Point-Based Computer Graphics

Transcription

1 Pont-Based Computer Graphcs Eurographcs 00 Tutoral T6 Organzers Markus Gross ETH Zürch Hanspeter Pfster MERL, Cambrdge Presenters Marc Alexa TU Darmstadt Markus Gross ETH Zürch Mark Pauly ETH Zürch Hanspeter Pfster MERL, Cambrdge Marc Stammnger Bauhaus-Unverstät Wemar Matthas Zwcker ETH Zürch

2 Contents Tutoral Schedule... Presenters Bographes...3 Presenters Contact Informaton...4 References...5 Project Pages...6 Tutoral Schedule 8:30-8:45 Introducton (M. Gross) 8:45-9:45 Pont Renderng (M. Zwcker) 9:45-10:00 Acquston of Pont-Sampled Geometry and Appearance I (H. Pfster) 10:00-10:30 Coffee Break 10:30-11:15 Acquston of Pont-Sampled Geometry and Appearance II (H. Pfster) 11:15-1:00 Dynamc Pont Samplng (M. Stammnger) 1:00-14:00 Lunch 14:00-15:00 Pont-Based Surface Representatons (M. Alexa) 15:00-15:30 Spectral Processng of Pont-Sampled Geometry (M. Gross) 15:30-16:00 Coffee Break 16:00-16:30 Effcent Smplfcaton of Pont-Sampled Geometry (M. Pauly) 16:30-17:15 Pontshop3D: An Interactve System for Pont-Based Surface Edtng (M. Pauly) 17:15-17:30 Dscusson (all)

3 Presenters Bographes Dr. Markus Gross s a professor of computer scence and the drector of the computer graphcs laboratory of the Swss Federal Insttute of Technology (ETH) n Zürch. He receved a degree n electrcal and computer engneerng and a Ph.D. on computer graphcs and mage analyss, both from the Unversty of Saarbrucken, Germany. From 1990 to 1994 Dr. Gross was wth the Computer Graphcs Center n Darmstadt, where he establshed and drected the Vsual Computng Group. Hs research nterests nclude physcs-based modelng, pont based methods and multresoluton analyss. He has wdely publshed and lectured on computer graphcs and scentfc vsualzaton and he authored the book "Vsual Computng", Sprnger, Dr. Gross has taught courses at major graphcs conferences ncludng SIGGRAPH, IEEE Vsualzaton, and Eurographcs. He s assocate edtor of the IEEE Computer Graphcs and Applcatons and has served as a member of nternatonal program commttees of major graphcs conferences. Dr. Gross was a papers co-char of the IEEE Vsualzaton '99 and Eurographcs 000 conferences. Dr. Hanspeter Pfster s Assocate Drector and Senor Research Scentst at MERL - Mtsubsh Electrc Research Laboratores - n Cambrdge, MA. He s the chef archtect of VolumePro, Mtsubsh Electrc's real-tme volume renderng hardware for PCs. Hs research nterests nclude computer graphcs, scentfc vsualzaton, and computer archtecture. Hs work spans a range of topcs, ncludng pont-based renderng and modelng, 3D scannng, and computer graphcs hardware. Hanspeter Pfster receved hs Ph.D. n Computer Scence n 1996 from the State Unversty of New York at Stony Brook. He receved hs M.S. n Electrcal Engneerng from the Swss Federal Insttute of Technology (ETH) Zurch, Swtzerland, n He s Assocate Edtor of the IEEE Transactons on Vsualzaton and Computer Graphcs (TVCG), member of the Executve Commttee of the IEEE Techncal Commttee on Graphcs and Vsualzaton (TCVG), and member of the ACM, ACM SIGGRAPH, IEEE, the IEEE Computer Socety, and the Eurographcs Assocaton. Mark Pauly s currently a PhD student at the Computer Graphcs Lab at ETH Zurch, Swtzerland. He s workng on pont-based surface representatons for 3D dgtal geometry processng, focusng on spectral methods for surface flterng and resamplng. Further research actvtes are drected towards multresoluton modelng, geometry compresson and texture synthess of pont-sampled objects. Dr. Marc Stammnger receved hs PhD n computer graphcs n 1999 from the Unversty of Erlangen, Germany, for hs work about fnte element methods for global llumnaton computatons. After that he worked at the Max-Planck-Insttut for Computer Scence (MPII) n Saarbrücken, Germany, where he headed the global llumnaton group. As a PostDoc n Sopha-Antpols n France he worked on the nteractve renderng and modelng of natural envronments. Snce 001 he s an assstant professor at the Bauhaus-Unversty n Wemar. Hs current research nterests are pont-based methods for complex, dynamc scenes, and nteractve global llumnaton methods. Matthas Zwcker s n hs last year of the PhD program at the Computer Graphcs Lab at ETH Zurch, Swtzerland. He has developed renderng algorthms and data 3

4 structures for pont-based surface representatons, whch he presented n the papers sessons of SIGGRAPH 000 and 001. He has also extended ths work towards hgh qualty volume renderng. Other research nterests concern compresson of pont-based data structures, acquston of real world objects, and texturng of pont-sampled surfaces. Dr. Marc Alexa leads the project group 3d Graphcs Computng wthn the Interactve Graphcs System Group, TU Darmstadt. He receved hs PhD and MS degrees n Computer Scence wth honors from TU Darmstadt. Hs research nterests nclude shape modelng, transformaton and anmaton as well as conversatonal user nterfaces and nformaton vsualzaton. Presenters Contact Informaton Dr. Markus Gross Professor Department of Computer Scence Swss Federal Insttute of Technology (ETH) CH 809 Zürch Swtzerland Phone: FAX: grossm@nf.ethz.ch Dr. Hanspeter Pfster Assocate Drector MERL - A Mtsubsh Electrc Research Lab 01 Broadway Cambrdge, MA 0139 USA Phone: (617) Fax: (617) pfster@merl.com Matthas Zwcker Department of Computer Scence Swss Federal Insttute of Technology (ETH) CH 809 Zürch Swtzerland Phone: FAX: zwcker@nf.ethz.ch Mark Pauly Department of Computer Scence Swss Federal Insttute of Technology (ETH) CH 809 Zürch 4

5 Swtzerland Phone: FAX: pauly@nf.ethz.ch Dr. Marc Stammnger Bauhaus-Unverstät Wemar Bauhausstr Wemar Germany Phone: FAX: Marc.Stammnger@meden.un-wemar.de Dr. Marc Alexa Interactve Graphcs Systems Group Technsche Unverstät Darmstadt Fraunhoferstr Darmstadt Germany Phone: FAX: alexa@grs.nformatk.tu-darmstadt.de References M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D. Levn, C. Slva. Pont set surfaces. Proceedngs of IEEE Vsualzaton 001, p. 1-8, San Dego, CA, October 001. O. Deussen, C. Coldtz, M. Stammnger, G. Drettaks, Interactve vsualzaton of complex plant ecosystems. Proceedngs of IEEE Vsualzaton 00, to appear, Boston, MA, October 00. W. Matusk, H. Pfster, P. Beardsley, A. Ngan, R. Zegler, L. McMllan, Imagebased 3D photography usng opacty hulls. Proceedngs of SIGGRAPH 00, to appear, San Antono, TX, July 00. W. Matusk, H. Pfster, A. Ngan, R. Zegler, L. McMllan, Acquston and renderng of transparent and refractve objects. Thrteenth Eurographcs Workshop on Renderng, to appear, Psa, Italy, June 00. M. Pauly, M. Gross, Spectral processng of pont-sampled geometry. Proceedngs of SIGGRAPH 001, p , Los Angeles, CA, August 001. M. Pauly, M. Gross, Effcent Smplfcaton of Pont-Sampled Surfaces. IEEE Proceedngs of Vsualzaton 00, to appear, Boston, MA, October 00. 5

6 H. Pfster, M. Zwcker, J. van Baar, M. Gross, Surfels - surface elements as renderng prmtves. Proceedngs of SIGGRAPH 000, p , New Orleans, LS, July 000. M. Stammnger, G. Drettaks, Interactve samplng and renderng for complex and procedural geometry, Renderng Technques 001, Proceedngs of the Eurographcs Workshop on Renderng 001, June 001. L. Ren, H. Pfster, M. Zwcker, Object space EWA splattng: a hardware accelerated approach to hgh qualty pont renderng. Proceedngs of the Eurographcs 00, to appear, Saarbrücken, Germany, September 00. M. Zwcker, H. Pfster, J. van Baar, M. Gross, EWA volume splattng. Proceedngs of IEEE Vsualzaton 001, p. 9-36, San Dego, CA, October 001. M. Zwcker, H. Pfster, J. van Baar, M. Gross, Surface splattng. Proceedngs of SIGGRAPH 001, p , Los Angeles, CA, August 001. M. Zwcker, H. Pfster, J. van Baar, M. Gross, EWA splattng. IEEE Transactons on Vsualzaton and Computer Graphcs, to appear. M. Zwcker, M. Pauly, O. Knoll, M. Gross, Pontshop 3D: an nteractve system for pont-based surface edtng. Proceedngs of SIGGRAPH 00, to appear, San Antono, TX, July 00 Project Pages Renderng Acquston Dynamc samplng Processng, samplng and flterng Pontshop3D 6

7 Surf. Reps. for Graphcs Pont-Based Computer Graphcs Eurographcs 00 Tutoral T6 Marc Alexa, Markus Gross, Mark Pauly, Hanspeter Pfster, Marc Stammnger, Matthas Zwcker Rase degree Mesh processng methods Subdvson schemes Herarchcal splnes Add operators Wavelets Dscrete (pont based) representatons Add connectvty Trangle meshes Polynomals... Polynomals -> Trangles Rgorous mathematcal concept Robust evaluaton of geometrc enttes Shape control for smooth shapes Advanced physcally-based modelng Pecewse lnear approxmatons Irregular samplng of the surface Forget about parameterzaton Requre parameterzaton Dscontnuty modelng Topologcal flexblty Refne h rather than p! 3 Trangle meshes Multresoluton modelng Compresson Geometrc sgnal processng 4 Trangles... Trangles -> Ponts Smple and effcent representaton Hardware ppelnes support Advanced geometrc processng s beng n sght The wdely accepted queen of graphcs prmtves From pecewse lnear functons to Delta dstrbutons Forget about connectvty Sophstcated modelng s dffcult (Local) parameterzatons stll needed Complex LOD management Compresson and streamng s hghly non-trval Pont clouds Ponts are natural representatons wthn 3D acquston systems Meshes provde an artcfcal enhancement of the acqured pont samples Remove connectvty! 5 6 1

8 Hstory of Ponts n Graphcs The Purpose of our Course s Partcle systems [Reeves 1983] Ponts as a dsplay prmtve [Whtted, Levoy 1985] Orented partcles [Szelsk, Tonnesen 199] Partcles and mplct surfaces [Wtkn, Heckbert 1994] Dgtal Mchelangelo [Levoy et al. 000] Image based vsual hulls [Matusk 000] Surfels [Pfster et al. 000] QSplat [Rusnkewcz, Levoy 000] Pont set surfaces [Alexa et al. 001] Radal bass functons [Carr et al. 001] Surface splattng [Zwcker et al. 001] Randomzed z-buffer [Wand et al. 001] Samplng [Stammnger, Drettaks 001] Opacty hulls [Matusk et al. 00] Pontshop3D [Zwcker, Pauly, Knoll, Gross 00]...? 7 I) to ntroduce ponts as a versatle and powerful graphcs prmtve II) to present state of the art concepts for acquston, representaton, processng and renderng of pont sampled geometry III) to stmulate YOU to help us to further develop Pont Based Graphcs 8 Taxonomy Mornng Schedule Renderng (Zwcker) Acquston (Pfster, Stammnger) 8:30-8:45 8:45-9:45 Introducton (M. Gross) Pont Renderng (M. Zwcker) Pont-Based Graphcs 9:45-10:00 10:00-10:30 Acquston of Pont-Sampled Geometry and Appearance I (H. Pfster) Coffee Break Representaton (Alexa) Processng & Edtng (Gross, Pauly) 10:30-11:15 11:15-1:00 Acquston of Pont-Sampled Geometry and Appearance II (H. Pfster) Dynamc Pont Samplng (M. Stammnger) 9 10 Afternoon Schedule 14:00-15:00 15:00-15:30 15:30-16:00 16:00-16:30 16:30-17:15 17:15-17:30 Pont-Based Surface Representatons (M. Alexa) Spectral Processng of Pont-Sampled Geometry (M. Gross) Coffee Break Effcent Smplfcaton of Pont-Sampled Geometry (M. Pauly) Pontshop3D: An Interactve System for Pont- Based Surface Edtng (M. Pauly) Dscusson (all) 11

9 Pont-Based Renderng Pont-Based Renderng Matthas Zwcker Computer Graphcs Lab ETH Zürch Introducton and motvaton Surface elements Renderng Antalasng Hardware Acceleraton Conclusons Pont-Based Computer Graphcs Your Name 1 Pont-Based Computer Graphcs Your Name Motvaton 1 Motvaton 1 Quake 1998 Nvda GeForce4 00 Performance of 3D hardware has exploded (e.g., GeForce4: 136 mllon vertces per second) Projected trangles are very small (.e., cover only a few pxels) Overhead for trangle setup ncreases (ntalzaton of texture flterng, rasterzaton) A smpler, more effcent renderng prmtve than trangles? Pont-Based Computer Graphcs Your Name 3 Pont-Based Computer Graphcs Your Name 4 Motvaton Modern 3D scannng devces (e.g., laser range scanners) acqure huge pont clouds Generatng consstent trangle meshes s tme consumng and dffcult Ponts as Renderng Prmtves Pont clouds nstead of trangle meshes [Levoy and Whtted 1985, Grossman and Dally 1998, Pfster et al. 000] A renderng prmtve for drect vsualzaton of pont clouds, wthout the need to generate trangle meshes? 4 mllon pts. [Levoy et al. 000] trangle mesh (wth textures) pont cloud Pont-Based Computer Graphcs Your Name 5 Pont-Based Computer Graphcs Your Name 6 1

10 Pont-Based Surface Representaton Ponts are samples of the surface The pont cloud descrbes: 3D geometry of the surface Surface reflectance propertes (e.g., dffuse color, etc.) There s no addtonal nformaton, such as connectvty (.e., explct neghborhood nformaton between ponts) texture maps, bump maps, etc. Surface Elements - Surfels Each pont corresponds to a surface element, or surfel, descrbng the surface n a small neghborhood Basc surfels: BascSurfel { poston; color; } y z x poston color Pont-Based Computer Graphcs Your Name 7 Pont-Based Computer Graphcs Your Name 8 Surfels How to represent the surface between the ponts? holes between the ponts Surfels need to nterpolate the surface between the ponts A certan surface area s assocated wth each surfel Pont-Based Computer Graphcs Your Name 9 Surfels Surfels can be extended by storng addtonal attrbutes Ths allows for hgher qualty renderng or advanced shadng effects ExtendedSurfel { poston; color; normal; radus; etc... } color radus surfel dsc normal poston Pont-Based Computer Graphcs Your Name 10 Surfels Surfels store essental nformaton for renderng Surfels are prmarly desgned as a pont renderng prmtve They do not provde a mathematcally smooth surface defnton (see [Alexa 001], pont set surfaces) Model Acquston 3D scannng of physcal objects See Pfster, acquston Drect renderng of acqured pont clouds No mesh reconstructon necessary [Matusk et al. 00] Pont-Based Computer Graphcs Your Name 11 Pont-Based Computer Graphcs Your Name 1

11 Model Acquston Samplng synthetc objects Effcent renderng of complex models Dynamc samplng of procedural objects and anmated scenes (see Stammnger, dynamc samplng) Model Acquston Processng and edtng of pont-sampled geometry [Zwcker et al. 001] [Stammnger et al. 001] spectral processng [Pauly, Gross 00] (see Gross, spectral processng) pont-based surface edtng [Zwcker et al. 00] (see Pauly, Pontshop3D) Pont-Based Computer Graphcs Your Name 13 Pont-Based Computer Graphcs Your Name 14 Pont Renderng Ppelne Pont Renderng Ppelne Pont Cloud Forward Warpng Flterng and Shadng Vsblty Framebuffer Image Reconstructon Smple, pure forward mappng ppelne Surfels carry all nformaton through the ppelne ( surfel stream ) No texture look-ups Framebuffer stores RGB, alpha, and Z Pont-Based Computer Graphcs Your Name 15 Forward Warpng Flterng and Shadng Vsblty Image Reconstructon Perspectve projecton of each pont n the pont cloud Analogous to projecton of trangle vertces homogeneous matrx-vector product perspectve dvson Pont-Based Computer Graphcs Your Name 16 Pont Renderng Ppelne Pont Renderng Ppelne Forward Warpng Flterng and Shadng Vsblty Image Reconstructon Forward Warpng Flterng and Shadng Vsblty Image Reconstructon Per-pont shadng Conventonal models for shadng (Phong, Torrance-Sparrow, reflectons, etc.) Hgh qualty antalasng s an advanced topc dscussed later n the course Vsblty and mage reconstructon s performed smultaneously Dscard ponts that are occluded from the current vewpont Reconstruct contnuous surfaces from projected ponts Pont-Based Computer Graphcs Your Name 17 Pont-Based Computer Graphcs Your Name 18 3

12 Overvew Vsblty and Image Reconstructon Forward Warpng Flterng and Shadng Vsblty Image Reconstructon wthout vsblty and mage reconstructon wth vsblty and mage reconstructon. 1. foreground pont occluded background pont surface dscontnuty ( hole ) Pont-Based Computer Graphcs Your Name 19 Pont-Based Computer Graphcs Your Name 0 Image Reconstructon Goal: avod holes Use surfel dsc radus r to cover surface completely normal surfel dsc radus r 3D object space Quad Renderng Prmtve Draw a colored quad centered at the projected pont The quad sde length s h, where h = * r * s The scalng factor s gven by perspectve projecton and vewport transformaton Hardware mplementaton: screen space OpenGL GL_POINTS colored quad projected pont y } h Pont-Based Computer Graphcs Your Name 1 x Pont-Based Computer Graphcs Your Name Projected Dsc Renderng Prmtve Project surfel dscs from object to screen space Projectng dscs results n ellpses n screen space Ellpses adapt to the surface orentaton screen space object space normal y x projected surfel dsc Pont-Based Computer Graphcs Your Name 3 y z x surfel dsc Comparson Quad prmtve Low mage qualty (prmtves do not adapt to surface orentaton) Effcent renderng Supported by conventonal 3D accelerator hardware (OpenGL GL_POINTS) Projected dsc prmtve Hgher mage qualty (prmtves adapt to surface orentaton) Not drectly supported by graphcs hardware Hgher computatonal cost Pont-Based Computer Graphcs Your Name 4 4

13 Vsblty: Z-Bufferng No blendng of renderng prmtves framebuffer z1 z1 >z{ z pxel Splattng y A splat prmtve conssts of a colored pont prmtve and an alpha mask y * = y z y x x colored pont prmtve c x alpha mask w(x,y) (often a D Gauss functon) x splat prmtve c * w(x,y) Pont-Based Computer Graphcs Your Name 5 Pont-Based Computer Graphcs Your Name 6 Splattng Splattng The fnal color c(x,y) s computed by addtve alpha blendng,.e., by computng the weghted sum color of splat c( x, y) = c w ( x, y) alpha of splat at poston (x,y) w ( x, y) Normalzaton s necessary, because the weghts do not sum up to one wth rregular pont dstrbutons w ( x, y) 1 wthout normalzaton varyng brghtness because of rregular pont dstrbuton wth normalzaton no artfacts Pont-Based Computer Graphcs Your Name 7 Pont-Based Computer Graphcs Your Name 8 Splattng Extended z-bufferng z-buffer pxel z-threshold accumulate splats surface 1 surface dscard splats surfel dsc z Pont-Based Computer Graphcs Your Name 9 Extended Z-Bufferng DepthTest(x,y) { f (abs(splat z z(x,y)) < threshold) { c(x,y) = c(x,y) + splat color w(x,y) = w(x,y) + splat w(x,y) } else f (splat z < z(x,y)) { z(x,y) = splat z c(x,y) = splat color w(x,y) = splat w(x,y) } } Pont-Based Computer Graphcs Your Name 30 5

14 Splattng Comparson Hgh Qualty Splattng mnf. ellptcal splats crcular splats wth mn. radus surface splattng Hgh qualty splattng requres careful analyss of alasng ssues Revew of sgnal processng theory Applcaton to pont renderng Surface splattng [Zwcker et al. 001] magnf. 18 x x x 19 Pont-Based Computer Graphcs Your Name 31 Pont-Based Computer Graphcs Your Name 3 Alasng n Computer Graphcs Alasng = Samplng of contnuous functons below the Nyqust frequency To avod alasng, samplng rate must be twce as hgh as the maxmum frequency n the sgnal Alasng effects: Loss of detal More patterns, jagged edges Dsntegraton of objects or patterns Alasng n Computer Graphcs Texture Mappng Scan converson of geometry Alasng n Computer Graphcs Alasng: hgh frequences n the nput sgnal appear as low frequences n the reconstructed sgnal Pont-Based Computer Graphcs Your Name 33 Pont-Based Computer Graphcs Your Name 34 Occurrence of Alasng Alasng-Free Reconstructon Spatal Doman Frequency Doman Spatal Doman Frequency Doman Spatal Doman Frequency Doman Spatal Doman Frequency Doman Pont-Based Computer Graphcs Your Name 35 Pont-Based Computer Graphcs Your Name 36 6

15 Antalasng Preflterng Band-lmt the contnuous sgnal before samplng Elmnates all alasng (wth an deal low-pass flter) Closed form soluton not avalable n general Supersamplng Rase samplng rate Reduces, but does not elmnate all alasng artfacts (n practce, many sgnals have nfnte frequences) Smple mplementaton (hardware) Resamplng dscrete nput sgnal warp resamplng dscrete output sgnal 4. Pont-Based Computer Graphcs Your Name 37 Pont-Based Computer Graphcs Your Name 38 Resamplng Flters Object Space Resamplng Flters Object Space Screen Space color poston reconstructed nput reconstructon kernels. Warp Screen Space Screen Space 4. Sample rregular spacng Pont-Based Computer Graphcs Your Name Flter Pont-Based Computer Graphcs Your Name 40 Resamplng Flters Resamplng. Warp Object Space Screen Space warped reconstructon kernel Screen Space sum of resamplng flters Screen Space resamplng flters 4. Sample Resamplng n the context of surface renderng Dscrete nput functon = surface texture (dscrete D functon) Warpng = projectng surfaces to the mage plane (D to D projectve mappng) low-pass flter convoluton 3. Flter Pont-Based Computer Graphcs Your Name 41 Pont-Based Computer Graphcs Your Name 4 7

16 D Reconstructon Kernels Warpng a D reconstructon kernel s equvalent to projectng a surfel dsc wth alpha mask y screen space x warped reconstructon kernel y object space Pont-Based Computer Graphcs Your Name 43 z x normal surfel dsc wth alpha mask = reconstructon kernel Resamplng Flters A resamplng flter s a convoluton of a warped reconstructon flter and a low-pass flter warped reconstructon kernel screen space pxel grd convoluton low-pass flter (determned by pxel grd) no nformaton falls nbetween the pxel grd resamplng flter ( blurred reconstructon kernel ) Pont-Based Computer Graphcs Your Name 44 Mathematcal Formulaton 1 c ( x, y) = c r ( m ( x, y)) h( x, y) k k k pxel color warpng functon low pass flter reconstructon kernel reconstructon kernel color Gaussan Resamplng Flters Gaussans are closed under lnear warpng and convoluton Wth Gaussan reconstructon kernels and low-pass flters, the resamplng flter s a Gaussan, too Effcent renderng algorthms (surface splattng [Zwcker et al. 001]) Pont-Based Computer Graphcs Your Name 45 Pont-Based Computer Graphcs Your Name 46 Mathematcal Formulaton Mathematcal Formulaton 1 c ( x, y) = c r ( m ( x, y)) h( x, y) k k k Gaussan reconstructon kernel Gaussan low-pass flter 1 c ( x, y) = c r ( m ( x, y)) h( x, y) k k = k c kgk ( x, y) k Gaussan resamplng flter screen space screen space Pont-Based Computer Graphcs Your Name 47 Pont-Based Computer Graphcs Your Name 48 8

17 Algorthm for each pont P { project P to screen space; shade P; determne resamplng kernel G; splat G; } for each pxel { normalze; } Propertes of D Resamplng Flters warped reconstructon kernel low-pass flter resamplng flter mnfcaton Pont-Based Computer Graphcs Your Name 49 magnfcaton Pont-Based Computer Graphcs Your Name 50 Hardware Implementaton Based on the object space formulaton of EWA flterng Implemented usng textured trangles All calculatons are performed n the programmable hardware (extensve use of vertex shaders) Presented at EG 00 ([Ren et al. 00]) Surface Splattng Performance Software mplementaton splats/sec on 866 MHz PIII splats/sec on GHz P4 Hardware mplementaton [Ren et al. 00] Uses texture mappng and vertex shaders splats/sec on GeForce4 T 4400 Pont-Based Computer Graphcs Your Name 51 Pont-Based Computer Graphcs Your Name 5 Conclusons Ponts are an effcent renderng prmtve for hghly complex surfaces Ponts allow the drect vsualzaton of real world data acqured wth 3D scannng devces Hgh performance, low qualty pont renderng s supported by 3D hardware (tens of mllons ponts per second) Hgh qualty pont renderng wth ansotropc texture flterng s avalable 3 mllon ponts per second wth hardware support 1 mllon ponts per second n software Antalasng technque has been extended to volume renderng Applcatons Drect vsualzaton of pont clouds Real-tme 3D reconstructon and renderng for vrtual realty applcatons Hybrd pont and polygon renderng systems Renderng anmated scenes Interactve dsplay of huge meshes On the fly samplng and renderng of procedural objects Pont-Based Computer Graphcs Your Name 53 Pont-Based Computer Graphcs Your Name 54 9

18 Future Work Dedcated renderng hardware Effcent approxmatons of exact EWA splattng Renderng archtecture for on the fly samplng and renderng References [Levoy and Whtted 1985] The use of ponts as a dsplay prmtve, techncal report, Unversty of North Carolna at Chapel Hll, 1985 [Heckbert 1986] Fundamentals of texture mappng and mage warpng, Master s Thess, 1986 [Grossman and Dally 1998] Pont sample renderng, Eurographcs workshop on renderng, 1998 [Levoy et al. 000] The dgtal Mchelangelo project, SIGGRAPH 000 [Rusnkewcz et al. 000] Qsplat, SIGGRAPH 000 [Pfster et al. 000] Surfels: Surface elements as renderng prmtves, SIGGRAPH 000 [Zwcker et al. 001] Surface splattng, SIGGRAPH 001 [Zwcker et al. 00] EWA Splattng, to appear, IEEE TVCG 00 [Ren et al. 00] Object space EWA splattng: A hardware accelerated approach to hgh qualty pont renderng, Eurographcs 00 Pont-Based Computer Graphcs Your Name 55 Pont-Based Computer Graphcs Your Name 56 10

19 Acquston of Pont-Sampled Geometry and Appearance Hanspeter Pfster, MERL The Goal: To Capture Realty Fully-automated 3D model creaton of real objects. Fathful representaton of appearance for these objects. Wojcech Matusk, MIT Addy Ngan, MIT Paul Beardsley, MERL Remo Zegler, MERL Leonard McMllan, MIT Pont-Based Computer Graphcs Hanspeter Pfster, MERL 1 Pont-Based Computer Graphcs Hanspeter Pfster, MERL Image-Based 3D Photography An mage-based 3D scannng system. Handles fuzzy, refractve, transparent objects. Robust, automatc Pont-sampled geometry based on the vsual hull. Objects can be rendered n novel envronments. Prevous Work Actve and passve 3D scanners Work best for dffuse materals. Fuzzy, transparent, and refractve objects are dffcult. BRDF estmaton, nverse renderng Image based modelng and renderng Reflectance felds [Debevec et al. 00] Lght Stage system to capture reflectance felds Fxed vewpont, no geometry Envronment mattng [Zongker et al. 99, Chuang et al. 00] Capture reflectons and refractons Fxed vewpont, no geometry Pont-Based Computer Graphcs Hanspeter Pfster, MERL 3 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 4 Outlne Overvew System Geometry Reflectance Renderng Results The System Lght Array Cameras Mult-Color Montors Rotatng Platform Pont-Based Computer Graphcs Hanspeter Pfster, MERL 5 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 6

20 Outlne Overvew System Geometry Reflectance Renderng Results Acquston For each vewpont ( 6 cameras x 7 postons ) Alpha mattes Use multple backgrounds [Smth and Blnn 96] Reflectance mages Pctures of the object under dfferent lghtng (4 lghts x 11 postons) Envronment mattes Use smlar technques as [Chuang et al. 000] Pont-Based Computer Graphcs Hanspeter Pfster, MERL 7 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 8 Geometry Opacty Hull Vsual hull augmented wth vew-dependent opacty. Approxmate Geometry The approxmate vsual hull s augmented by radance data to render concavtes, reflectons, and transparency. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 9 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 10 Geometry Example Surface Lght Felds A surface lght feld s a functon that assgns a color to each ray orgnatng on a surface. [Wood et al., 000] Pont-Based Computer Graphcs Hanspeter Pfster, MERL 11 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 1

21 Shadng Algorthm A vew-dependent strategy. Color Blendng Blend colors based on angle between vrtual camera and stored colors. Unstructured Lumgraph Renderng [Buehler et al., SIGGRAPH 001] Vew-Dependent Texture Mappng [Debevec, EGRW 98] Pont-Based Computer Graphcs Hanspeter Pfster, MERL 13 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 14 Pont-Based Renderng Pont-based renderng usng LDC tree, vsblty splattng, and vew-dependent shadng. Geometry Opacty Hull Store the opacty of each observaton at each pont on the vsual hull [Matusk et al. SIG00]. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 15 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 16 Geometry Opacty Hull Example Assgn vew-dependent opacty to each ray orgnatng on a pont of the vsual hull. A B C (θ,φ) φ Photo Surface Lght Feld A B C Red = nvsble Whte = opaque Black = transparent Pont-Based Computer Graphcs Hanspeter Pfster, MERL 17 θ Vsual Hull Opacty Hull Pont-Based Computer Graphcs Hanspeter Pfster, MERL 18

22 Results Pont-based renderng usng EWA splattng, A-buffer blendng, and edge antalasng. Opacty Hull Dscusson Vew dependent opacty vs. geometry trade-off. Smlar to radance vs. geometry trade-off. Sometmes acqurng the geometry s not possble (e.g. resoluton of the acquston devce s not adequate). Sometmes representng true geometry would be very neffcent (e.g. har, trees). Opacty hull stores the macro effect. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 19 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 0 Pont-Based Models No need to establsh topology or connectvty. No need for a consstent surface parameterzaton for texture mappng. Represent organc models (feather, tree) much more readly than polygon models. Easy to represent vew-dependent opacty and radance per surface pont. Outlne Overvew Prevous Works Geometry Reflectance Renderng Results Pont-Based Computer Graphcs Hanspeter Pfster, MERL 1 Pont-Based Computer Graphcs Hanspeter Pfster, MERL Lght Transport Model Assume llumnaton orgnates from nfnty. The lght arrvng at a camera pxel can be descrbed as: C(x,y) E W C ( x, y) = W ( ω) E( ω) dω Ω - the pxel value - the envronment -the reflectance feld Surface Reflectance Felds 6D functon: ω ω ω r W P, ω, ω ) = W ( u, v ; θ, Φ ; θ, Φ ) ( r r r r r P Pont-Based Computer Graphcs Hanspeter Pfster, MERL 3 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 4

23 Reflectance Functons For each vewpont, 4D functon: W xy ( ω ) = W ( x, y; θ, Φ ) Reflectance Feld Acquston We separate the hemsphere nto hgh resoluton Ω h and low resoluton Ω l [Matusk et al., EGRW00]. (θ,φ ) φ T Ωh Ω l Pont-Based Computer Graphcs Hanspeter Pfster, MERL 5 θ L(ω C( x, y) = Wh ( ξ ) T ( ξ ) dξ + W ) l ( ω) L( ω ) dω Ωh Pont-Based Computer Graphcs Hanspeter Pfster, MERL 6 Ωl Acquston For each vewpont ( 6 cameras x 7 postons ) Alpha mattes Use multple backgrounds [Smth and Blnn 96] Reflectance mages Low resoluton Pctures of the object under dfferent lghtng (4 lghts x 11 postons) Envronment mattes Hgh resoluton Use smlar technques as [Chuang et al. 000] Pont-Based Computer Graphcs Hanspeter Pfster, MERL 7 Low-Resoluton Reflectance Feld C( x, y) = W l sampled by takng pctures wth each lght turned on at a tme [Debevec et al 00]. Ωh W ( ξ ) T ( ξ dξ + h ) W ( ω ) L( ω ) dω l Ω = 1 l Pont-Based Computer Graphcs Hanspeter Pfster, MERL 8 n Ωl W L W ( ω ) L( ω ) dω l for n lghts Compresson Subdvde mages nto 8 x 8 pxel blocks. Keep blocks contanng the object (avg. compresson 1:7) PCA compresson (avg. compresson 1:10) PCA a 0 a 1 a a 3 a 4 a 5 Hgh-Resoluton Reflectance Feld C( x, y) = Ωh W ( ξ ) T ( ξ dξ + h ) W ( ω ) L( ω ) dω Use technques of envronment mattng [Chuang et al., SIGGRAPH 00]. Approxmate W h by a sum of up to two Gaussans: N G Reflectve G 1. 1 Refractve G. Ωl l 1 1 G W h ( ξ ) = a G + a G Pont-Based Computer Graphcs Hanspeter Pfster, MERL 9 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 30

24 Surface Reflectance Felds Work wthout accurate geometry. Surface normals are not necessary. Capture more than reflectance: Inter-reflectons Subsurface scatterng Refracton Dsperson Non-unform materal varatons Smplfed verson of the BSSRDF [Debevec et al., 00]. Outlne Overvew Prevous Works Geometry Reflectance Renderng Results Pont-Based Computer Graphcs Hanspeter Pfster, MERL 31 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 3 Renderng Input: Opacty hull, reflectance data, new envronment Create radance mages from envronment and low-resoluton reflectance feld. Reparameterze envronment mattes. Interpolate data to new vewpont. 1 st Step: Relghtng Ω l Compute radance mage for each vewpont. x Downsample New Illumnaton = The sum s the radance mage of ths vewpont n ths envronment. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 33 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 34 nd Step: Reproject Ω h Project envronment mattes onto the new envronment. Envronment mattes acqured was parameterzed on plane T (the plasma dsplay). We need to project the Gaussans to the new envronment map, producng new Gaussans. Ω h T 3 rd Step: Interpolaton From new vewpont, for each surface pont, fnd four nearest acqured vewponts. Store vsblty vector per surface pont. Interpolate usng unstructured lumgraph nterpolaton [Buehler et al., SIGGRAPH 01] or vewdependent texture mappng [Debevec 96]. Opacty. Contrbuton from low-res reflectance feld (n the form of radance mages). Contrbuton from hgh-res reflectance feld. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 35 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 36

25 3 rd Step: Interpolaton For low-res reflectance feld, we nterpolate the RGB color from the radance mages. For hgh-resoluton reflectance feld: V1 Interpolate drecton of ~ N reflecton/refracton. ~ V Interpolate other parameters of the ~ Gaussans. ~ Convolve wth the envronment. G 1r G r Gt G 1t Outlne Overvew Prevous Works Geometry Reflectance Renderng Results Pont-Based Computer Graphcs Hanspeter Pfster, MERL 37 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 38 Results Results Performance for 6x7 = 43 vewponts 337,84 mages taken n total!! Acquston (47 hours) Alpha mattes 1 hour Envronment mattes 18 hours Reflectance mages 8 hours Processng Opacty hull ~ 30 mnutes PCA Compresson ~ 0 hours (MATLAB, unoptmzed) Renderng ~ 5 mnutes per frame Sze Opacty hull ~ MB Envronment mattes ~ GB Reflectance mages ~ Raw 370 GB / Compressed - 4 GB Pont-Based Computer Graphcs Hanspeter Pfster, MERL 39 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 40 Results Results Hgh-resoluton Ω h Low-resoluton Ωl Combned Pont-Based Computer Graphcs Hanspeter Pfster, MERL 41 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 4

26 Results Results Ω h Pont-Based Computer Graphcs Hanspeter Pfster, MERL 43 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 44 Results Ω l Results Combned Pont-Based Computer Graphcs Hanspeter Pfster, MERL 45 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 46 Results Results Pont-Based Computer Graphcs Hanspeter Pfster, MERL 47 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 48

27 Conclusons A fully automatc system that s able to capture and render any type of object. Opacty hulls combned wth lghtfelds / surface reflectance felds provde realstc 3D graphcs models. Pont-based renderng offers easy surface parameterzaton of acqured models. Separaton of surface reflectance felds nto hghand low-resoluton areas s practcal. New renderng algorthm for envronment matte nterpolaton. Future Drectons Use more than Gaussans for the envronment mattes. Better compresson. Real-tme renderng. Pont-Based Computer Graphcs Hanspeter Pfster, MERL 49 Pont-Based Computer Graphcs Hanspeter Pfster, MERL 50 Acknowledgements Colleagues: MIT: Chrs Buehler, Tom Buehler. MERL: Bll Yerazuns, Darren Legh, Mchael Stern. Thanks to: Davd Tames, Jennfer Roderck Pfster. NSF grants CCR and EIA Papers avalable at: Pont-Based Computer Graphcs Hanspeter Pfster, MERL 51

28 motvaton dynamc pont samplng tree created by AMAP 150,000 trangles 8 fps Marc Stammnger Pont-Based Computer Graphcs Pont-Based Computer Graphcs Marc Stammnger motvaton pont renderng ppelne level of detal 100 trees 70,000 ponts 0 fps scene descrpton vrml fle mgf fle procedural model pont set (3D-coordnates, normal, materal) screen pont generaton pont renderng Pont-Based Computer Graphcs Marc Stammnger 3 Pont-Based Computer Graphcs Marc Stammnger 4 pont generaton Surfels (Pfster et al., SIG000) (orthographc) vews Q-Splat (Rusnkewcz et al.,sig000) fltered trangle mesh herarchy Randomzed z-buffer (Wand et al., SIG001) random ponts pont renderng n software flterng texturng hole fllng n hardware as ponts as polygonal dsks as splats Pont-Based Computer Graphcs Marc Stammnger 5 Pont-Based Computer Graphcs Marc Stammnger 6 1

29 our approach dynamc pont generaton for procedural objects terrans complex dynamc objects pont renderng wth OpenGL s GL_POINT very fast (> 10 7 ponts per second) OpenGL does lghtng results ponts are well suted for procedural geometry Pont-Based Computer Graphcs Marc Stammnger 7 Pont-Based Computer Graphcs Marc Stammnger 8 results ponts are well suted for procedural geometry terrans results ponts are well suted for procedural geometry terrans complex geometry Pont-Based Computer Graphcs Marc Stammnger 9 Pont-Based Computer Graphcs Marc Stammnger 10 results ponts are well suted for procedural geometry terrans complex geometry combnatons results ponts are well suted for procedural geometry terrans complex geometry combnatons eco systems Pont-Based Computer Graphcs Marc Stammnger 11 Pont-Based Computer Graphcs Marc Stammnger 1

30 complex polygonal geometry generate lst of randomly dstrbuted samples for every frame: compute n, render the frst n complex polygonal geometry easy speed / qualty trade off frame rate control 100,000 10,000 1, ,000 10,000 1,000 Pont-Based Computer Graphcs Marc Stammnger 13 Pont-Based Computer Graphcs Marc Stammnger 14 sample denstes adapt pont denstes to mage space (D) or: adapt to post-perspectve space (3D) denstes complex geometry world space -> post-perspectve: area decreases by squared dstance goal: unform post-perspectve pont densty pont number ~ area/d Pont-Based Computer Graphcs Marc Stammnger 15 Pont-Based Computer Graphcs Marc Stammnger 16 modfed complex geometry smple modfcatons on the fly complex geometry vdeo complex geometry download at 30 fps Pont-Based Computer Graphcs Marc Stammnger 17 Pont-Based Computer Graphcs Marc Stammnger 18 3

31 dsplaced geometry dsplaced geometry 5,000 ponts 5,000 ponts 5,000 ponts 100,000 ponts Pont-Based Computer Graphcs Marc Stammnger 19 Pont-Based Computer Graphcs Marc Stammnger 0 adaptve samplng undersamplng factor < 1 > 1 undersamplng n D mage space, not post-perspectve! Pont-Based Computer Graphcs Marc Stammnger 1 Pont-Based Computer Graphcs Marc Stammnger undersamplng factor adaptve pont generaton adaptve sample pattern Pont-Based Computer Graphcs Marc Stammnger 3 Pont-Based Computer Graphcs Marc Stammnger 4 4

32 5 samplng 5 samplng (/5,1/5) ntal samples, all undersampled newly nserted samples ntal samples, all undersampled newly nserted samples Pont-Based Computer Graphcs Marc Stammnger 5 Pont-Based Computer Graphcs Marc Stammnger 6 5 samplng 5 samplng undersampled samples newly nserted samples newly nserted samples Pont-Based Computer Graphcs Marc Stammnger 7 Pont-Based Computer Graphcs Marc Stammnger 8 5 samplng 5 samplng undersampled samples newly nserted samples newly nserted samples Pont-Based Computer Graphcs Marc Stammnger 9 Pont-Based Computer Graphcs Marc Stammnger 30 5

33 5 samplng procedural modfers rotated, nested grds grd dstance decreases by 1/sqrt(5) rotaton angle 7 o specal attenton to boundares orgnal geometry: square orgnal geometry: truncated cone Pont-Based Computer Graphcs Marc Stammnger 31 Pont-Based Computer Graphcs Marc Stammnger 3 vdeo terrans vdeo 5 samplng download at Pont-Based Computer Graphcs Marc Stammnger 33 Pont-Based Computer Graphcs Marc Stammnger 34 terrans terran parameterzaton parameterze sector by (d,u) u d terran d u screen Pont-Based Computer Graphcs Marc Stammnger 35 Pont-Based Computer Graphcs Marc Stammnger 36 6

34 terran parameterzaton lookng straght ahead 1 d( v) = v dmn dmax d lookng up mn lookng down terran algorthm 5 samplng scheme undersamplng factor parameterzaton dstortons perspectve dstortons dsplacement Pont-Based Computer Graphcs Marc Stammnger 37 Pont-Based Computer Graphcs Marc Stammnger 38 terran occluson cullng terran occluson cullng elevaton drecton n mage space along v smplfes occluson cullng elevaton occluson cullng, regular samplng occluson cullng, wth adaptve samplng Pont-Based Computer Graphcs Marc Stammnger 39 Pont-Based Computer Graphcs Marc Stammnger 40 vdeo vdeo terran renderng download at eco systems level of detal: polygonal model replace polygons by ponts and lnes reduce number of ponts and lnes Pont-Based Computer Graphcs Marc Stammnger 41 Pont-Based Computer Graphcs Marc Stammnger 4 7

35 eco systems example ponts lnes eco systems modeller (xfrog) delvers: trangle set T p random pont set representng T p trangle sett l randomlne setl representng T l ( L < T l ) polygons Pont-Based Computer Graphcs Marc Stammnger 43 Pont-Based Computer Graphcs Marc Stammnger 44 eco systems eco systems level-of-detal 1 level-of-detal trangles trangles ponts lnes ponts lnes trangles ponts lnes ponts lnes trangles ponts lnes ponts lnes trangles ponts lnes Pont-Based Computer Graphcs Marc Stammnger 45 Pont-Based Computer Graphcs Marc Stammnger 46 eco systems crteron for pont / lne number (per object) user parameter: pont sze d p / lne wdth d l approxmate screen space area of object: A = A * 0.5 / d #ponts~ A / d p #lnes~ A / d p eco systems vdeo eco system renderng download at Pont-Based Computer Graphcs Marc Stammnger 47 Pont-Based Computer Graphcs Marc Stammnger 48 8

36 Motvaton Surfaces from Pont Samples Marc Alexa TU Darmstadt Many applcatons need defnton of surface based on pont samples Reducton Up-samplng Interrogaton (e.g. ray tracng) Desrable surface propertes Manfold Smooth Local (effcent computaton) Pont-Based Computer Graphcs Pont-Based Computer Graphcs Marc Alexa Overvew Introducton & Bascs Fttng Implct Surfaces Projecton-based Surfaces Introducton & Bascs Regular/Irregular Approxmaton/Interpolaton Global/Local Standard technques LS, RBF, MLS Problems Sharp edges, feature sze/nose Functonal/Manfold Pont-Based Computer Graphcs Marc Alexa 3 Pont-Based Computer Graphcs Marc Alexa 4 Regular/Irregular Regular Requres to store only values Approxmaton/Interpolaton Nosy data -> Approxmaton Irregular Requres to store locatons p Perfect data -> Interpolaton p x p y Pont-Based Computer Graphcs Marc Alexa 5 Pont-Based Computer Graphcs Marc Alexa 6 1

37 Global/Local Global approxmaton Local approxmaton Least Squares Fts a prmtve to the data Mnmzes squared dstances between the p s and prmtve g g ( x) = a + bx + cx Localty comes at the expense of smoothness Pont-Based Computer Graphcs Marc Alexa 7 mn g ( p g( p ) y x Pont-Based Computer Graphcs Marc Alexa 8 Least Squares - Example Prmtve s a polynomal mn 0 = ( 1, x, x T ) c ( 1, p, p,...) g( x) =,... T ( p c ) y x x j T p ( p ( 1, p, p,...) c ) x y Lnear system of equatons Pont-Based Computer Graphcs Marc Alexa 9 x x Least Squares - Example Resultng system j 0 = p p ( 1, p, p,...) c x y x x 1 x x M x x x 3 T ( ) x x x 3 4 K c0 y c1 yx = c yx O M M Pont-Based Computer Graphcs Marc Alexa 10 Movng Least Squares Compute a local LS approxmaton at t Weght data ponts based on dstance to t Movng Least Squares The set f t = g ( t), g : mn p g p θ () ( ( ) ( t p ) t t g y x s a smooth curve, ff θ s smooth x ( p g( p ) ( t p ) mn θ y x t x g ( x) = a + bx + cx Pont-Based Computer Graphcs Marc Alexa 11 Pont-Based Computer Graphcs Marc Alexa 1

38 Movng Least Squares Typcal choces for θ: r θ( d ) = d d / h θ d = e ( ) ( ) Note: θ s fxed = θ t p x For each t Standard weghted LS problem Lnear ff correspondng LS s lnear Radal Bass Functons Representnterpolantas Sum of radal functons r Centered at the data ponts p f x = w r p x ( ) ( ) Pont-Based Computer Graphcs Marc Alexa 13 Pont-Based Computer Graphcs Marc Alexa 14 Radal Bass Functons Solvep jy ( p p ) = w r x jx to compute weghts w Lnear system of equatons r r r( 0) r( p0 p ) ( ) x 1 r p x 0 p x x ( p1 p ) ( ) ( ) x 0 r 0 r p p x 1x x ( p p ) r( p p ) r( 0) x M 0x x 1x L w0 p0 y w1 p1 y = w p y O M M Radal Bass Functons Solvablty depends on radal functon Several choces assure solvablty r d = d log (thn plate splne) ( ) d ( ) / h d r d = e (Gaussan) h s a data parameter h reflects the feature sze or antcpated spacng among ponts Pont-Based Computer Graphcs Marc Alexa 15 Pont-Based Computer Graphcs Marc Alexa 16 Typcal Problems Sharp corners/edges Functonal/Manfold Standard technques are applcable f data represents a functon Nose vs. feature sze Manfolds are more general Pont-Based Computer Graphcs Marc Alexa 17 Pont-Based Computer Graphcs Marc Alexa 18 3

39 Implcts Each orentable n-manfold can be embedded n n+1 space Idea: Represent n-manfold as zeroset of a scalar functon n n+1 space Insde: On the manfold: f ( x) < 0 f ( x) = 0 Outsde: f x > ( ) 0 Implcts - Illustraton Image courtesy Greg Turk Pont-Based Computer Graphcs Marc Alexa 19 Pont-Based Computer Graphcs Marc Alexa 0 Implcts from pont samples Implcts from pont samples Functon should be zero n data ponts f ( p ) = 0 Use standard approxmaton technques to fnd f Trval soluton: f = 0 Addtonal constrants are needed 0 Constrants defne nsde and outsde Smple approach (Turk, O Bren) Sprnkle addtonal nformaton manually Make addtonal nformaton soft constrants Pont-Based Computer Graphcs Marc Alexa 1 Pont-Based Computer Graphcs Marc Alexa Implcts from pont samples Use normal nformaton as + constrant + f ( p + n ) = 1 Normals could be + computed from scan Or, normals have to be estmated Estmatng normals Two problems Normal drecton and Orentaton n (Implcts are sgned!) Normal drecton by fttng a tangent LS ft to nearest neghbors Weghted LS ft MLS ft q Pont-Based Computer Graphcs Marc Alexa 3 Pont-Based Computer Graphcs Marc Alexa 4 4

40 Estmatng normals General fttng problem mn q p, n θ q, p n = 1 ( ) Problem s non-lnear because n s constraned to unt sphere n q Estmatng normals The constraned mnmzaton problem mn n = 1 q p, n s solved by the egenvector correspondng to the smallest egenvalue of ( qx p ) θ ( qx p ) θ ( qx p ) θ x y z ( q ) ( ) ( ) y p θ qy p θ qy p θ x y z ( q ) ( ) ( ) z p θ x qz p θ y qz p θ z θ Pont-Based Computer Graphcs Marc Alexa 5 Pont-Based Computer Graphcs Marc Alexa 6 Estmatng normals Consstent orentaton Problem s NP-hard Greedy approach (Hoppe) Compute spannng tree based on graph of k-nearest neghbors Orent consstently along spannng tree Computng Implcts Gven N ponts and normals p, n and constrants f p = 0, f p + n = Let p + N = p + n An RBF approxmaton f x = w r x ( ) ( ) 1 ( ) ( ) leads to N lnear equatons n N unknowns (a N N matrx) p Pont-Based Computer Graphcs Marc Alexa 7 Pont-Based Computer Graphcs Marc Alexa 8 Computng Implcts Computng Implcts Practcal problems: N > Matrx soluton becomes dffcult Two solutons Sparse matrces allow teratve soluton Smaller number of RBFs Sparse matrces Needed: ( 0) r( p p ) r( p p ) r( p1 p ) r( 0) r( p1 p ) 0 r( p p0 ) r( p p1 ) r( 0) c c Compactly supported RBFs r M d > c r( d) = 0, r'( c) = L O Pont-Based Computer Graphcs Marc Alexa 9 Pont-Based Computer Graphcs Marc Alexa 30 5

41 Computng Implcts Smaller number of RBFs Greedy approach (Carr et al.) Start wth random small subset Add RBFs where approxmaton qualty s not suffcent RBF Implcts - Results Images courtesy Greg Turk Pont-Based Computer Graphcs Marc Alexa 31 Pont-Based Computer Graphcs Marc Alexa 3 RBF Implcts - Results Images courtesy Greg Turk Implcts - Conclusons Scalar feld s underconstraned Constrants only defne where the feld s zero, not where t s non-zero Sgned felds restrct surfaces to be unbounded All mplct surfaces defne solds Pont-Based Computer Graphcs Marc Alexa 33 Pont-Based Computer Graphcs Marc Alexa 34 Projecton Idea: Map space to surface Surface s defned as fxponts of mappng r r Surface defnton Projecton procedure (Levn) Local polyonmal approxmaton Inspred by dfferental geometry Implct surface defnton Infntely smooth & Manfold surface r r Pont-Based Computer Graphcs Marc Alexa 35 Pont-Based Computer Graphcs Marc Alexa 36 6

42 Surface Defnton Local Reference Plane Constructve defnton Input pont r Compute a local n reference plane H r =<q,n> Compute a local polynomal over the plane G r Project pont r =G r (0) Estmate normal r q H r G r Fnd plane H r = q, n + D mn q p, n θ q p q, n = 1 ( ) / d h θ d = e h s feature sze/ pont spacng H r s ndependent of r s dstance Manfold property ( ) n r q Weght functon based on dstance to q, not r H r Pont-Based Computer Graphcs Marc Alexa 37 Pont-Based Computer Graphcs Marc Alexa 38 Local Reference Plane Local Reference Plane Computng reference plane Non-lnear optmzaton problem Mnmze ndependent varables: Over n for fxed dstance r q Along n for fxed drecton n q changes -> the weghts change Only teratve solutons possble n r n r q H r H r q Practcal computaton Mnmze over n for fxed q Egenvalue problem Translate q so that r = q + r q n Effectvely changes r q Mnmze along n for fxed drecton n Explot partal dervatve n r n r q H r H r q Pont-Based Computer Graphcs Marc Alexa 39 Pont-Based Computer Graphcs Marc Alexa 40 Projectng the Pont MLS polyonomal over H r mn q p, n G p G Π d LS problem r =G r (0) ( ( ) θ( q p ) Estmate normal n H r r q H r Spatal data structure Regular grd based on support of θ Each pont nfluences only 8 cells Each cell s an octree r Dstant octree cells are approxmated by one pont n center of mass G r Pont-Based Computer Graphcs Marc Alexa 41 Pont-Based Computer Graphcs Marc Alexa 4 7

43 Error bounds Paradgm: Gven surface S Pont set P = sampled from S { } p ( r S ) defnes S R Error bounds Approxmaton error of S P to S MLS error approxmatng a functon f wth m+1 a polynomal g: f g M h ( m+1) M O( f ) m = degree of polynomal S P s approxmated by a polynomal n each pont m+1 S S M h p Pont-Based Computer Graphcs Marc Alexa 43 Pont-Based Computer Graphcs Marc Alexa 44 Error bounds Conclusons Remark: Curvature s a useful crteron only for pecewse lnear surfaces Generally: Hgher order dervatves are not accessble Qualty of representaton s manly dctated by h Number of ponts control h Increase/decrease number of ponts to adjust the qualty of representaton Conclusons Projecton-based surface defnton Surface s smooth and manfold Surface may be bounded Representaton error manly depends on pont densty Adjustable feature sze h allows to smooth out nose Pont-Based Computer Graphcs Marc Alexa 45 Pont-Based Computer Graphcs Marc Alexa 46 Some References Alexa, Behr, Cohen-Or, Fleshman, Levn, Slva. Pont Set Surfaces. IEEE Vsualzaton 00, pp. 1-8, 00 Carr, Beatson, Cherre, Mtchell, Frght, McCallum, Evans. Reconstructon and Representaton of 3D Objects wth Radal Bass Functons. SIGGRAPH 001 Proc., pp , 001 Hoppe, DeRose, Duchamp, McDonald, Stuetzle. Surface Reconstructon from unorganzed ponts. SIGGRAPH 199 Proc., pp , 199 Levn. The approxmaton power of movng least-squares. Math. Comp. 67(4): , 1998 Levn. Mesh-ndependent surface nterpolaton. Curves & Surfaces 000 Savchenko, Pasko, Okunev, Kun. Functon representaton of solds reconstructed from scattered surface ponts and contours. Computer Graphcs Forum, 14(4): , 1995 Turk, O Bren. Shape transformaton usng varatonal mplct surfaces. SIGGRAPH 1999 Proc., pp , 1999 Turk, O Bren. Varatonal mplct surfaces. Techncal Report GITGVU 9915, Georga Insttute of Technology, 1999 Pont-Based Computer Graphcs Marc Alexa 47 8

44 Overvew Spectral Processng of Pont- Sampled Geometry Introducton Fourer transform Spectral processng ppelne Applcatons Spectral flterng Adaptve subsamplng Summary Pont-Based Computer Graphcs Markus Gross 1 Pont-Based Computer Graphcs Markus Gross Introducton Idea: Extend the Fourer transform to manfold geometry Spectral representaton of pont-based objects Powerful methods for dgtal geometry processng Introducton Applcatons: Spectral flterng: Nose removal Mcrostructure analyss Enhancement Adaptve resamplng: Complexty reducton Contnuous LOD Pont-Based Computer Graphcs Markus Gross 3 Pont-Based Computer Graphcs Markus Gross 4 Fourer Transform 1D example: N nk j π N X n = xke spectral bass functon k = 1 output sgnal nput sgnal Benefts: Sound concept of frequency Extensve theory Fast algorthms Fourer Transform Requrements: Fourer transform defned on Eucldean doman we need a global parameterzaton Bass functons are egenfunctons of Laplacan operator requres regular samplng pattern so that bass functons can be expressed n analytcal form (fast evaluaton) Lmtatons: Bass functons are globally defned Lack of local control Pont-Based Computer Graphcs Markus Gross 5 Pont-Based Computer Graphcs Markus Gross 6 1

45 Approach Spectral Ppelne Splt model nto patches that: are parameterzed over the unt-square mappng must be contnuous and should mnmze dstorton are re-sampled onto a regular grd adjust samplng rate to mnmze nformaton loss provde suffcent granularty for ntended applcaton (local analyss) process each patch ndvdually and blend processed patches Pont-Based Computer Graphcs Markus Gross 7 Pont-Based Computer Graphcs Markus Gross 8 Patch Layout Creaton Clusterng Optmzaton Samples Clusters Patches Patch Layout Creaton Iteratve, local optmzaton method Merge patches accordng to qualty metrc: Φ = Φ S Φ NC Φ S patch Sze Φ NC curvature Φ B Φ Reg Φ B patch boundary Φ Reg sprng energy regularzaton Pont-Based Computer Graphcs Markus Gross 9 Pont-Based Computer Graphcs Markus Gross 10 Patch Layout Creaton Parameterze patches by orthogonal projecton onto base plane Bound normal cone to control dstorton of mappng usng smallest enclosng sphere Patch Resamplng Patches are rregularly sampled: Pont-Based Computer Graphcs Markus Gross 11 Pont-Based Computer Graphcs Markus Gross 1

46 Patch Resamplng Resample patch onto regular grd usng herarchcal push-pull flter (scattered data approxmaton) Spectral Analyss D dscrete Fourer transform (DFT) Drect manpulaton of spectral coeffcents Flterng as convoluton: F( x y) = F( x) F( y) Convoluton: O(N ) multplcaton: O(N) Inverse Fourer transform Fltered patch surface Pont-Based Computer Graphcs Markus Gross 13 Pont-Based Computer Graphcs Markus Gross 14 Spectral Flters Spectral Flters Smoothng flters deal low-pass Gaussan low-pass orgnal Mcrostructure analyss and enhancement transfer functon: spectral doman transfer functon: spatal doman Pont-Based Computer Graphcs Markus Gross 15 Pont-Based Computer Graphcs Markus Gross 16 Spectral Resamplng Low-pass flterng Band-lmtaton Regular Resamplng Optmal samplng rate (samplng theorem) Reconstructon Flterng can lead to dscontnutes at patch boundares Create patch overlap, blend adjacent patches Samplng rates regon of overlap Error control (Parseval s theorem) Power Spectrum Pont postons Normals Pont-Based Computer Graphcs Markus Gross 17 Pont-Based Computer Graphcs Markus Gross 18 3

47 Reconstructon Tmngs Blendng the samplng rate Tme Clusterng 9% Patch Mergng 38% SDA 3% blended samplng rate n regon of patch overlap dscretzed samplng rate on regular grd pre-computed samplng patterns Analyss Reconstructon 4% 6% Pont-Based Computer Graphcs Markus Gross 19 Pont-Based Computer Graphcs Markus Gross 0 Applcatons Surface Restoraton Applcatons Interactve flterng Orgnal Gaussan low-pass Wener flter Patch layout Pont-Based Computer Graphcs Markus Gross 1 Pont-Based Computer Graphcs Markus Gross Applcatons Adaptve Subsamplng Summary Versatle spectral decomposton of pontbased models Effectve flterng Adaptve resamplng Effcent processng of large pont-sampled models 4,18,614 pts. = 100% 87,163 pts. = 6.9% Pont-Based Computer Graphcs Markus Gross 3 Pont-Based Computer Graphcs Markus Gross 4 4

48 Reference Pauly, Gross: Spectral Processng of Pont-sampled Geometry, SIGGRAPH 001 Pont-Based Computer Graphcs Markus Gross 5 5

49 Overvew Effcent Smplfcaton of Pont-sampled Surfaces Introducton Local surface analyss Smplfcaton methods Error measurement Comparson Pont-Based Computer Graphcs Mark Pauly 1 Pont-Based Computer Graphcs Mark Pauly Introducton Pont-based models are often sampled very densely Many applcatons requre coarser approxmatons, e.g. for effcent Storage Transmsson Processng Renderng we need smplfcaton methods for reducng the complexty of pont-based surfaces Introducton We transfer dfferent smplfcaton methods from trangle meshes to pont clouds: Incremental clusterng Herarchcal clusterng Iteratve smplfcaton Partcle smulaton Dependng on the ntended use, each method has ts pros and cons (see comparson) Pont-Based Computer Graphcs Mark Pauly 3 Pont-Based Computer Graphcs Mark Pauly 4 Local Surface Analyss Cloud of pont samples descrbes underlyng (manfold) surface We need: mechansms for locally approxmatng the surface MLS approach fast estmaton of tangent plane and curvature prncpal component analyss of local neghborhood Neghborhood No explct connectvty between samples (as wth trangle meshes) Replace geodesc proxmty wth spatal proxmty (requres suffcently hgh samplng densty!) Compute neghborhood accordng to Eucldean dstance Pont-Based Computer Graphcs Mark Pauly 5 Pont-Based Computer Graphcs Mark Pauly 6 1

50 Neghborhood k-nearest neghbors Neghborhood Improvement: angle crteron (Lnsen) can be quckly computed usng spatal datastructures (e.g. kd-tree, octree, bsp-tree) requres sotropc pont dstrbuton project ponts onto tangent plane sort neghbors accordng to angle nclude more ponts f angle between subsequent ponts s above some threshold Pont-Based Computer Graphcs Mark Pauly 7 Pont-Based Computer Graphcs Mark Pauly 8 Neghborhood Local Delaunay trangulaton (Floater) Covarance Analyss Covarance matrx of local neghborhood N: p p p p 1 1 C = L L, p p n p p n T j N project ponts nto tangent plane compute local Vorono dagram wth centrod 1 p = N p N Pont-Based Computer Graphcs Mark Pauly 9 Pont-Based Computer Graphcs Mark Pauly 10 Covarance Analyss Consder the egenproblem: C v l = λ v, l {0,1,} l C s a 3x3, postve sem-defnte matrx All egenvalues are real-valued The egenvector wth smallest egenvalue defnes the least-squares plane through the ponts n the neghborhood,.e. approxmates the surface normal l Covarance Analyss The total varaton s gven as: N We defne surface varaton as: λ0 σ n ( p) =, λ0 λ1 λ λ + λ + λ 0 p p = λ + λ + λ 1 measures the fracton of varaton along the surface normal,.e. quantfes how strong the surface devates from the tangent plane estmate for curvature 0 1 Pont-Based Computer Graphcs Mark Pauly 11 Pont-Based Computer Graphcs Mark Pauly 1

51 Covarance Analyss Comparson wth curvature: Surface Smplfcaton Incremental clusterng Herarchcal clusterng Iteratve smplfcaton Partcle smulaton orgnal mean curvature varaton n=0 varaton n=50 Pont-Based Computer Graphcs Mark Pauly 13 Pont-Based Computer Graphcs Mark Pauly 14 Incremental Clusterng Clusterng by regon-growng: Start wth random seed pont Successvely add nearest ponts to cluster untl cluster reaches maxmum sze Choose new seed from remanng ponts Growth of clusters can also be bounded by surface varaton Curvature adaptve clusterng Incremental Clusterng Incremental growth leads to nternal fragmentaton assgn stray samples to closest cluster Note: ths can ncrease maxmum sze and varaton bounds! Pont-Based Computer Graphcs Mark Pauly 15 Pont-Based Computer Graphcs Mark Pauly 16 Incremental Clusterng Replace each cluster by ts centrod Herarchcal Clusterng Top-down approach usng bnary space partton: Splt the pont cloud f: Sze s larger than user-specfed maxmum or Surface varaton s above maxmum threshold orgnal model wth color-coded clusters (34,384 ponts) smplfed model (1,000 ponts) Splt plane defned by centrod and axs of greatest varaton (= egenvector of covarance matrx wth largest assocated egenvector) Leaf nodes of the tree correspond to clusters Pont-Based Computer Graphcs Mark Pauly 17 Pont-Based Computer Graphcs Mark Pauly 18 3

52 Herarchcal Clusterng D example Herarchcal Clusterng Adaptve clusterng orgnal model wth color-coded clusters (34,384 ponts) smplfed model (1,000 ponts) Pont-Based Computer Graphcs Mark Pauly 19 Pont-Based Computer Graphcs Mark Pauly 0 Iteratve Smplfcaton Iteratvely contracts pont pars Each contracton reduces the number of ponts by one Contractons are arranged n prorty queue accordng to quadrc error metrc (Garland and Heckbert) Quadrc measures cost of contracton and determnes optmal poston for contracted sample Equvalent to QSlm except for defnton of approxmatng planes Iteratve Smplfcaton Quadrc measures the squared dstance to a set of planes defned over edges of neghborhood plane spanned by vectors e p p and e e e e 1 1 = = 1 n n p p Pont-Based Computer Graphcs Mark Pauly 1 Pont-Based Computer Graphcs Mark Pauly Iteratve Smplfcaton Partcle Smulaton Resample surface by dstrbutng partcles on the surface Partcles move on surface accordng to nterpartcle repellng forces Partcle relaxaton termnates when equlbrum s reached (requres dampng) Can also be used for up-samplng! orgnal model (187,664 ponts) smplfed model (1,000 ponts) remanng pont par contracton canddates Pont-Based Computer Graphcs Mark Pauly 3 Pont-Based Computer Graphcs Mark Pauly 4 4

53 Partcle Smulaton Intalzaton randomly spread partcles Repulson lnear repulson force F ( p) = k( r p p ) ( p p ) only need to consder neghborhood of radus r Projecton keep partcles on surface by projectng onto tangent plane of closest pont apply full MLS projecton at end of smulaton Pont-Based Computer Graphcs Mark Pauly 5 Partcle Smulaton Adaptve smulaton Adjust repulson radus accordng to surface varaton more samples n regons of hgh varaton orgnal model (75,781 ponts) smplfed model (6,000 ponts) Pont-Based Computer Graphcs Mark Pauly 6 Partcle Smulaton User-controlled smulaton Adjust repulson radus accordng to user nput Measurng Error Measure the dstance between two pont-sampled surfaces usng a samplng approach Maxmum error: max ( S, S ) = max d(, S q Q q ) Two-sded Hausdorff dstance 1 Mean error: S S = d q S avg (, ) (, ) Q q Q Area-weghted ntegral of pont-to-surface dstances Q s an up-sampled verson of the pont cloud that descrbes the surface S Pont-Based Computer Graphcs Mark Pauly 7 Pont-Based Computer Graphcs Mark Pauly 8 Measurng Error d ( q, S ) measures the dstance of pont q to surface S usng the MLS projecton operator wth lnear bass functons Comparson Error estmate for Mchelangelo s Davd smplfed from,000,000 ponts to 5,000 ponts Pont-Based Computer Graphcs Mark Pauly 9 Pont-Based Computer Graphcs Mark Pauly 30 5

54 Comparson Executon tme as a functon of target model sze (nput: dragon, 535,545 ponts) Comparson Executon tme as a functon of nput model sze (reducton to 1%) Pont-Based Computer Graphcs Mark Pauly 31 Pont-Based Computer Graphcs Mark Pauly 3 Comparson Summary Pont-based vs. Mesh Smplfcaton Incremental Clusterng Herarchcal Clusterng Iteratve Smplfcaton Partcle Smulaton Effcency o Surface Error Control - - o + Implementaton + + o - pont-based smplfcaton wth subsequent mesh reconstructon mesh reconstructon wth subsequent mesh smplfcaton (QSlm) pont-based smplfcaton saves an expensve surface reconstructon on the dense pont cloud! Pont-Based Computer Graphcs Mark Pauly 33 Pont-Based Computer Graphcs Mark Pauly 34 References Pauly, Gross: Effcent Smplfcaton of Pontsampled Surfaces, IEEE Vsualzaton 00 Shaffer, Garland: Effcent Adaptve Smplfcaton of Massve Meshes, IEEE Vsualzaton 001 Garland, Heckbert: Surface Smplfcaton usng Quadrc Error Metrcs, SIGGRAPH 1997 Turk: Re-Tlng Polygonal Surfaces, SIGGRAPH 199 Alexa et al. Pont Set Surfaces, IEEE Vsualzaton 001 Pont-Based Computer Graphcs Mark Pauly 35 6

55 Overvew An Interactve System for Pont-based Surface Edtng Introducton Pontshop3D System Components Pont Cloud Parameterzaton Resamplng Scheme EdtngOperators Summary Pont-Based Computer Graphcs Mark Pauly 1 Pont-Based Computer Graphcs Mark Pauly PontShop3D Interactve system for pont-based surface edtng Generalzes D photo edtng concepts and functonalty to 3D pont-sampled surfaces Uses 3D surface pxels (surfels) as versatle dsplay and modelng prmtve Concept Parameterzaton v u Resamplng Edtng Operator Pont-Based Computer Graphcs Mark Pauly 3 Pont-Based Computer Graphcs Mark Pauly 4 Key Components Pont cloud parameterzaton Φ brngs surface and brush nto common reference frame Dynamc resamplng Ψ creates one-to-one correspondence of surface and brush samples Edtng operator Ω combnes surface and brush samples S = Ω( Ψ( Φ( S)), Ψ( B)) Parameterzaton Constraned mnmum dstorton parameterzaton of pont clouds u x( u) u = u z( u) 3 [ 0,1] X ( ) y( ) = P R x modfed surface orgnal surface brush Pont-Based Computer Graphcs Mark Pauly 5 Pont-Based Computer Graphcs Mark Pauly 6 1

56 Pont-Based Computer Graphcs Mark Pauly 7 Parameterzaton contrants = matchng of feature ponts mnmum dstorton = maxmum smoothness Pont-Based Computer Graphcs Mark Pauly 8 Parameterzaton Fnd mappng X that mnmzes objectve functon: { + = M j P j j d X X C u u x p ) ( ) ) ( ( ) ( γ ε { fttng constrants dstorton surface ponts brush ponts Pont-Based Computer Graphcs Mark Pauly 9 Parameterzaton Measurng dstorton Integrates squared curvature usng local polar re-parameterzaton θ θ γ θ d r X r ), ( ) ( = u u θ u r + = ) sn( ) cos( ), ( θ θ θ r X r X u u Pont-Based Computer Graphcs Mark Pauly 10 Parameterzaton Dscrete formulaton: Approxmaton: mappng s pecewse lnear 1 ~ ) ( ) ( ) ( ) ( ~ = + = M j n N j j j j j U U U C v x v x p ε u Pont-Based Computer Graphcs Mark Pauly 11 Parameterzaton Drectonal dervatves as extenson of dvded dfferences based on k-nearest neghbors Pont-Based Computer Graphcs Mark Pauly 1 Parameterzaton Multgrd solver for effcent computaton of resultng sparse lnear least squares problem 1, ) ( ~ u b u b A a U C j n j j = = =

57 Reconstructon Reconstructon Parameterzed scattered data approxmaton fttng functons Φ ( u) r ( u) X ( u) = r ( u) weght functons normalzaton factor Fttng functons Compute local fttng functons usng local parameterzatons Map to global parameterzaton usng global parameter coordnates of neghborng ponts reconstructon wth lnear fttng functons weght functons n parameter space Pont-Based Computer Graphcs Mark Pauly 13 Pont-Based Computer Graphcs Mark Pauly 14 Reconstructon Reconstructon wth lnear fttng functons s equvalent to surface splattng! we can use the surface splattng renderer to reconstruct our surface functon (see chapter on renderng) Ths provdes: Fast evaluaton Ant-alasng (Band-lmt the weght functons before samplng usng Gaussan low-pass flter) Dstortons of splats due to parameterzaton can be computed effcently usng local affne mappngs Samplng Three samplng strateges: Resample the brush,.e., sample at the orgnal surface ponts Resample the surface,.e., sample at the brush ponts Adaptve resamplng,.e., sample at surface or brush ponts dependng on the respectve samplng densty Pont-Based Computer Graphcs Mark Pauly 15 Pont-Based Computer Graphcs Mark Pauly 16 Edtng Operators Pantng Texture, materal propertes, transparency Edtng Operators Sculptng Carvng, normal dsplacement texture map dsplacement maps carved and texture mapped pont-sampled surface Pont-Based Computer Graphcs Mark Pauly 17 Pont-Based Computer Graphcs Mark Pauly 18 3

58 Edtng Operators Flterng Scalar attrbutes, geometry Summary Pontshop3D provdes sophstcated edtng operatons on pont-sampled surfaces ponts are a versatle and powerful modelng prmtve Lmtaton: only works on clean models suffcently hgh samplng densty no outlers lttle nose requres model cleanng (ntegrated or as preprocess) Pont-Based Computer Graphcs Mark Pauly 19 Pont-Based Computer Graphcs Mark Pauly 0 Reference Zwcker, Pauly, Knoll, Gross: Pontshop3D: An nteractve system for Pont-based Surface Edtng, SIGGRAPH 00 check out: Pont-Based Computer Graphcs Mark Pauly 1 4