Mesh Simplification. Mesh Simplification. Mesh Simplification Goals. Mesh Simplification Motivation. Mesh Simplification Overview.

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1 Mesh Simlifition Mesh Simlifition homs Funkhouser Prineton University CS 56, Fll 6 ringles : 41,855 7,97,9 1,939 8,385 4,766 Division, Viewoint, Cohen Mesh Simlifition Motivtion Intertive visuliztion Store nd drw simler version for distnt ojets Simultion roxies Store nd roess simler version for roximte solutions first, nd then refine detils for hits Mesh Simlifition Gols Redue numer of olygons Less storge Fster rendering Simler mniultion Desirle roerties Generlity, effiieny, slility Produes good roximtion Geometri Visul Stnford Grhis L Mesh Simlifition Overview Some lgorithms Vertex lustering Mesh retiling Mesh otimiztion Mesh deimtion Considertions Seed of lgorithm Qulity of roximtion Generlity (tyes of meshes) oology modifitions Control of roximtion ulity Continuous LOD Smooth trnsitions Vertex Clustering Prtition verties into lusters nd rele ll verties in eh luster y one reresenttive 1,18 olys 1,383 olys 474 olys 46 olys Rossign 1

2 Vertex Clustering Exmle lgorithm [Rossign93]: 1. Build grid ontining verties. Merge verties in sme grid ell. Selet new osition for reresenttive vertex. Collse degenerte edges nd fes Mesh Re-iling Resmle mesh with uniformly sed verties Comments: Fst Collses toology Low ulity Hrd to ontrol Rossign urk Mesh Re-iling Exmle lgorithm [urk9]: Generte rndom oints on surfe Use diffusion/reulsion to sred them uniformly essellte verties (mny detils here) Mesh Otimiztion Aly otimiztion roedure to minimize n ojetive funtion E(K,V) E(K,V) E dist (K,V) + E re (K) + E sring (K,V) Comments: Slow Blurs shr fetures urk Hoe Mesh Otimiztion Exmle lgorithm [Hoe9]: Iterte with deresing sring term 1. Rndomly modify toology with edge ollse, edge sw, or edge slit. Move verties to minimize E(K,V) 3. Kee toologil hnge if redue overll ojetive funtion Mesh Otimiztion Initil mesh (3 verties) Smle Points (675 verties) Hoe re 1-5 (487 verties) re 1-4 (39 verties) Hoe

3 Mesh Deimtion Aly itertive, greedy lgorithm to grdully redue omlexity of mesh Mesure error of ossile deimtion oertions Ple oertions in ueue ording to error Perform oertions in ueue suessively After eh oertion, re-evlute error metris Mesh Deimion Oertions Generl ide: Eh oertions simlifies model y smll mount Aly mny oertions in suession yes of oertions Vertex remove Edge ollse Vertex luster Vertex Remove Method Remove vertex nd djent fes Fill hole with new tringles (redution of ) Reuires mnifold surfe round vertex Preserves lol toologil struture Edge Collse Method Merge two edge verties to one Delete degenerte tringles Reuires mnifold surfe round vertex Preserves lol toologil struture Allows smooth trnsition Vertex Cluster Method Merge verties sed on roximity ringles with reeted verties eome edge or oint Generl nd roust Allows toologil hnges Not est ulity Oertion Considertions oology onsidertions Attention to toology romotes etter erne Allowing non-mnifolds inreses roustness nd ility to simlify Oertion onsidertions Collse-tye oertions llow smooth trnsitions Vertex remove ffets smller ortion of mesh thn edge ollse 3

4 Mesh Deimtion Error Metris Motivtion Promote urte 3D she reservtion Preserve sreen-se silhouettes nd ixel overges Vertex-Vertex Distne E mx( v3-v1, v3-v ) Rossign nd Borrel 93 Lueke nd Erikson 97 yes Vertex-Vertex Distne Surfe-Surfe Distne Point-Surfe Distne Vertex-Plne Distne v Vertex-Vertex Distne Surfe-Surfe Distne E mx( v3-v1, v3-v ) Rossign nd Borrel 93 Lueke nd Erikson 97 v Error is mximum distne etween originl nd simlified surfe olerne Volumes - Guézie 96 Simlifition Enveloes - Cohen/Vrshney 96 Husdorf Distne - Klein 96 Ming Distne - Bjj/Shikore 96, Cohen et l. 97 v v Point-Surfe Distne Error is sum of sured distnes from originl verties to losest oint on simlified surfe Hoe et l. 9 Vertex-Plne Distne Error is sed on distnes from originl verties to lnes of fes in simlified surfe Mx distne to lne Mintin set of lnes for eh vertex [Ronfrd96] Sum of sured distnes Aroximted y udri t eh vertex [Grlnd97] 4

5 Qudri Error Metri Error is sum of sured distnes from originl verties to lnes of fes in simlified surfe How omute When verties re merged, merge sets Qudri Error Metri Sum of sured distnes from vertex to lnes: v x y v, z 1 Dist( v, ) v d Dist( v, ) x + y + z + d v Qudri Error Metri Using Qudri Error Metri Common mthemtil trik: udrti form symmetri mtrix Q multilied twie y vetor v ( v v Qv v v ) v Q d d d d d d d Aroximte error of edge ollses Eh vertex v i hs ssoited udri Q i Error of ollsing nd v to v is v Q 1 v +v Q v Qudri for new vertex v is Q Q 1 +Q v v Q 1 Q Q Q Q 1 + Q Using Qudri Error Metri Find otiml lotion v fter ollse: ' Q min v' Q' v': v' x y z Using Qudri Error Metri Find otiml lotion v fter ollse: v' v'

6 Qudri Error Visuliztion Ellisoids: iso-error surfes Smller ellisoids reresent greter error for given motion Lower error for motion rllel to surfe Lower error in flt regions thn t orners Elongted in ylindril regions ner ridges Qudri Error Visuliztion Ellisoids: iso-error surfes Smller ellisoids reresent greter error for given motion Lower error for motion rllel to surfe Lower error in flt regions thn t orners Elongted in ylindril regions ner ridges Qudri Error Metri Results Qudri Error Metri Results Qudri Error Metri Detils Boundry reservtion: dd lnes erendiulr to oundry edges Prevent foldovers: hek for norml fliing Crete virtul edges etween verties loser thn some threshold t Look in Grlnd nd Hekert, SIGGRAPH 1997 Mesh Deimtion Summry Fst (with udri error metri) Good ulity roximtion Only onneted meshes Allows toology modifitions (if llow vertex merging) Allows ontrol over mount of simlifition Continuous LOD Smooth trnsitions 6