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by: Ransom Blanda
Ransom Blanda
GPA 3.63


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This 45 page Class Notes was uploaded by Ransom Blanda on Monday October 19, 2015. The Class Notes belongs to CSCI 6962 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/224853/csci-6962-rensselaer-polytechnic-institute in ComputerScienence at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15
Sampling Aliasing amp Mipmaps Last Time CSCLEEZAdmedCmnyutnxzeyhcs Cndu Optimally eombining Importance Sampling samplingteenniqnesror Veaeh amp Guibas SIGGRAPH 1995 a Chosen uniformly b Chosen With e Weighted Within the eone of probability eombination of direetions subtende proportional a amp b by eaeh light souree to the 4 samples plxel 4 samples pixel 8 samples pixel CSCLEEZAdmedCmnyutnxzeyhcs Cndu Today What is a Pixel Examples of Aliasing Sampling amp Reconstruction Filters in Computer Graphics AntiAliasng for Texture Maps CSCLEEZAdmedCmnyutnxzeyhcs Cndu What is a Pixel More on Samples I A pixel is not 7 abox r a disk 7 ateeny tiny little light Apier looks different on different display devices I A pixel is a sample 7 it has no dimension 7 it occupies no area 7 it cannot be seen 7 it has a coordinate r it has avalue CSCLEEZAdmedCmnyutnxzeyhcs Cndu Most things in the real world are mnnnmus yet everything in a computer is dzscmte The process ofmapping acontinuous function to a discrete one is called sampling The process ofmapping acontinuous variable to adiscrete one is called quantization To represent or render an image using a computer we must both sample and quantize diserete posiaon An Image is a 2D Function Sampling Grid An 11211 Image is a cuntlnuuus functmn 1xy ufintensitzes We can generate thetable values by multiplying the cuntlnuuus n can be phase as sham eld image functzun by a sampling gnd umeneekerdelta functmns In ganeral an mags M mmmiwown mum m m 4 mtneeue We cannut be represented mm e I m t a 2 m ll lllM MWEW L ou t39w gtlll t l Emmi nlltcvwixc i l es Huw an we ll Wuhan this table7 M H ZZE1m t eseteemecwms m ESEIMZMrredcmnmwms m Sampling an lm age Questions The result is a set ofpoint samples or pixels MN m town1 hm Esclm mmadcw w ls m cscszmacmpmme m Today Examples of Aliasing What is aPixel Examples ofAliasing Sampling amp Reconstruction Filters in Computer Graphics AntiAliasng for Texture Maps Huguul lnmg sampte IlLnnmlll lmn Aliasng occurs because ofmmplmg and recumtmctzurz cs Esclm mmadcw w ls m crammmedcmnm qm m Examples of Aliasing Examples of Aliasing Jagged boundaries Improperly rendered detail 050176962 Advancedeme Graphms cm cscimz Advanced Carnme Graphics Curler Examples of Aliasing Questions Texture Errors g gi axx 050176962 Advancedeme Graphms cm point sampling cscimz Advanced Carnme Graphics Curler Today Sampling Density What is a Pixel Examples of Aliasing Sampling amp Reconstruction 7 Sampling Density 7 Fourier Analysis amp Convolution Filters in Computer Graphics AntiAliasing for Texture Maps 050176962 Advancedeme Graphms cm How densely must we sample an image in order to capture its essence If we undersample the signal we won39t be able to accurately reconstruct it cscimz Advanced Carnme Graphics Curler Sampling Density Sampling Density If we insuf ciently sample the signal it may be mistaken for something simpler during reconstruction that39s aliasing image mm RubenL Cuuk i Stuehastie Sampimg and c A T 39r 7 eemiemmaemg I I i T T T T a An lntmdumun tn Ray Tmnng Andrew Giassner ed AcadmeFmssLimited i989 W I 39 l U T T h A I cm m Aliasing in 2D because of insufficient sampling density CSCL Q ZAd medenpuuanpl ncs ijer Remember Fourier Analysis Remember Fourier Analysis All periodic signals can be represented as a quot of l sinusoidal waves l images mm hllp aan physics uh mm lr lfuunafuunerhtml Every periodic signal in the spatial domain has a dual in the frequency domain x F u gt x u Spau39ai domain frequency domain This particular signal is band limited meaning it has no frequencies above some threshold CSCL Q ZAd medenpuuanpl ncs ijer Remember Fourier Analysis We can transform from one domain to the other using the Fourier Transform frequency domain Spau39ai domain Flu 9 39jwuw39dvrb39 Invurse 39 I i quot1 Y YV I Y i Fourier I xy um e ditdi39 Transform ml 050176962 Advanced Carnme Graphics Cutler Remember Convolution iiimuliiiiuli iicniil cx Tum a mum mm mummi rmymiunuii hunk w wpiml m f1 hm j jmiu 7 nine f f 3 f images mm Mark Meyer mp www gg ealteeh eeuesi74ia Remember Convolution Sampling in the Frequency Domain I Some operations that are dif cult to compute in the spatial domain can be simpli ed by tmnsforming to its dual representation in the frequency domain I For example convolution in the spatial domain is the same as multiplication in the frequency domain fx W gt FuHu I And convolution in the frequency domain is the same as multiplication in the spatial domain F u H04 3 f W06 CSCL62ancedenputszmphws Cutler In Fauna T H Transform Original 39 r A 1 519131 39 v t u 5 quot Fauna Slnl Transform sampling O gtd 39 u l multiplication 1convolution Unix Fauna AI39tlll39AYul 7mm 1 sampled Q 39 t signal f E H I I quot 0516962 Advancedenputszm ms Cutler Reconstruction Guaranteeing Proper Reconstruction I If we can extract a copy of the original signal from the frequency domain of the sampled signal we can reconstruct the original signal I But there may be overlap between the copies LPnFurSu CSCL62ancedenputszmphws Cutler I Separate by removing high I nlll39mh39rul frequencles from the ongmal signal low pass pre ltering 3 4 MIA Kn I Separate by increasing the sampling density HuiStu I If We can39t sepamte the copies We will have overlapping frequency spectrum during reconstruction gt alim39 g 0516962 Advancedenputszm ms Cutler Sampling Theorem Questions When sampling a signal at discrete intervals the sampling frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version Shannon Nyquist CSCL62ancedenputszmphws Cutler 0516962 Advancedenputszm ms Cutler Today Filters What is aPixel Examples ofAliasing Sampling amp Reconstruction Filters in Computer Graphics 7 PreFiltering PostFiltering 7 Ideal Gaussian Box Bilinear Bicubic AntiAliasing for Texture Maps cscr m Mme cummm W m Weighting function or a convolution kem 7 Cunstantbnghmess as abject muves acmss the screen 0 No negative weightscolors 60ml cscxmzmmedcmnm W m Filters Filters are used to econstzruct a continuous signal from a sampled signal reconstruction lters 7 an limit continuous signals to avoid aliasing during sampling lowpass lters Desired frequency domain properties are the same for both types of lters Often the same lters are used as reconstruction and lowpass lters cscr m Advtrad cummm W m PreFiltering Filter eentmueus pnmmves Treat apixel as an area cempute Weighted amuunt e abject uverlap What Weighting functiun shuul a we use7 PostFiltering The Ideal Filter Filter samples Compute the Weighted average ofmany samples Regular orjittered sampling better mam mzemza cscr m Mme cummm W m Unfortunately it has m mte spatial extent 7 Every sample cunmbutes tn every inta39pulated permt Expensiveimpossible to compute I mm mm 1 cscr mammed mm W m Problems with Practical Filters Gaussian Filter Many visible artifacts in resampled images are caused by poor reconstruction lters Excessive passband attenuation results in blurry images Excessive highfrequency leakage causes quotringingquot an can accentuate tt samplin rid anisotropy quotPIES301Mquot A elmalian Highfrequellqv leamgr 050176962 Advanced c amputei Graphms Cutler This is what a CRT does for free I Pm CSCL g zAd uncedCamputJGraphms Cutler Box Filter Nearest Neighbor Tent Filter BiLinear Interpolation Pretending pixels are little squares iiaiii I 39i spatj a1 rind 2 E rimIx 7 i 39 2 V x 39 frequency 050176962 Advanced c mpwei Graphms Cutler Simple to implement Reasonably smooth CSCL g zAd uncedCamputJGraphms Cut BiCubic Interpolation Begins to approximate 39 39 39 the ideal spatial lter 39 39 f the sine function z z 04 frequency 050176962 Advanced c amputei Graphms Cutler Why is the Box Filter Bad Why is it bad to think of pixels as squares Original 11511 Downsampled with Downsampled with esolutjm image a 5x5 box lter a 5x5 Gaussian lter uniform weights nonuniform weights notice the ugly horizontal banding CSCL y lAdvancedcamputanphms Cutler Questions Today cscieiiaanmencmnurnrehrs mm What is a Pixel Examples ofAliasing Sampling amp Reconstruction Filters in Computer Graphics AntiAliasing for Texture Maps 7 Magni cation amp Mini cation eMipmaps e Anisotropic Mipmaps csci m zmmedcmm W Emu Sampling Texture Maps Linear Interpolation When texture mapping it is rare that the screen space sampling density matches the sampling density of the texture 64x64 pix els Ongnal Texaire Mignitieitianrarnispliy Minineitianrarnispliy far which we must use i recanstruetian lter cscieiiaanmencmnurnrehrs mm Tell OpenGL to use a tent filter instead of a box lter Magni cation looks better but bl e texture is underrsampled for this resolution Im csci m zmmedcmm W Emu Spatial Filtering MIP Mapping Remove the high frequencies which cause artifacts in texture miniiicatidn Compute a spatial integration overthe extent ofthe 39 1 equivalent to convolving pineeteetextureinimigepiine the texture with a lter kernel centered at the sample ie pixel center Expensive to do during rastenzatidn but an apprdin39matidn it can be precomputed t box lter intexdire plme cscieiiaanmencmnurnrehrs mm Consauet apyramid anon we compute the index ofthe deeimatedimage that is sampled at arate closest to the density of our desired sampling rate MI eans stands for mnltnm in pawn whieh m many m a smll place csci m zmmedcmm W Emu MIP Mapping Example MIP Mapping Example Thin lines may become disconnected disappear meanting my Mapped hhnear cscx mm mm cummm mm m Small details may quotpopquot in and out ofview NexustNexg uar MIP Mapped ermear gamma um mm m Examples of Aliasing Storing MIP Maps Texture Errors I fx f t39e ygigfg itnx mpm um My cscx mm mm cummm mm m Can be stored compactly Illustrates the 13 overhead ofmaintaining the MIP map Havel my map 7 Mammy Iarmu on min map Anisotropic MIPMapping What happens when the surface is tilted meanting my Mapped ermear cscx am my cummm arm m circular pler windaw mug pun Square MIPmap area is abad approximation cscxszmdcmm mm m Anisotropic MIP Mapping different Questions m r the 2 directions Additional extensions can We can use i an e non axisaligned Views m m mp Maw s9 mmsamwpw mmeayxmsmdsmim cscx mzmmad mm mm m Texture Synthesis Last Time CSCL y lAd mncedenpuuanphcs Cutler Today Texture Tiling Texture Tiling Texture Synthesis Challenge Markov Model Constrained Texture Synthesis Image Completion Wang Tiles for Texture Synthesis Volumetric Texture Synthesis CSC176962Ad mncedCampuurGrsphics cm Specify a texture coordinate uv at each vertex Canonical texture coordinates 00 4 11 quotiii u u 3 tiles with visible seams seamless tiling repeating CSClr g lAd mncedenpuhanphcs Cutler Texture Synthesis Challenge Questions input tiled synthe sis CSC176962Ad mncedCampuurGrsphics cm CSCL y lAd mncedenpuuanphcs Cutler Today Markov Random Field I Texture Tiling I Texture Synthesis Challenge I Markov Model I Constrained Texture Synthesis I Image Completion I Wang Tiles for Texture Synthesis I Volumetric Texture Synthesis cs mszmemdcmmx vmhks m I English words and sentences can be modeled as a Markov Random Field I spent an interesting evening recently with a grain afralt esemez We CW new um Fast Tmure symhees usmg Treerstructured Vector Quenuzeuen We at Levey SIGGRAFH 2999 w Neighborhood N p Template cs mszmemdcmmx vmhks m Alternate Synthesis Order Texture Synthesis by Nonparametric Sampling39 E us at Leung xccv 1999 esemez We CW new um Failure Examples x fmmWei ampLevuy esemez We CW new um Questions Today cscr am my cummm mm m Texture Tiling Texture Synthesis Challenge Markov Model Constrained Texture Synthesis Image Completion Wang Tiles for Texture Synthesis Volumetric Texture Synthesis gamma um mm m Constrained Texture Synthesis quotmm 33 Examplesfmm Efxask Lung mp Hgaphms cs cmuadvpeapleefmsresurchEfmsLemghmd cscr mm mm cummm mm an Image Inpainting Image inpmung Eemlmx a up mango Emma SIGGRAPH mun Fragnentbased image camplmanquot Image Completion D Mommas sigmanun Image Completion m i cscr am my cummm mm Questions Today csci on Mixed CuminA mam mu Texture Tiling Texture Synthesis Challenge MarxoyModel Constrained Texture Synthesis Image Completion Wang Tiles for Texture Synthesis Volumetric Texture Synthesis csciiiizaaamacmnm mm mm Wang Tile Texture Synthesis Wang Tiles 1ll II Ilil Cu 0161 mm III39I l Align tiles to match edge color I ll I to create nonperiodic tilings Wang Tiles fax image and Texoire Genznhmquot Cohen 3422 mm DCMSEH SIGGRAPH mu As aprecomputati39on ll the tiles with texture Then create in nite amounts ofnonpen39odie texture mum minus Wang Tiles fax image and Texoire Generationquot ohm Sandal Inna DCMSEH SIGGRAPH mu Questions Today csci on Mixed CuminA mam mu Texture Tiling Texture Synthesis Challenge MarxoyModel Constrained Texture Synthesis Wang Tiles for Texture Synthesis Image Completion Volumetric Texture Synthesis csciiiizaaamacmnm mm mm b sunningquTeamwesmsmmmm es39 O Jective Jaguar DaxseyampRushmexeleGGRAPIlznm Recovering Sphere Distributions Given a 2D slice through an aggregate material create a 3D volume with a comparable appearance cscxmzmmdcmm 5mm mm sire rm Rab Jagmw NA Pronie dener number or crrcies per unit area NV Panrcie density number or spneres per unit volume E Mean Caiiper particle diameter Tne fundamental relatinnsnip nfslerenlugy NA HNV cscxmzmmedcmm npms Ma Shdzfmmhblagmw Recovering Sphere Distributions Pro le Statistics 006 0000 56 9 NA K NV cscxmzmmdcmm 5mm mm sire rm Rab Jagmw Segment input rrnage to obtain pronie densitiesNA 39 I 42 1 input Segmentation Brn pronies accordrng to their area jAAm cx wzmmadcmm apm Ma Shdzfmmhblagmw Recovering Color Select mean particle colors from segmented regions in the input image Mean Coiors svnrnenc Volume cscxmzmmdcmm 5mm mm sire rm Rab Jagmw Recovering Noise How can We replicate the noisy appearance oftne rnpum input 39 Mean Culuvs Residual Tne nors resrduai rs less structured and responds Well to Heeger amp Bergen s rnetnod Svmneslzed Residual Ma cscx m Admtrad Baum 5mm Shdz rmm Rab Jagmw Putting It All Together Re sults 5mm vmume wthnm nmse cscx am my cummm mm m Svmhelm vnmme wth nmse 51m rm m Jagmw gamma um mm m shag rm m Jagmw Questions cscx am my cummm mm m Last Time Distribution Ray Tracing cscma Adm commending cudquot Today Shadows Distribution Ray Tracing one shadow ray per 7 Soft shadows intersection per point 7 Antialiasing getting rid of jaggies light source 7 Glossy re ection 7 Motion blur w V 7 Depth of eld focus MonteCarlo Integration Probabilities and variance Analysis of MonteCarlo Integration nue shadnwxay cscma Admed Curranm Gnyhcs my cscma Adm commending cum Shadows amp Light Sources Soft Shadows multiple shadow rays to sample area light source H mm mm V pmnum mum m lemngwummmlm im at shadnw rays cscma Adm commending cudquot cscma Admed Curranm Gnylrncs Cndu Antialiasing 7 Supersampling Re ection I multiple jawES mys per 39 el pnintlight Glossy Re ection multiple re ection rays i e pollshed surface rsrl aoazaamn Emu minkz mum I one re ection ray per intersection p erfect mirror rsrl aoazaamn Emmx myhus nun Motion Blur I Sample objects temporally rsrl aoazaamn Emmx myhus nun Depth of Field Ray Tracing Algorithm Analysis Ray casting cost n Lots ofprimitives urn mitives Recursive intersection cost size 0 recursive ray tree it Distributed Ray num shadow rays Tracing Effects nurn supersamples 7 Sea shadows num glossy rays um temporal samples it Antialiasing num focal sarnples e Glossy re ection 7 Motion blur 7 Depth of eld can we reduce this rsrl aoazaamn Emmx myhus nun Spatial Data Structures Questions regular gnd nested gnds entree kd tree bsp tree boundmg volume memehy etc Last Time Cloth Simulation amp Collisions 050176962Ad mm2dcampmemphcs cm CSCL Q ZAd mmedCampmnanhws cmquot Today How would you simulate a string 0 Mass Spring Systems 7 String 0 StretchStiffness Discretization Collisions lmplic it Integration 050176962Ad mm2dcampmemphcs cm 0 Each particle is linked to two particles 0 Forces try to keep the distance between particles constant 0 What force CSCL Q ZAd mmedCampmnanhws cmquot Spring forces How would you simulate a string 0 Force in the direction of the spring and proportional to difference with rest length PT Iquot 1 I K L 7 39I l 7 lt lt o n i HHHPJH 0 K is the stiffness of the spring 7 When K gets bigger the spring really wants to keep is rest length Springs link the particles 0 Springs try to keep their rest lengths and preserve the length of the string 0 Problems 7 Stretch actual length will be greater than rest length 7 Numerical oscillation How would you simulate hair Massspring Similar to string Deformation forces proponional to the angle between segments csci om Admired um um mu Interaction between panicles Create anetwoik of spring forces that link pairs ofparticles Used for string hair cloth Image by Earner witior Kass Questions Today csci om Ammo um um mu Mass Spring Systems 7 txing rHair r Cloth StretchStiffness Discretization Collisions Implicit Integration cscioooziomrrucmum mom our Three types of forces Cloth modeled with MassSpring Structural forces 7 Try to enforce invariant properties ofthe system 7 Eg force the distance between two particles to be constant 7 Ideally these should be constraints not forces Internal Deformation forces 7 Eg a string deforms a spring board tries to remain at External forces 7 Gravity etc csci om Admired um um mu Network ofmasses and springs Structural springs 7 link Li ampil and Li amp i 11 Shear springs 7 link or ampil 11 Flexion springs 7 link Li amp 2 and m amp iJZ Cloth External forces W Gravity Gm Viscous damping CV wind etc csci mzmmed cmnm mm me Provost 95 e mm Cloth simulation Then the all tn39ck is to sett Contact forces Reachm force the stiffness of all springs to get realistic motionl t Remember that forces depend on other particles coupled system But it is sparse only neighbors csci m muted cummm W 11am gm 2 contact points stay fixed What does itmean 7 Sum ofthe forces is zero How so 7 Because those point undergo external force that balances What is the force at the cont t e Depends on all other fo 7 Gravity wind etc csci mzmmed cmnm W cm Hanging cu 39 summit an the system rces in the system mm How can we compute the Contact forces Questions external Contact force 7 Inverse dynamicsl 7 Sum all other forces applied to point 7 Take negative Do we really need to compute this force ot really just ignore the other forces appliedto this point csci m muted cummm W ml cscienaMmmcmnm W mm Today lmplem enting Cloth Mass Spring Systems 7 tring rHair r Cloth StretchStiffness Discretization Collisions Implicit Integration csci m Mme cummm W m Excessive deformation t springs are not stiff enough 1mm paahm csci mzmmed mm W m The Stiffness Issue Cloth is only abit elastic shouldn t sti39etch so much We use springs where we re ly Want a constraint 7 What relative suf qess du we Want furthethe different spnngs m the netwurk7 Inverse relationship between stiffness ampAt Many um 39cal solutions 7 reduce t 7 use cunstxaints e implicit integrahun csci m Advtrad cummm W m One Solution Constrain length to increase by less than 10 w more MHz2 any proved solution simpie massspnng system lm see Pravarapl39ncslntzrface 1995 cscxmzmmedcmnpm W m The Discretization Problem Questions What happens ifWe discretize our cloth more finely or with a di erent mesh structure Do We get the same behavior Usually not It takes a lot ofeffort to design a scheme that does not depend on the discretization csci m Mme cummm W m cscxmzmmedcmnpm W m Last Time Ray Tracing cscx m Mm cummm mm m gamma um mm m Quiz Today Tuesday Oct 18Lh In class lpage ofnotes cscx m Mm cummm mm m Ray Casting r RayPlane Intersection r RaySphere Intersection Recursive Ray Tracing gamma um mm m Durer s Ray Casting Machine Ray Casting Albrecht Durer 163911 century cscx m Mm cummm mm m For avary plxe Construct a ray m tha aya and nurmmsm For avary objact n tha S p a ntersactlon wlth the ray 1 closast zap Shada dep ndlng on 11g and normal vactor Finding themtersecuun 3 central pan may casting A Note on Shading Ray Representation SurfaceScene Characteristics 7 surface immisi 7 diissuin in light t 7 Viewpmn 7 Specular Shiny More later 047mg phere spssuzm phere csci m Mimi cm W mi Two Ve ctors 7 Origin 7 Direction normalized is hem Parametric line explicit representation 7 Pt 7 origin c direction assimmimicmm W m 3D Plane Representation Explicit vs Implicit Plane de ned by 1K d gt0 0 P Pa Kw I Implicit plane equation P AxBy 0 P D PointPlane distance 7 Ifn is normalized distance to plane d HP r d is the signed distance csci m Mimi cm W mi Ray equation is explicit Pt Rn t Rd 7 Parametric 7 Generates points 7 Hard to Verify dis a point is on me is Plane equation is implicit HP nP D 0 7 Solution ofan equation 7 Does not generate points 7 Veri es dis a point is on me plane Exercise Explicit plane and implicit ray assimmimicmm W m RayPlane Intersection Additional Housekeeping Intersection means both are satis ed So insert explicit equation ofray into implicit equation ofplane amp solve fort Pt RD t Rd H nRutRdD0 csci m Mimi cm W mi Verify that intersection is closer than previous lt tsmsm Verify that it is not out ofrange behind eye Pt assimmimicmm W m Normal RayPolygon Intersection For shading diffuse dotpmduct between light and nomal Test ifintersection is in the polygon Normal is constant 7 Solve in the 2D plane nnrmal I b cscx m mm cummm mm m Rayplane intersection gamma um mm m Point InsideOutside Polygon Sphere Representation Ray intersection de nition 7 C Implicit sphere equation est my m my dm39mn 7 Assume centered at origin easy to translate m mgndxsmanzr 7 7 Cnuntintersecnnns 7 HQ 71 P 39 r2 7 lfndd number pmntls inside Works for concave and starshaped Special case fortxiangle Esclm mmsdcw w ls m cscxmzmmsdcmpm nmxs m RaySphere Intersection RaySphere Intersection Insert explicit equation ofray into implicit equation of sphere amp solve fort PtRntRd HPPPr2 0 RnthRnthr2 o awn RdRdtl ZRdRntRnRn r2 0 Quadratic at2 ht c 0 7 a 7 1 remember HRdH 1 ZRd39Rs with discriminant I 7 113 7 law Amium anm 39 and solutions Rd Ru cscx m mm cummm mm m gamma um mm m Questions Today cscx am my cummm mm m Ray Casting r RayPlane Intersection r RaySphere Intersection Recursive Ray Tracing gamma um mm m How Can We Add Shadows Mirror Re ection If no one ear Add concxxbucxon from light cscx am um um i cm ray symmetric with Spec to the normal ul p y by re ection coef cient color Re ection Amount of Re ection Re ection angle View angle Traditional ray tracing hack i RV72VNN rCunstant reflectioncolor More r 39 cscx am my cummm mm m an term mu re re ectmn at grazing angle 7 Schlick39s appmxlmatmn RSRoerolrcus er cscxmzmnmslipmms mP Elecm 355 Transparency Qualitative Refraction Cast ray in refracted direction Multiply by transparency coef cient color cscx m Adriana cummm mm pm chm Muhammw mm mtmw mu csmmhhchmm mm Refraction Refraction amp the Sidedness of Objects x Ncase7MsnS Makes M mmerhms you know whether you re ehtm39hg or leaving the transmissive material V anthem a M of 53min 1 rcas r rm rcasi smwm VLN U WHNWIJ dlzsluznmulising39mry Dun39l ugelmnnmal39nel cscxmaMmthW m What about intersecting transparent objects cscxmmhuacmhm mm m Total Internal Reflection Refraction and the Lifeguard Problem Running is faster than swimming megqu me c alar and Light m Namequot by cs 1 m Adriana cummm mm Lynch and Livinan m Questions Today cm can Advtrad swamA W ml Ray Casting e RayPlane Intersection e RaySphere Intersection 7 Point in Polygon Recursive Ray Tracing cscimemraacmnm W m Recap Ray Tracing ace ray Intersect all objects color e ament term For every llght t naaew ray sm ping Enten If chiral local shadlng tern Recursmn mm color e colorm e 7 mp ranunbe trace reflected ray beamces If transparent Ray mnmbuum color e coloru e t race transmtted ray Slaprre ected Hammad cmmbuh Dmil mrznd becamesm small cm can Advtrad swamA W ml Recursion For Re ection n retursinn l retursinn z retursinns cscimemraacmnm W m The Ray Tree L shaduw ray T transmitted re ected ray csc Complexity ism Advtrad swamA W ml Ray Debugging Visualize the ray tree for single image pixel xe ukd ny lnaew my rimmed relmeal my csci mama minn name mm Does Ray Tracing Simulate Physics Photons go from the light to the eye not the other way What we do is backward ray tracing cscrmz Advanced c ampuler Graphms Cutler Forward Ray Tracing Start from the light source 7 But low probability to reach the eye What can we do about it 7 Always send a ray to the eye still not ef cient Transparent Shadows What to do if ray to light source intersects a transparent object 7 Pretend it s opaque 7 multiply by transparency color ignores refraction amp does not produce caustics Ray Tracing is full of ding tricks Images y enrik Warm Jmsen 39 No Refraction and complex re ection for illumination are not handled properly in traditional backward ray tracing cscmzszAamuacampmxcnpms cunt What makes a Rainbow Refraction is wavelengthdependent 7 Refraction increases as thewavelength of light decreases 7 violet and blue experience more bending than orange and red Usually ignored in graphics Rainbow is caused by refraction internal re ection refraction Pink Flvyd m Dnvksde om Moon me Culur and Light Nature by Lynch and Livingstune The Rendering Equation Clean mathematical framework for light transport simulation At each point outgoing light in one direction is the integral of incoming light in all directions multiplied by re ectance property CSCL Q ZAdvancedcampulerGraphacs cmm Questions 050176962Ad mm2dcampmemphcs cm Last Time Fracture amp Deformation cscr mm mm cm W m Rigid Body Dynamics gamma um mm m Today Rigid Body Dynamics Rigid Body Dynamics Finite Element Method Deformation Could use particles for all points on the object 7 But rigid body does not deform 7 Few degrees of freedom 1 Use on y o e Fracture particle at the K v center ofmass 7 Net Torque Compute 11 Net Force amp Net Torque Net Fume cscrmammedcwms m v an Milenknvica Harald 5mm Rigid Body Dynamics ColliSions opmamaasmmm SlGGRAPH mm Physics We know how to simulate city bouncing really Well 7 Acceleration But resting collisions are 7 ar Momentum Collisions Friction mm Darren Lewis my chsgudm Stanford eduNdalewrscyMXangdbady mmi cscr mm mm c Wsm 1 2 ii Guenrtelman avmsnn a ream Nuncnnvex Rigid Homes mm Statmg SlGGRAPH Inna Que stions Today Rigid Body Dynamics Finite Element Method Deformation Fracture cscimzmadcmmmmxs mm cscimzmedcmm nmrs mm Simulation of NonRigid Objects Finite Element Method We modeled string amp cloth using massspring To solve the continuous problem sys ems Can we do me same deformation of all points ofthe object Yes 7 Discrenze the lam quot39 mess the interrelationship But a more physically accurate model uses a ahglme volumetric elements m m M1 I csci ammoma r W r mmm szwvm My mum 7 normal stress amp shear stress Strain mmmmmmmm 7 material deformation caused by stress e sured by m change inlength Al ofa um or by the change in angle r between two um cscxmzmmdcmm mm mm mm mteelements large quotEmma systEm WNW DugmmfmmDebnrmeetal 2cm Gilles Debunne Malhleu Desbvun Stram amp Stress Level of Deta1l Mawm Cm MW Eaquot warm ReaJune Delormahons usmg Space e We Adaptive Sampmg Interactive shape S GG RAW 2mm deformatio e eme deformation Q m cscxmzmadcmmmmxs Jaw 3M2 Tree Stump Multiple Materials MueHev Duvs McMwHan s b Sympusmm un Cumpme Ammanunzuuz 3 a 3 19 A 4 cscx mzmmad mm Questions gamma um mm m Today FraCture James O Enen ampJessma Hudgms GraphicsModeling and Ammanon oBIRIeFrache Rigid Body Dynamics S GGWH 1999 Finite Element Method Fm39mthreshht ld 39 Deformation 39 Remeshing 39 Material properties mere ineed connectivity info Parameter tuning cscx am my cummm mm m gamma um mm m Fracture Opening Modes Fracture Propagation mumd by i umk ml Mm Ill 0mm cscx mm mm cummm mm m K2 1quot 00 quot5 w manm mmomtiu 7 aged m3 gamma um mm m Fracture Propagation cscx am my cummm mm m L ocal Mesh Refinem ent Imagesm v mm M 1999 Managing Fracture Adj acency Questions cscx mm mm cummm mm m gamma um mm m Today Computer Animation amp Particle Systems Some slides courtesy of Jovan Popovic amp Ronen Barzel cscx the muted mumA W mu How do We specify or generate motion e Keyframing r ocedural Animation e PhysicallyBased Animation 7 Forward and Inverse Kinematics e Motion Capture What is aPaiticle System Particle System Examples Advanced Particle Systems cscxmzmmedcmpm W m K eyfram ing Procedural Animation Use sp1me curves to automate the inbetweening 7 Ga u d cuntxul e Lesstedwus than drawing every ame Creating a good animation still requires considerable skill and talent Describes the motion algorithmically as a function ofsmall number of ameters Example a clock with second minute and hour hands e express the eieek meams m terms ufa sewnds vanahie e the cluckis animated by varying the secunds ameter par Example A bouncing ball e Ahsltsmlttttenek cscxmzmmedcmpm W m PhysicallyBased Animation Articulated Models Assign physical properties to objects masses forces inertial properties Simulate physics by solving equations Realistic but dif cult to control cscx the muted mumA W mu Articulated models e connected by joints They can be animated by specifying thejoint angles as functions oftime cscxmzmmedcmpm W m Forward Kinematics Skeleton Hierarchy 0 Each bone transformation described relative memzhlquhlsh to the parent in the hierarchy q f sr 121mg qg V 97 r r lDOF knee ZDOF wnst KDOF arm CSCL Q ZAd mncedCampumanhws Cutler Given skeleton xNththrfNSh parameters p and the position qirfrrsr of the effecter in local coordinates V5 qf w t is the position y of the effector in the v world coordinates VW Qflff quot x Vw TththZhRClhvfhv5hThRCltv rstTtRQeTeR lrvfrVs Vw SPVs CSCL Q ZAd mncedCampnexGmphws Omar Inverse Kinematics 1K 1K Challenge 0 Given the position of the effecter in local coordinates Vs meszMMsh and the desired position VW in world coordinates what in 39f 39S are the skeleton parameters p qt 0 Much harder requires solving the inverse of the nonlinear Lynf vs function findp st SpV5 VW Underdeterrnined problem with many solutions CSCL Q ZA medCamp MGmphws Cutler 0 Find a natural skeleton configuration for a given collection of pose constrains 0 A scalar objective tnction gp measures the quality of apose g is minimum for most natural poses iExample gp deviation from natural pose joint stiffness power consumption etc 0 A vector constraint function Cp 0 collecm all pose constraints CSCL Q ZAd mncedCampnexGmphws Omar Motion Capture 0 Optical markers highspeed cameras triangulation 4 3D position Captures style subtle nuances and realism 0 You must observe someone do something rr How Do They Animate Movies Keyrmm ing mostly Articulated gures inverse kinematics Skinning 7 Complex deformable skin muscle skin motion Hierarchical controls 7 Smile control eye blinking etc 7 Keyhames for thesehlgherelevel controls A huge time is spent building the 3D models its skeleton and its controls Physical simulation for secondary motion 7 Hair cloths water 7 Particle systems for fuzzy objects images m the Maya maul CSCL Q ZAd mncedCampnexGmphws Omar Questions Today cscx am my cummm mm m How do We specify or generate motion What is aPatticle System 7 Describe the external forces with rlntegrate the laws ofmechanics ODE Solvers 7 Collisions later Particle System Examples Advanced Particle Systems gamma um mm m Types of Dynamics Point r39 Rigid body Deformable body include clothes uids smoke etc mg m summon What is a Particle System Collection ofmany small simple particles Particle motion in uenced byfurce elds Particles created by generator Particles often have leIZWLES Used for eg 7 sand dust smoke Sparks ame water gamma um mm m Particle Motion mass m position x Velocity V equations ofmotion I quot1 v1 TI39 A v I Ordinary Differential Equation X7 X I 7 v quot gamma um mm m Numerical solutions to ODEs Solving ODEs for animation Analytic solutions can be found for some classes of differential equations but most can t be solved analytically eg 3body problem d x I wmzm I Given a function mm compute xn Typically mxtml Value problems 7 Given values Xzx constrained problems etc cscr mm mm cummm Bums m XUgt X i ix fxJ 2quot 39 For animation Want a series ofvalues XlJ 1 A Jm 7 samples ofthe continuous function x 7 ie frames Oran mimicquot cscxmzmmedcmpm Bums m Questions Path through a field cscr mm mm cummm Bums m xt is a vector eld de ned everywhere 7 Eg a velocity eld which may change over time Xt is apath through the eld ammmcmm Bums m Higher order ODEs Eg for a 3D particle Eg Mechanics has 2nd order ODE Express as 1 1 order ODE by defining Vt xm MI 39le39tv 7 x r 39 r 39 7 r l J IVn cscr mm mm cummm Bums m We have a 6 dimension ODE problem I quot 1 y quoty 7 l39 x7 V HI HM Iquot YIquotX u HAM cscxmzmmedcmpm Bums m and for a collection of 3D particles Intuitive solution take steps w l l an A lw l cscx mm mm cummm mm m Current state X Examine fXt at or near current state Take a step to new value ofX Most solvers do some form ofthis gamma um mm mm Euler s method Effect of step size Simplest andmost intuitive De ne step sizeh Given X0Xta take step r 1 7 Xl XuIXJu Piecewiselinear approximation to the curve cscx mm mm cummm mm m Step size controls accuracy Smaller steps more closely follow curve For animation may need to take many small steps per frame Euler s method inaccurate Euler s method unstable Moves along tangent can leave curve eg lXl F Exact solution is circle Agra 1 39l W K Euler s spirals outward no matter how small h is M cscx mm mm cummm mm m mm g k l fxt l39x Exact solution is decaying exponential rutquot 39 Limited step size Alil IS II ll ll oscillnlcst Y gt zl explodes Ifkis big h must be small summmacmm mm mm Analysis Taylor series Can we do better than Euler s method Expand exact solution Xt Problem fhas varied along the step xum7xupl xunl 77xunl re Idea look at fat the arrival ofthe step and Euler s method approximates COmPe sme fquot Variam quot Nu m x r I Xrll s 1W ycrmr l 7 I 2 m mm a mar4 perslepx nuce as many smps 7 L r39rm Firstorder method Accuracy Varies with h To get 100x better accuracy need 100x more steps gt s s V cscrmmmcwms mu ESEISMMmdcmmnumxs m 2ndorder methods Compariso Euler Midpoint RungeKutta Midpoint 7 V1 Euler step 7 evaluate f 7 full step using1I Same order of accuracy cscr m Advtrad cummm W m Questions What is a force Forces can depend on location time Velocity There can be multiple force sources csummmecwme m cscxmzmmedcmpm nmxs m depends only on particl Forces Gravity on Earth emass Forces Gravity for N body problem Xt constant for smoke ame make gravity point up V O F G mmJ r1 Gravin 0 P ml m o P summarich m ESEIWZAdrmdcmnmmmxs m Depends on all otherpanicles Opposite for pairs ofpanicles Force in the direction ofppJ with magnitude inversely proportional to square distance Forces Damping Forces Spatial fields fin dvm force opposes motion removes energy so system can settle cscr mm mm cummm mm m force on panicle i depends only on Velocity ofi small amount ofdamping can stabilize solver too much damping makes motion too gluelike force on particle i depends only on position ofi 7 wind 7 attractors 7 repulsers r vortices can depend on time note these add energy may need damping too gamma um mm m eg approximate uid Lenn Forces Spatial interaction Questions 0N2 to test all pairs 7 In practice onlylocal cscr mm mm cummm mm ardJones force jxmvvm A m gamma um mm m Today Particle Animation Reeves et a1 1983 Sun trek mu l How do We speclfy or generate motlon What is aParticle System IxhoIK Particle System Examples Advanced Particle Systems cscl om Advtrad cm W m How they did it One big particle system at impact Where do Particles come from Secondary systems for rings offlre O en created by generators or emitters 7 can be attachedtu abjects lnthe mude 7 recurd lm uflast particle re t 17 LU IAM rate 7 createnpamcles updatetmg n reate with random distribution ofinitial x and v 7 lfcreatlngn gt 1 pameles at unEE spread nut un pend csclmzmmedcmnm W m Particle Lifetimes Record time of birlh for each particle Specify lifetime Use particle age to 7 remove particles from system when too old 7 o en change color 7 o en change transparency old particles fade Sometimes also remove particles that are offscreen cscl om Advtrad cm W m Rendering and Motion Blur Particle Dreams by Karl Sims Particles are usually not shaded just emission Often rendered last with z buffer disabled 0 Draw a line for motion blur x X vdt 0 Sometimes use texture maps fire clouds CSCL62AdwncedCampltexanhws Cudzx 0 1988 0 A Connection Machine CM2 computer was used to perform physical simulations on thousands of particles simultaneously one processor for each particle 0 Karl Sims quotParticle Animation and Rendering Using Data Parallel Computation SIGGRAPH l 990 0 http wwwgenarts comkarl CSCL Q ZAdvumedCampume ms omquot Particle Modeling Questions grass made from particles Reeves et al 1983 5 CSCL62AdwncedCampltexanhws Cudzx CSCL Q ZAdvumedCampume ms omquot Today Advanced Particle Systems 0 How do we specify or generate motion 0 What is a Particle System 0 Particle System Examples 0 Advanced Particle Systems CSCL62AdwncedCampltexanhws Cudzx 0 Each bird modeled as a complex particle boid 0 A set of forces control its behavior based on location of other birds and control forces 1 aboid s nexgaborhood 0 Craig Reynolds h pIHWWW red3dcomcwrboids CSCL Q ZAdvumedCampume ms omquot Advanced Particle Systems Boid is an A abbreviation of quotbirdoidquot as the Snpnmlwn Ives m mm mm lam Bedanlkl rules apply equally A to simulated ocking birds and Mmmrmu schooling fish 3amp3 ng hum Craig Reynolds Cannu Anna a rm 1mg paxnon I haul Harbin Fish School Crowds 9 Hum av Helm s Drupi LOTR Finding Nemuquot Pixzr CSCL Q ZAmedcampmexanhcs cmquot Questions CSCL Q ZAd mncedCampmemphws cm


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