Parameterization II Some slides from the Mesh Parameterization Course from Siggraph Asia

Transcription

1 Parameterization II Some slides from the Mesh Parameterization Course from Siggraph Asia

2 Non-Convex Non Convex Boundary Convex boundary creates significant distortion Free boundary is better 2

3 Fixed vs Free Boundary 3

4 Fixed vs Free Boundary 4

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7 Cut graph 7

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9 Free Boundary Methods - Tactics Solve for the (u,v) coordinates using the discrete first fundamental form MIPS [Hormann et al., 2000] Stretch optimization [Sander et al., 2001] LSCM (conformal, linear) [Levy et al., 2002] DCP (conformal, linear) [Desbrun et al., 2002] Solve for the angles of the map (conformal) ABF [Sheffer et al., 2001], ABF++ [Sheffer et al., 2004] LinABF (linear) [Zayer et al., 2007] 9

10 Free Boundary Methods - Tactics Solve for the edge lengths of the map by yprescribing curvature Circle patterns [Kharevych et al., 2006] CPMS (linear) [Ben-Chen et al., 2008] CETM [Springborn et al., 2008] Balance area/conformality ARAP [Liu et al., 2008] More 10

11 Back to the First Fundamental Form 11

12 Linear Map Surgery Singular Value Decomposition (SVD) of with rotations and and scale factors (singular values) 12

13 Notion of Distortion isometric or length-preserving conformal or angle-preserving equiareal or area-preservingp g everything defined pointwise on 13

14 Computing the Stretch Factors first fundamental form eigenvalues of singular values of and 14

15 Measuring Distortion local distortion measure has minimum at isometric measure conformal measure overall distortion 15

16 Piecewise Linear Parameterizations piecewise linear atomic maps distortion constant per triangle overall distortion 16

17 Distortion Based Methods Define energy functional F as a function of J p, I p, σ 1,σ 2 Expand their expression in F in function of the unknown u i, v i Design an algorithm to find the u i,v i 'ss that minimizes F 17

18 Linear Methods the terms and are quadratic in the parameter points Dirichlet energy Conformal energy minimization yields linear problem [Pinkall & Polthier 1993] [Eck et al. 1995] [Lévy et al. 2002] [Desbrun et al. 2002] 18

19 Linear Methods both result in barycentric mappings with discrete harmonic weights for interior vertices Dirichlet maps require to fix all boundary vertices Conformal maps only two result depends on this choice best choice [Mullen et al. 2008] both maps not necessarily bijective 19

20 Non-linear Methods MIPS energy [Hormann &Greiner2000] Area-preserving MIPS [Degener et al. 2003] 20

21 Non-linear Methods Green-Lagrange deformation tensor [Maillot et al. 1993] Stretch energies (,, and symmetric stretch) [Sander et al. 2001] [Sorkine et al. 2002] 21

22 Conformal Parameterization - LSCM v u Cauchy Riemann equations: No Piecewise Linear solution in general v u = x y v u = y x 22

23 Conformal Parameterization LSCM [Levy et al., 2002] 2 Minimize Fix two vertices to determine rot transl scaling T v u x y v u determine rot,transl,scaling T u y x 23

24 Conformal Parameterization - LSCM Equivalent formulation, uses only edge lengths ratios and angles For each triangle: p 1 p 2 p p = R α p p l 1 ( ) ( ) l l 2 α l 1 p 3 24

25 Conformal Parameterization DCP x 0 α [Desbrun et al., 2002] β x 2 For one triangle: x 1 γ cot 90 γ ( x x ) + cot β( x x ) = R ( x x ) i γ i β i 0 For complete one-ring of triangles (interior vertex): n (cot γ + cot β ) x x = 0 i= 1 i i i ( ) 0 For incomplete one-ring of triangles (boundary vertex): n 90 i 1 γi + βi ( xi x0 ) = R xi x 1 i2 0 i= 1 (cot cot ) ( ) i 2 25

26 Isotropic Parameterizations Conformal = Harmonic 26

27 Angle Based Flattening (ABF) Fact: Triangular 2D mesh is defined by its angles (up to similarity) Define problem in angle space Angle based formulation: Distortion i as function of angles Validity - set of angle constraints 27

28 Constrained Minimization Notation: β i are (given) 3D angles. α i are (unknown) 2D angles. Objective: minimize (relative) deviation of angles: 3T i = 1 D ( α ) = ( α β ) i i i 2 28

29 Constraints All angles are positive (linear inequalities). Sum of angles in each triangle is π (linear equalities). Sum of angles around each vertex is 2π (linear equalities). All one-rings close properly (non-linear equalities). 29

30 Solving Use non-linear solver (Lagrange multipliers, Newton method) to solve for α i Use LSCM to embed in plane based on α i 30

31 Examples LSCM ABF 31

32 Examples LSCM ABF 32

33 Linear Methods uniform harmonic mean value conformal 33

34 Linear Methods mean value conformal 34

35 Non-Linear Non Linear Methods ABF++ circle p patterns MIPS stretch 35

36 Non-Linear Non Linear Methods ABF++ circle patterns MIPS stretch 36

37 Curvature Prescription LSCM [2002] ABF++ [2005] LinABF [2007] 1. Cut mesh to disk 2. Compute new angles/lengths 3. Embed in plane Discontinuities in scale (and many more ) Circle Patterns [2006] Ricci Flow [2007] CPMS, CETM [2008]

38 Curvature Prescription LSCM [2002] ABF++ [2005] LinABF [2007] (and many more ) Circle Patterns [2006] Ricci Flow [2007] 1. Compute new angles/lengths 2. Cut Mesh to disk 3. Embed in plane No discontinuities in scale CPMS, CETM [2008]

39 Gaussian Curvature Continuous definition Scale dependent κ 1/r 2 Gauss-Bonnet theorem: κ = 2πχ v α i Discrete definition: κ(v) = 2π Σα i Scale independent! Gauss-Bonnet theorem: κ = 2πχ κ=π/2

40 Discrete Metric Triangle edge lengths (a,b,c) are the discrete metric The metric defines the angles by the cosine law v α i The angles define the Gaussian curvature

41 CPMS [2008] Original edge lengths (metric) Conformal factor (per vertex) Extend φ to edge φ () t = φ t +φ (1 t ) 2 3 Integrate e φ on edge Scale to get new edge lengths l = l e φ 1 1 ( t )

42 What Happens to the Curvature? Continuous conformal map 2 φ 2 e κ = κ φ Curvature after map Curvature before map Conformal factor

43 What Happens to the Curvature? Continuous conformal map 2φ 2 e κ = κ φ Discrete conformal map (for small changes) κ κ 2 φ Where did the exponent go? Discrete curvature doesn t scale!

44 The Flattening Algorithm for a Topological Disk Set target curvature to 0 on interior vertices Set φ to 0 on the boundary Find φ on interior vertices by solving 2 φ = κ κ Scale edge lengths to get conformal metric Embed new edge lengths using LSCM

45 Some Results

46 Comparison with Non-Linear Methods ABF++ circle patterns MIPS stretch CPMS 46

47 Reducing Area Distortion Gather Gaussian curvature into cone points Target curvature not 0 everywhere First compute new edge lengths, then cut through cone points No discontinuities in scale 47

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49 Conformal Equivalence of Triangle Meshes [2008] 52

50 Conformal Equivalence of Triangle Meshes [2008] Given input mesh and target curvatures, find u st s.t. New mesh is conformally equivalent to source mesh Target curvatures are achieved Minimize i i convex energy global l minimum i First optimization step equivalent to CPMS 53

51 Conformal Equivalence of Triangle Meshes [2008] 54

52 Parameterization - Conclusions Many MANY methods out there e Fixed / free boundary Bijective / non bijective Conformal / area preserving Linear / non linear First cut then embed / cone points Best method depends on mesh Close to disk linear methods can work well Require large distortion prefer non-linear methods 58