: Mesh Processing. Chapter 6

Transcription

1 : Msh Procssing Chaptr 6

2 Quad-Dominant Rmshing Goal: Gnrat a rmshing of th surfac that consists mostly of quads whos dgs align with th principal curvatur dirctions. [Marinov t al. 04] [Alliz t al. 03]

3 Quad-Dominant Rmshing Approach: Whr w can, trac lins of minimal/maximal curvatur. Sinc ths ar orthogonal, thir intrsctions should giv quads. [Alliz t al. 03]

4 Quad-Dominant Rmshing Challngs: 1. What ar th principal curvatur dirctions?. What ar th principal curvatur lins? 3. Whr do w plac th lins and how long should thy b? 4. What happns whn principal dirctions ar not wll-dfind?

5 Principal Curvatur Dirctions Rcall: W can dfin curvaturs at an dg in trms of th angl () btwn curv sgmnts*: Th min/max curvatur is 0, with principal curvatur dirction along. Th max/min curvatur is qual to th dihdral angl (()=n 1 n ), with principal curvatur dirction along n x. f 1 n 1 n f n *This dfinition follows th dfinition of H v rathr than H v in [Cohn-Stinr t al. 03]

6 Principal Curvatur Dirctions Rcall: This allows us to dfin a 3x3 curvatur tnsor along th dg as th symmtric matrix with ignvalu () in th dirction across and ignvalus of 0 in prpndicular dirctions: 1 C ( p ) ( ) n n n t n f 1 n 1 f n

7 Principal Curvatur Dirctions Rcall: This, in turn, allows us to dfin th curvatur tnsor around a vrtx v, avrag ovr a nighborhood B around v: 1 1 C ( v) B ( ) B n v n n v B t

8 Rcall: Principal Curvatur Dirctions 1 1 C ( v) B ( ) B n n n Computing th ign-dcomposition of th curvatur tnsor w gt an stimat of: Th normal: Th ignvctor with smallst absolut ignvalu. Th principal dirctions and valus: Th othr two ignvctors and thir associatd ignvalus. t

9 Not: Principal Curvatur Dirctions 1 1 C ( v) B ( ) B n n n Whn th two principal dirctions hav th sam principal curvatur valus, th principal dirctions ar not wll dfind. t

10 Not: Principal Curvatur Dirctions 1 1 C ( v) B ( ) B n n n Whn th two principal dirctions hav th sam principal curvatur valus, th principal dirctions ar not wll dfind. Such points ar calld umbilical points. t

11 Principal Curvatur Lins What ar th principal curvatur lins? Assuming that w ar away from th umbilical points, w can dfin two vctor filds: 1. v min : Aligns with th min. curvatur. v max : Aligns with th max. curvatur

12 [Alliz t al. 03] Principal Curvatur Lins What ar th principal curvatur lins? Assuming that w ar away from th umbilical points, w can dfin two vctor filds: 1. v min : Aligns with th min. curvatur. v max : Aligns with th max. curvatur Givn a starting p, solv th diff. q.: min/ max '( t) v ( ( t)) s.t. (0) min/ max p

13 Principal Curvatur Lins How far should w intgrat? W should intgrat th min/max curvs until thy ar within a prscribd dnsity: 1. Accuracy of th rmsh. Local curvatur

14 Principal Curvatur Lins Q: If th usr wants th rmshd surfac to b within a distanc of from th original surfac, how far should th minimal/maximal curvatur lins b from ach othr?

15 Principal Curvatur Lins A: Considr th surfac btwn two lins of minimal/maximal curvatur: l 1 l

16 Principal Curvatur Lins A: Considr th surfac btwn two lins of minimal/maximal curvatur: Th curv btwn thm will follow th maximal/minimal curvatur dirction. l 1 l c

17 Principal Curvatur Lins A: Considr th surfac btwn two lins of minimal/maximal curvatur: Th curv btwn thm will follow th maximal/minimal curvatur dirction. Th curv will b, roughly, a circular arc with radius qual to on ovr th maximal/minimal curvatur. l 1 l c

18 Principal Curvatur Lins Looking at this in cross sction, w choos th distanc d btwn th curvs so that th distanc to th surfac is blow a thrshold. l 1 c d/ l 1/ l 1 l c

19 Principal Curvatur Lins Looking at this in cross sction, w choos th distanc d btwn th curvs so that th distanc to th surfac is blow a thrshold. Dnoting th distanc by w gt: l 1 l c 1 1 d l 1 l c 1/ d/

20 Principal Curvatur Lins Looking at this in cross sction, w choos th distanc d btwn th curvs so that th distanc to th surfac is blow a thrshold. Dnoting th distanc by w gt: l 1 l c 1 1 d l 1 l c 1/ d/ d

21 Variations on a Thm [Alliz t al. 03]: 1. Comput a conformal paramtrization of th surfac.. Idntify high curvatur umbilicals and start growing curvatur lins, adding candidat sd points into quu as th lins ar grown. 3. Uniformly sampl umbilicals in isotropic aras. 4. Us th quads in th anisotropic aras and us th dgs of a constraind Dlaunay triangulation to triangulat th isotropic points.

22 Variations on a Thm [Alliz t al. 03]: 1. Comput a conformal paramtrization of th surfac.. Smooth th curvatur tnsor ovr th paramtrization domain. (Gaussian convolution with radius wightd invrsly to th ara distortion.) 3. Idntify high curvatur umbilicals and start growing curvatur lins, adding candidat sd points into quu as th lins ar grown. 4. Uniformly sampl umbilicals in isotropic aras. 5. Us th quads in th anisotropic aras and us th dgs of a constraind Dlaunay triangulation to triangulat th isotropic points. [Alliz t al. 03]

23 Variations on a Thm [Marinov t al. 04]: 1. Work dirctly on th msh. Estimat pr-triangl confidncs for th curvatur tnsors by looking at th consistncy of th minimal curvatur dirctions ovr th thr vrtics. 3. Grow curvs from rgions of high confidnc, using th principal curvatur dirction in confidnt aras and continuing on along a straight lin in rgions of low confidnc. 4. Comput intrsctions and polygoniz.

24 Variations on a Thm [Marinov t al. 04]: [Marinov t al. 04]

25 Variations on a Thm Distinctions: Paramtrization [Alliz t al. 03]: Conformal paramtrization, limiting th approach to ithr disk-lik objcts or patching. [Marinov t al. 04]: Non

26 Variations on a Thm Distinctions: Smoothing [Alliz t al. 03]: Gaussian smoothing with spatially varying radius. [Marinov t al. 04]: Confidnc-wightd Laplacian smoothing.

27 Variations on a Thm Distinctions: Intgration [Alliz t al. 03]: Prformd in th D paramtrization domain. [Marinov t al. 04]: Prformd on msh by locally flattning and walking along a straight lin. [Marinov t al. 04]

28 Variations on a Thm Distinctions: Proximity Quris [Alliz t al. 03]: Maintain (and updat) a D Constraind Dlaunay Triangulation as nw dg sgmnts ar introducd. [Marinov t al. 04]: Hash curv dgs with associatd triangls, find adjacnt triangls and xhaustivly tst dgs.

29 Variations on a Thm Distinctions: Isotropic Rgions [Alliz t al. 03]: Chang from quadrangulation to triangulation. [Marinov t al. 04]: Attmpt to continu going in th sam dirction (may switch curvs from minimal to maximal). [Alliz t al. 03] [Marinov t al. 04]