Geometric Modeling Assignment 5: Shape Deformation

Transcription

1 Geometric Modeling Assignment 5: Shape Deformation Acknowledgements: Olga Diamanti, Julian Panetta

2 Shape Deformation

3 Step 1: Select and Deform Handle Regions Draw vertex selection with mouse H 2 Move one handle H at a time to the deform mesh Leave some handles undeformed to fix vertices. H 1 Mesh S Code provided for the picking/dragging

4 Step 2: Smooth Mesh Remove high-frequency details from H 2 unconstrained vertices. This smooth mesh will be deformed; details are added back afterward Mesh B Smooth by solving a bi-laplacian system H 1 (minimize the Laplacian Energy)

5 Step 2: Smooth Mesh Remove high-frequency details from H 2 unconstrained vertices. This smooth mesh will be deformed; details are added back afterward Mesh B Smooth by solving a bi-laplacian system H 1 (minimize the Laplacian Energy): min v v T L! M 1 L! v s.t. v Hi = o Hi 8i Original vertex positions of handles

6 Step 3: Encode Displacements Compute the per-vertex displacements from B to S: Mesh B d i = v S i v B i These represent the details of the original surface. We will use these to add back (transformed) details after the deformation

7 Step 3: Encode Displacements d i = v S i v B i We want these details to rotate with Mesh B Mesh B as it is later deformed into Mesh B To do this, we express the displacements in a local frame on B, which is then rotated to align with B We just need to define the basis vectors for this frame

8 Step 3: Encode Displacements Construct the local frame for vertex i: 1. Calculate normal ni (for surface B) n di i

9 Step 3: Encode Displacements Construct the local frame for vertex i: 1. Calculate normal ni (for surface B) 2. Project all neighboring vertices to the tangent plane (perpendicular to ni) n di i

10 Step 3: Encode Displacements Construct the local frame for vertex i: 1. Calculate normal ni (for surface B) 2. Project all neighboring vertices to the tangent plane (perpendicular to ni) n di 3. Find neighbor j* for which projected edge (i, j) is longest. Normalize this edge vector and call it xi. i j

11 Step 3: Encode Displacements Construct the local frame for vertex i: 1. Calculate normal ni (for surface B) 2. Project all neighboring vertices to the tangent plane (perpendicular to ni) n di 3. Find neighbor j* for which projected edge (i, j) is longest. Normalize this edge vector and call it xi. xi i yi 4. Construct yi using the cross product, completing orthonormal frame (xi, yi, ni)

12 Step 3: Encode Displacements Decompose the displacement vectors in the frame s basis: d i = d x i x i + d y i y i + d n i n i (The basis is orthonormal, so you can do this just with inner products.) n di yi xi i

13 Step 4: Deform B H 2 User manipulates the handles You solve for the deformed smooth mesh, B Mesh B Solve bi-laplacian system again, using the new handle position as constraints: min v v T L! M 1 L! v s.t. v Hi = t(o Hi ) 8i H 1 Transformed vertex positions at handles.

14 Step 5: Add Transformed Detail Compute for each vertex vi the new local frame on B Calculate normal ni (for surface B ) Compute new frame basis (xi, yi, ni ) as before, but using the same edge (i, j*) as chosen for B (so frames are compatible). Construct the transformed displacement vectors: d 0 i = d x i x 0 i + d y i y0 i + d n i n 0 i

15 Step 5: Add Transformed Detail Compute for each vertex vi the new local frame on B Calculate normal ni (for surface B ) Compute new frame basis (xi, yi, ni ) as before, but using the same edge (i, j*) as chosen for B (so frames are compatible). Construct the transformed displacement vectors: d 0 i = d x i x 0 i + d y i y0 i + d n i n 0 i Add add them to B in to form S

16 Solving the Bi-Laplacian System min v v T L! M 1 L! v s.t. v Hi = o Hi 8i The positions of handle vertex must be imposed as hard constraints This can be done with the row/column removal trick from last assignment: A = L! M 1 L! = apple Aff A fc A cf A cc, b = 0 = apple bf b c Instead of Av = b, solve A ff v f = b A fc v c (So an efficient, sparse Cholesky factorization can be used)

17 Pre-factoring the System Eigen::SimplicialCholesky<SparseMatrixType, Eigen::RowMajor > solver; solver.compute(a_ff); // Factorize the free part of the system solver.compute() is slow. After the factorization is computed, solving for different vectors is fast. Factorization needs to be recomputed only when the matrix Aff changes (when new handles are drawn). For real-time performance and full credit on the assignment you must recompute the factorization only when necessary (not while the user is interactively transforming handles).