Discrete Geometry Processing

Transcription

1 Non Convex Boundary Convex boundary creates significant distortion Free boundary is better Some slides from the Mesh Parameterization Course (Siggraph Asia 008) 1 Fixed vs Free Boundary Fixed vs Free Boundary Copyright 011 Gotsman, Pauly, Ben-Chen Page 1

2 Cut graph 7 8 Back to the First Fundamental Form Linear Map Surgery Singular Value Decomposition (SVD) of with rotations and and scale factors (singular values) 9 10 Types of Distortion isometric or length preserving Computing the Stretch Factors first fundamental form conformal or angle preserving equiareal or area preserving everything defined pointwise on 11 eigenvalues of singular values of and 1 Copyright 011 Gotsman, Pauly, Ben-Chen Page

3 Measuring Distortion Piecewise Linear Parameterizations local distortion measure has minimum at isometric measure conformal measure overall distortion piecewise linear atomic maps distortion constant per triangle overall distortion Distortion based Methods Define distortion functional E as a function of 1 and Expand their expressions in E as function of the unknown u i, v i Design an algorithm to find the u i,v i 'sthat minimize E The terms and are quadratic in the parameter points, so Dirichlet energy Conformal energy minimization yields linear problem Both result in barycentric mappings with discrete harmonic weights for interiorvertices Dirichlet maps require to fix all boundary vertices Conformal maps only two Both maps not necessarily bijective MIPS energy Area preserving MIPS Non linear Methods Copyright 011 Gotsman, Pauly, Ben-Chen Page 3

4 Non linear Methods Green Lagrange deformation tensor Output Conformally flattened D mesh Stretch energies (,, and symmetric stretch) similarity Input 3D mesh 19 0 Conformal Parameterization Least Squares Conformal Map (LSCM) Conformal Parameterization LSCM v u t t Minimize å ç - ç = åat ( s1-s) tît æ vö æ uö ç- ç x ç y v u èç yy ø èç x ø tît Cauchy Riemann equations: No piecewise linear solution in general ì v u =- ï x y í ï v u ï = ï ïî y x 1 Fix two vertices to determine rot, transl, scaling Linear system in (u,v) LSCM Angle Based Flattening (ABF) Fact: Triangular D mesh is defined by its angles (up to similarity) Define problem in angle space Angle based formulation: Distortion as function of angles Validity set of angle constraints 3 4 Copyright 011 Gotsman, Pauly, Ben-Chen Page 4

5 Constrained Minimization Notation: i are (given) 3D angles. i are (unknown) D angles. Objective: minimize i i (relative) )deviation of angles: 3T i i i i 1 D( ) ( ) Constraints All angles are positive (linear inequalities). Sum of angles in each triangle is (linear equalities). Sum of angles around each vertex is (linear equalities). All one rings close properly (non linear equalities). 5 6 Solving Use non linear solver (Lagrange multipliers, Newton method) to solve for a i Examples Embed in plane based on a I by unfolding LSCM 7 ABF 8 Examples LSCM uniform harmonic mean value conformal ABF 9 30 Copyright 011 Gotsman, Pauly, Ben-Chen Page 5

6 Non mean value conformal ABF++ circle patterns MIPS min stretch 31 3 Non Parameterization Conclusions ABF++ circle patterns MIPS min stretch Many, MANY methods out there Fixed / free boundary Bijective / non bijective Conformal / Non conformal Linear / non linear solver Best method depends on input mesh Close to disk linear methods can work well Require large distortion prefer non linear methods Copyright 011 Gotsman, Pauly, Ben-Chen Page 6