Geometric Modeling Assignment 3: Discrete Differential Quantities

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1 Geometric Modeling Assignment : Discrete Differential Quantities Acknowledgements: Julian Panetta, Olga Diamanti

2 Assignment (Optional) Topic: Discrete Differential Quantities with libigl Vertex Normals, Curvature, Smoothing (Tutorials ) per-face per-vertex, uniform per-vertex, quadratic fit

3 Vertex Normals uniform average area-weighted average mean-curvature based libigl tutorial #201, igl::principal_curvature PCA on k-nearest neighbors quadratic fitting on k-nearest neighbors

4 Curvature libigl tutorials #202, #20 min max mean gauss

5 X Curvature 2 P Mean Curvature Sp = 2Hn k H(v i )=0.5kL c (v i )k k (v i ) 2 P j Gaussian Curvature G(v i ) = j A(v i ) (v j ) P j Min Curvature p apple p 1 = H + H 2 G G = apple 1 apple 2 Max Curvature p apple 1 = H H 2 p G 2H = apple 1 + apple 2

6 Data Smoothing: Grid Case Image/2D data smoothing: Filter out noise/rapid oscillations Solve the heat equation over some = f = Intuition: move toward the average of neighboring

7 Data Smoothing: = f Approach: discretize laplacian with finite differences Now we have an ordinary differential equation (ODE) Integrate (time-step) the ODE, e.g., with forward/backwards Euler

8 Data Smoothing: Surface Case Laplace-Beltrami operator lets us smooth functions on = Sf P E.g., with forward Euler (explicit smoothing) on a triangle mesh, M: f t+dt (v i )=f t (v i )+ dt f t (v M i ) 0 X = f t (v i )+ v j 2N(v i )[{v i } 1 w ij f t (v j ) A (v i ) vertices adjacent vi discrete Laplacian weights (v j )

9 Mesh Smoothing We can view the vertex positions as a (vector-valued) function to smooth! 2 f(v i )= x i y i z i 5 P Here, first-order explicit time stepping looks like: f t+dt (v i )=f t (v i )+ dt M f t (v i ) =) (v i ) vi new = v i + dt ( M v) i 0 1 X = v i + w ij v A j (v j ) v j 2N(v i )[{v i } Note: Laplacian weights generally change when mesh changes!

10 6 5 (I dtl)v = v Diffusion Flow Meshes Diffusion Flow on Meshes Diffusion Flow onon Meshes n+1 n ) (I + dtl)v vf (vn= n+1 n n n+1 v = v + dtlv = p = p = p Continuous Continuous In Continuous L(v) @t n+1 n n n+1 n n+1 v v = dtlv v = v + dtlv p p + t p p p + t p p p + t p Discretization Discretization Discretization Intuition: vertices@f moving neighbor i i i i i itoward i i i average. (I Explicit Smoothing n+1 v v n+1 n+1 n+1 n =n v n = (I + = 0 Iterations 0 Iterations 0 Iterations f v n+1 = (I + dtl)v = 10 Iterations Iterations 1010 Iterations 100 Iterations 100 Iterations 100 Iterations n

11 6 5 Implicit Smoothing Backward Euler is more stable Take larger time steps without mesh blowing up (becoming jagged) Laplacian at next (unknown) step! v n+1 = v n + dtlv n+1 Must solve a linear system: (I dtl)v n+1 = v n libigl tutorial #205 original explicit, 1000 iterations, λ = 0.01 implicit, 1 iteration, λ = 20

12 2D Rectangular Grid Laplacian h (x j ) 20 1 X (x i ) h hf(x i ):= 1 h f(x j ) A f(x i ) 5 x j 2N(x i ) Measures difference from neighbor s average (if we divide by /h^2): 1 Weights can be represented by a stencil: 1-1 1

13 2D Rectangular Grid Laplacian 20 1 (x j ) X 1 (x i ) hf(x i ):= 1 h f(x j ) A f(x i ) x j 2N(x i ) 1 Derivation: Compute second derivatives by finite-differencing the forward difference approximation to first derivative: Sum second partial derivative approximations for each 2 2 (x, y) 1 h f(x + h, y) f(x, y) h f(x, y) f(x h, y) h f(x + h, y)+f(x h, y) 2f(x, y) = h 2 =) f(x, y) 2 (x, f y)+@2 (x, f(x + h, y)+f(x h, y)+f(x, y + h)+f(x, y h) f(x, y) h 2

14 Irregular Grid (Mesh) Laplacian First approach: umbrella operator assume all edges are unit length depends only on connectivity, not vertex locations Notice, when edge length h = 1 the 2D grid laplacian is: 20 hf(x i )= h 2 X 1 f(x j ) A f(x i ) 5 X = f(x j ) f(x i ) x j 2N(x i ) x j 2N(x i ) This version easily generalizes to an arbitrary mesh, M: uniform X M f(v) = = v j 2N(v) X v j 2N(v) f(v j ) 1 f(v j ) A f(v) N(v) f(v) #neighbors

15 Y = The Umbrella (Uniform) Laplacian In Matrix Form: 2 6 f(v 1 ) f(v 2 ) f(v ) f(v N ) 5 w ij = 8 >< 2 6 L = 8 < : uniform M f(v) = 6 X 5 Y = LF w 11 w 12 w 1N w 21 w 22 w 2N w N1 w N2 w NN 8 < = {w ij} 0 i 6= edge (i, j) 1 i 6= j, 9 edge (i, j) N(v i ) v j 2N(v) i = j 1 f(v j ) A N(v) f(v) F = f(v 1 ) f(v 2 ) f(v ) 5 f(v N ) 5

16 The Cotangent Laplacian cotan f(v) = 1 M 2A(v) X v i 2N(v) (cot( i ) + cot( i )) f(v i ) f(v) Accounts for edges differing lengths. Converges to the continuous Laplace-Beltrami operator S with mesh refinement Derivation in next recitation lecture.

17 The Cotangent Laplacian Y = In Matrix Form: 2 6 f(v 1 ) f(v 2 ) f(v ) f(v N ) cotan f(v) = 1 M 2A(v) 5 L = w ij = >< >: Y = LF w 11 w 12 w 1N w 21 w 22 w 2N < : w N1 w N2 w NN 8 < 5 5 = {w ij} 0 i 6= edge (i, j) cot j +cot j i 6= j, 9 edge (i, j) 2A(v P i ) i = j v k 2N(v i ) X v i 2N(v) w ik (cot( i ) + cot( i )) 2 F = f(v i ) f(v 1 ) f(v 2 ) f(v ) 5 f(v N ) 5 f(v)

18 Eigen Sparse Matrix Full-sized Laplacian can be huge for large meshes but most elements are zero! Instead, only store non-zero elements: Sparse Matrix #include <Eigen/Sparse> Eigen::SparseMatrix<double> Laplacian;

19 Eigen Sparse Matrix How to construct a sparse matrix in Eigen: //declare size of matrix L = SparseMatrix<double> (V.rows(), V.rows()); //declare list of non-zero elements (row, column, value) std::vector<eigen::triplet<double> > tripletlist; //insert element to the list //if multiple triplets exist with the same row and column, values will be *added* tripletlist.push_back(eigen::triplet<double>(source,dest,c(i,e))); //construct matrix from the list L.setFromTriplets(tripletList.begin(), tripletlist.end()); see igl::cotmatrix (V,F)

20 How to build cotangent matrix cotan f(v) = 1 M 2A(v) X v i 2N(v) Iterating over vertices is slow (cot( i ) + cot( i )) f(v i ) f(v) Instead, iterate over faces and add cot terms to the vertices incident each face edge for i = 1:number_of_faces for j = 1:face_valence source_vertex = faces(i,j); destination_vertex = faces(i,(j+1) % face_valence); weight =...; //laplacian weight for edge (source_vertex, destination_vertex)(cotan or uniform) Laplacian(source_vertex, destination_vertex) += weight; Laplacian(destination_vertex, source_vertex) += weight; Laplacian(destination_vertex, destination_vertex) -= weight; Laplacian(source_vertex, source_vertex) -= weight; end end

21 Averaged or Summed? We ll see next time that the cotan laplacian we introduced: cotan M f(v) = 1 2A(v) X v i 2N(v) (cot( i ) + cot( i )) f(v i ) f(v) computes the average Laplacian over some area A around v (hence the division by A) This gives an asymmetric matrix. But if we instead compute the summed laplacian (i.e., don t divide by A), we end up with a symmetric matrix. This summed, symmetric Laplacian is more standard in the finite element community, M = and is related to our Laplacian by a lumped (diagonal) mass matrix holding the areas: 2 6 lumped mass matrix (averaging region areas) A(0) A(1) A(N) 5 (L averaged corresponds to formula above) L averaged = M 1 L summed [L summed ] ij = 8 1 >< 2 cot( ij ) + cot( ij ) if j 2 N(i) P j2n(i) >: [Lsummed ] ij if i = j 0 otherwise

22 Averaged or Summed? Suppose we have a linear system: L averaged x = b i.e. using the asymmetric cotangent Laplacian we introduced earlier. Right hand side vector b represents a scalar field, storing the desired Laplacian value (not the summed value!) at each vertex. Solving symmetric sparse linear systems matrices is much more efficient, so we prefer to solve with the summed quantities: ML averaged x = L summed x = Mb

23 How to compute A(v) Barycentric area Connect edge midpoints and triangle barycenters Each of the incident triangles contributes 1/ of its area to all its vertices, regardless of the placement + Simple to compute + Always positive weights - Heavily connectivity-dependent - Changes if edges are flipped

24 How to compute A(v) Voronoi area Connect edge midpoints and triangle circumcenters The resulting quadrilaterals are not equareal Sum contributions from incident triangles + Only depends on vertex positioning - More complicated computations

25 How to compute A(v) Voronoi area Connect edge midpoints and triangle circumcenters The resulting quadrilaterals are not equiareal Sum contributions from incident triangles For obtuse triangles, circumcenter is outside the triangle -> negative areas! + Only depends on vertex positioning - More complicated computation - May introduce negative weights (obtuse triangles)

26 How to compute A(v) Voronoi area - compromise Connect edge midpoints with: triangle circumcenters, for non-obtuse triangles midpoint of opposite edge, for obtuse ones Sum contributions from incident triangles + Only depends on vertex positioning - More complicated computations

27 Linear Systems with Eigen #include <Eigen/SparseCholesky> //... SparseMatrix<double> A; // fill A VectorXd b, x; // fill b // solve Ax = b SimplicialLDLt<SparseMatrix<double> > solver; solver.compute(a); if(solver.info()!=success) return; // decomposition failed x = solver.solve(b); if(solver.info()!=success) return; // solving failed // solve for another right hand side: x1 = solver.solve(b1); Cholesky factorization works only for symmetric, positive definite A! Other solvers available, e.g. SparseLU:

28 Questions?