Simplification of 3D Meshes
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1 Smplfcaton of 3D Meshes Addy Ngan /4/00 Outlne Motvaton Taxonomy of smplfcaton methods Hoppe et al, Mesh optmzaton Hoppe, Progressve meshes Smplfcaton of 3D Meshes 1
2 Motvaton Hgh detaled meshes becomng wdely avalable n computer vson, scentfc vsualzaton, terran data from satellte etc. Need to store, transmt, analyze, edt and dsplay them effcently. > 1M trangles! Goals Reduce number of polygons Faster renderng Less storage Smpler manpulaton Other qualtes General (non-manfold) Effcency Preserve attrbutes other than geometry Smplfcaton of 3D Meshes
3 Taxonomy of methods Manfold Smplfcaton Vertex decmaton Wavelet Edge collapse Non-manfold Smplfcaton Vertex clusterng Mesh Decmaton [Schroeder et al 9] Multple passes For each pass remove all vertces that matches certan crtera and retrangulate Crtera: dstance to plane/edge Advantage: smple algorthm, fast Dsadvantage: bg memory Smplfcaton of 3D Meshes 3
4 Re-Tlng [Turk 9] Steps: Generate random ponts on surface Iteratve repulson spread the ponts out unformly Add new set of ponts to the surface and mutual tessellate Remove old vertces one by one yeldng a new trangulaton Varant to put ponts adaptvely dependng on curvature Advantage : mantan topology Dsadvantage : complex algorthm, blur sharp features. Re-Tlng (con t( con t) Smplfcaton of 3D Meshes 4
5 3D Grd Method [Rossgnac[ Rossgnac-Borrel 93] Steps: Subdvde the boundng volume nto regular grd Merge all vertces wthn each cell together nto a new vertex Advantage: Form trangles very general, accordng to orgnal topology. fast Dsadvantage : low qualty, non-adaptve Optmzaton [Hoppe et al 93] Optmzaton based smplfcaton. Mnmze energy functon. Repeat sem-random changes to topology and optmze the geometry. Work on manfold Advantages: Hgh qualty, less senstve to nose Dsadvantages: slow Smplfcaton of 3D Meshes 5
6 Mesh Optmzaton Hugues Hoppe, Tony DeRose, Tom Duchamp, John McDonald, Werner Stuetzle, SIGGRAPH 93 Optmzaton Smlar problem to smplfcaton Startng wth sample data ponts from the surface and a ntal mesh, fnd a smpler mesh Mnmze energy functon that descrbes the concseness and accuracy of the mesh Smplfcaton of 3D Meshes 6
7 Mesh Representaton A mesh s represented by a par (K,V) K s a smplcal complex representng the connectvty of the vertces/edges/faces. V={v 1,,, v m } s a set of m vertex postons defnng the shape of the mesh n R 3 If mesh s not self-ntersectng, every pont can be represented by a barycentrc vector Energy Functon E(K,V) = E dst (K,V) + E rep (K) + E sprng (K,V) E dst (K,V) = E rep (K) = c rep m E sprng (K,V) = n = 1 d { j, k} K ( x, ϕ ( K )) j v v V k Wthout E sprng, spkes possble for regon wth no data ponts Smplfcaton of 3D Meshes 7
8 Smplfcaton of 3D Meshes 8 Mnmzng the Energy Functon Mnmzng the Energy Functon A bgger c rep sparser representaton. Outer loop optmze over K (Dscrete) Inner loop optmze over V for a fxed K (Contnuous) = Mnmze + For each x, dstance = New objectve functon : E(K,V,B)= Optmzaton for fxed K Optmzaton for fxed K = n K d 1 V )) (, ( ϕ x }, { k j K k j v v }, { V 1 ) ( mn k j K k j K n v v b x b + = V ) ( mn K b x b
9 Optmzaton over fxed K Two subproblems : For fxed vertces V, fnd optmal barycentrc coordnate vectors B by projecton For fxed B, fnd optmal V by solvng a lnear least squares problem Fnd optmal solutons to both subproblems, so E(K,V,B) must converge. Projecton Subproblem Fnd optmal B by projectng x onto the mesh. To accelerate, buld a spatal parttonng data structure so that for each pont only consder nearby subset of faces Assume a pont s projecton les n the neghborhood of ts projecton n prevous teraton (perform well n practce) Smplfcaton of 3D Meshes 9
10 Lnear Least Squares Subproblem Mnmze for x,y,z coordnates, v m-vector A (n+e) m matrx d (n+e)-vector For m mesh vertces, n data ponts, e edges Frst n rows of d contans the n data ponts Next e rows of d are zeroes v contans the m mesh vertces Av d Lnear Least Squares (con t( con t) Av d j j Frst n rows of A contans the barycentrc coordnates computed n projecton (at most 3 non-zero entres) Next e rows represent the sprng energy: each contans an κ and - κ entres n columns correspondng to ndces of edge s endponts (exactly non-zero entres) A s sparse. Use conjugate gradent method to solve n O(n+m) tme Smplfcaton of 3D Meshes 10
11 Optmzaton over K Take a legal move and accept f t gves lower energy. Termnate f a number of trals faled to gve a lower energy. Legal move : applcaton of one of these that leaves the topologcal type of K unchanged Three elementary transformatons: edge collapse/splt/swap Legal move Splt Always legal Collapse : f and only f For all vertces {k} adjacent to both {} and {j}, {,j,k} s a face If {} and {j} are both boundary vertces, {,j} s a boundary edge K has more than 4 vertces f {} and {j} both are not boundary vertces, or K has more than 3 vertces f ether {} or {j} are boundary vertces Swap f and only f {k,l} K Proof n Hoppe et al, TR Unversty of Washngton Smplfcaton of 3D Meshes 11
12 Evaluaton of a legal move Instead of re-computng the global energy after a legal move, only compute the change n energy for the local submesh For s K, defne star(s;k) = {s K : s nonempty subset of s } Edge Collapse To evaluate collapsng of an edge {,j}, take the submesh to be star({};k) star({j};k). Optmze over the new vertex h whle holdng all other constant Attempt optmzatons startng at v, v j, and ½(v +v j ). Accept the best one. Instead of checkng for self-ntersecton after collapse whch s expensve, use a threshold value for the maxmum dhedral angle of edges n star({h};k ) Smplfcaton of 3D Meshes 1
13 Edge splt/swap Edge splt Same procedure as collapse, usng the submesh to be star({,j}; K). Intal value of v h be ½(v +v j ). Edge swap- For a swap that replace an edge {,j} wth {k,l}, choose the best of the two optmzatons, one wth submesh star({k};k ) varyng vertex v k, another wth submesh star({l};k ) varyng vertex v l. Results Intal mesh (03 vertces) Sample Ponts (675 vertces) c rep =10-5 (487 vertces) c rep =10-4 (39 vertces) Smplfcaton of 3D Meshes 13
14 Results Results - Segmentaton Smplfcaton of 3D Meshes 14
15 Progressve Meshes Hugues Hoppe SIGGRAPH 96 Addtonal Goal Lossless,, contnuous-resoluton, progressve representaton More compact representaton Apart from geometry, also preserve other attrbutes (colors, normals,, materals, texture coordnates ) Smplfcaton of 3D Meshes 15
16 Mesh Representaton Edge collapse only s suffcent! n An ntal mesh Mˆ = M can be smplfed nto a coarser mesh M 0 by a sequence of edge collapse. (Mˆ = M The sequence s nvertble n ) ecoln ecol1 ecol n M M 0 Progressve Mesh (Mˆ = M n ) ecoln ecol1 ecol n M M 0 Smplfcaton of 3D Meshes 16
17 Energy Functon Old E(K,V) = E dst (K,V) + E rep (K) + E sprng (K,V) New E(M) = E dst (M) + E sprng (M) + E scalar (M) +E dscrete (M) where M = (K,V,D,S) E rep goes away! And so does the parameter c rep Preservng geometry E dst + E sprng Same as before, but only consder edge collapse n the outer loop. The possble legal collapses are placed n a prorty queue wth ts estmated energy change E For an edge collapse K K, E = E K -E K After each collapse, the cost of ts neghborhood edges n the prorty queue are updated. Smplfcaton of 3D Meshes 17
18 Preservng Scalar attrbutes For d scalar attrbutes, we could have added d dmenson to E dst. For effcency purpose, we separate t as a n dfferent term E (V) = (c ) x scalar scalar V ( ) = 1 b Frst solve for the vertex postons. Then usng the same b to fnd vertex attrbutes mnmzng E scalar by lnear least square Preservng Scalar Attrbutes 00x00 vertces 400 vertces Smplfcaton of 3D Meshes 18
19 Preservng dscontnuty We want to preserve dscontnuty because they often form notceable features sample an addtonal set of ponts X dsc from sharp edges of ntal mesh. Compute E dsc by projectng X dsc onto the correspondng sharp edges dsallow/penalze collapse of boundary and dscontnuty edges Wthout E dsc References Survey of Polygonal Surface Smplfcaton Algorthms, Paul Heckbert and Mchael Garland, tech. report, CS Dept.,Carnege Mellon U., to appear H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, Mesh Optmzaton, SIGGRAPH 93, Hugues Hoppe, Progressve Meshes, SIGGRAPH 96, Hugues Hoppe, Effcent mplementaton of progressve meshes, Computers & Graphcs, Vol., No. 1, 1998, pages Greg Turk, Re-Tlng Polygonal Surfaces, SIGGRAPH 9, Wllam J. Schroeder, Jonathan Zarge, and Wllam Lorensen, Decmaton of Trangle Meshes, SIGGRAPH 9, J. Rossgnac and P. Borrel, Mult-resoluton 3D approxmatons for renderng complex scenes Smplfcaton of 3D Meshes 19
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