insde a cubc cell a local trangulaton s aled deendng on the bnary classcaton o the corner onts The obtaned geometrcal model s nteractvely rendered by the ncremental mage synthess aroach 33 / 44

12 The Marchng Cubes algorthm Seccaton o an sosurace by a threshold t Bnary classcaton o the voels volume element Trangulaton o the ntersected cells: an 8 bt long nde s calculated based on the bnary classcaton o the corner onts readng an edge lst rom a recalculated loou table LUT calculaton o ntersecton onts along the edges usng a lnear nterolaton lnear nterolaton o the normals estmated at the lattce onts 34 / 44 Classcaton nde calculaton 35 / 44 Trangulaton o the cells The 8 bt long nde addresses 256 cases There are only 4 toologcally nvarant cases the recalculated LUT contans only 4 edge lsts The other cases can be retreved by rotaton or mrrorng transormatons 36 / 44 2

13 Elementary trangulatons 37 / 44 Ambguous trangulatons 38 / 44 Alternatve trangulatons 39 / 44 3

14 Calculaton o the trangle vertces Verte calculaton: V t P P + P P t P P Normal calculaton: N t P N + P P P t N 40 / 44 A reconstructed sosurace 4 / 44 Drawbacs Huge number o trangles s generated Trangle decmaton s needed Trangles reresentng a surace regons o low curvature are contracted Octree-based herarchcal aroach Pecewse lnear aromaton the edges o the trangles are aarent rom a short dstance Iteratve smoothng o the trangular mesh curvature mnmzaton based on a enalty uncton Today the dslay o a trangular mesh s already slower than a GPU-accelerated drect volume renderng 42 / 44 4

15 Alcaton n C AD systems 43 / 44 Mechancal smulaton Fnte element analyss 44 / 44 5

Motivation. Motivation. Monte Carlo. Example: Soft Shadows. Outline. Monte Carlo Algorithms. Advanced Computer Graphics (Fall 2009)

Motivation. Motivation. Monte Carlo. Example: Soft Shadows. Outline. Monte Carlo Algorithms. Advanced Computer Graphics (Fall 2009) Advanced Comuter Grahcs Fall 29 CS 294, Renderng Lecture 4: Monte Carlo Integraton Rav Ramamoorth htt://nst.eecs.berkeley.edu/~cs294-3/a9 Motvaton Renderng = ntegraton Relectance equaton: Integrate over