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269 lines
10 KiB
269 lines
10 KiB
1 year ago
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// This file is part of libigl, a simple C++ geometry processing library.
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//
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// Copyright (C) 2020 Xiangyu Kong <xiangyu.kong@mail.utoronto.ca>
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//
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// This Source Code Form is subject to the terms of the Mozilla Public License
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// v. 2.0. If a copy of the MPL was not distributed with this file, You can
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// obtain one at http://mozilla.org/MPL/2.0/.
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#include "direct_delta_mush.h"
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#include "cotmatrix.h"
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template <
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typename DerivedV,
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typename DerivedOmega,
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typename DerivedU>
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IGL_INLINE void igl::direct_delta_mush(
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const Eigen::MatrixBase<DerivedV> & V,
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const std::vector<Eigen::Affine3d, Eigen::aligned_allocator<Eigen::Affine3d> > & T,
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const Eigen::MatrixBase<DerivedOmega> & Omega,
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Eigen::PlainObjectBase<DerivedU> & U)
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{
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using namespace Eigen;
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// Shape checks
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assert(V.cols() == 3 && "V should contain 3D positions.");
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assert(Omega.rows() == V.rows() && "Omega contain the same number of rows as V.");
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assert(Omega.cols() == T.size() * 10 && "Omega should have #T*10 columns.");
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typedef typename DerivedV::Scalar Scalar;
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int n = V.rows();
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int m = T.size();
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// V_homogeneous: #V by 4, homogeneous version of V
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// Note:
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// In the paper, the rest pose vertices are represented in U \in R^{4 x #V}
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// Thus the formulae involving U would differ from the paper by a transpose.
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Matrix<Scalar, Dynamic, 4> V_homogeneous(n, 4);
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V_homogeneous << V, Matrix<Scalar, Dynamic, 1>::Ones(n, 1);
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U.resize(n, 3);
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for (int i = 0; i < n; ++i)
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{
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// Construct Q matrix using Omega and Transformations
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Matrix<Scalar, 4, 4> Q_mat(4, 4);
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Q_mat = Matrix<Scalar, 4, 4>::Zero(4, 4);
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for (int j = 0; j < m; ++j)
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{
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Matrix<typename DerivedOmega::Scalar, 4, 4> Omega_curr(4, 4);
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Matrix<typename DerivedOmega::Scalar, 10, 1> curr = Omega.block(i, j * 10, 1, 10).transpose();
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Omega_curr << curr(0), curr(1), curr(2), curr(3),
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curr(1), curr(4), curr(5), curr(6),
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curr(2), curr(5), curr(7), curr(8),
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curr(3), curr(6), curr(8), curr(9);
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Affine3d M_curr = T[j];
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Q_mat += M_curr.matrix() * Omega_curr;
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}
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// Normalize so that the last element is 1
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Q_mat /= Q_mat(Q_mat.rows() - 1, Q_mat.cols() - 1);
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Matrix<Scalar, 3, 3> Q_i = Q_mat.block(0, 0, 3, 3);
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Matrix<Scalar, 3, 1> q_i = Q_mat.block(0, 3, 3, 1);
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Matrix<Scalar, 3, 1> p_i = Q_mat.block(3, 0, 1, 3).transpose();
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// Get rotation and translation matrices using SVD
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Matrix<Scalar, 3, 3> SVD_i = Q_i - q_i * p_i.transpose();
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JacobiSVD<Matrix<Scalar, 3, 3>> svd;
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svd.compute(SVD_i, ComputeFullU | ComputeFullV);
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Matrix<Scalar, 3, 3> R_i = svd.matrixU() * svd.matrixV().transpose();
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Matrix<Scalar, 3, 1> t_i = q_i - R_i * p_i;
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// Gamma final transformation matrix
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Matrix<Scalar, 3, 4> Gamma_i(3, 4);
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Gamma_i.block(0, 0, 3, 3) = R_i;
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Gamma_i.block(0, 3, 3, 1) = t_i;
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// Final deformed position
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Matrix<Scalar, 4, 1> v_i = V_homogeneous.row(i);
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U.row(i) = Gamma_i * v_i;
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}
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}
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template <
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typename DerivedV,
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typename DerivedF,
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typename DerivedW,
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typename DerivedOmega>
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IGL_INLINE void igl::direct_delta_mush_precomputation(
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const Eigen::MatrixBase<DerivedV> & V,
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const Eigen::MatrixBase<DerivedF> & F,
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const Eigen::MatrixBase<DerivedW> & W,
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const int p,
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const typename DerivedV::Scalar lambda,
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const typename DerivedV::Scalar kappa,
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const typename DerivedV::Scalar alpha,
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Eigen::PlainObjectBase<DerivedOmega> & Omega)
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{
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using namespace Eigen;
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// Shape checks
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assert(V.cols() == 3 && "V should contain 3D positions.");
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assert(F.cols() == 3 && "F should contain triangles.");
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assert(W.rows() == V.rows() && "W.rows() should be equal to V.rows().");
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// Parameter checks
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assert(p > 0 && "Laplacian iteration p should be positive.");
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assert(lambda > 0 && "lambda should be positive.");
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assert(kappa > 0 && kappa < lambda && "kappa should be positive and less than lambda.");
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assert(alpha >= 0 && alpha < 1 && "alpha should be non-negative and less than 1.");
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typedef typename DerivedV::Scalar Scalar;
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// lambda helper
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// Given a square matrix, extract the upper triangle (including diagonal) to an array.
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// E.g. 1 2 3 4
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// 5 6 7 8 -> [1, 2, 3, 4, 6, 7, 8, 11, 12, 16]
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// 9 10 11 12 0 1 2 3 4 5 6 7 8 9
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// 13 14 15 16
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auto extract_upper_triangle = [](
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const Matrix<Scalar, Dynamic, Dynamic> & full) -> Matrix<Scalar, Dynamic, 1>
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{
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int dims = full.rows();
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Matrix<Scalar, Dynamic, 1> upper_triangle((dims * (dims + 1)) / 2);
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int vector_idx = 0;
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for (int i = 0; i < dims; ++i)
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{
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for (int j = i; j < dims; ++j)
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{
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upper_triangle(vector_idx) = full(i, j);
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vector_idx++;
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}
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}
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return upper_triangle;
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};
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const int n = V.rows();
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const int m = W.cols();
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// V_homogeneous: #V by 4, homogeneous version of V
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// Note:
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// in the paper, the rest pose vertices are represented in U \in R^{4 \times #V}
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// Thus the formulae involving U would differ from the paper by a transpose.
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Matrix<Scalar, Dynamic, 4> V_homogeneous(n, 4);
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V_homogeneous << V, Matrix<Scalar, Dynamic, 1>::Ones(n);
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// Identity matrix of #V by #V
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SparseMatrix<Scalar> I(n, n);
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I.setIdentity();
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// Laplacian matrix of #V by #V
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// L_bar = L \times D_L^{-1}
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SparseMatrix<Scalar> L;
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igl::cotmatrix(V, F, L);
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L = -L;
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// Inverse of diagonal matrix = reciprocal elements in diagonal
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Matrix<Scalar, Dynamic, 1> D_L = L.diagonal();
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// D_L = D_L.array().pow(-1); // Not using this since not sure if diagonal contains 0
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for (int i = 0; i < D_L.size(); ++i)
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{
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if (D_L(i) != 0)
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{
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D_L(i) = 1 / D_L(i);
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}
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}
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SparseMatrix<Scalar> D_L_inv = D_L.asDiagonal().toDenseMatrix().sparseView();
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SparseMatrix<Scalar> L_bar = L * D_L_inv;
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// Implicitly and iteratively solve for W'
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// w'_{ij} = \sum_{k=1}^{n}{C_{ki} w_{kj}} where C = (I + kappa L_bar)^{-p}:
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// W' = C^T \times W => c^T W_k = W_{k-1} where c = (I + kappa L_bar)
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// C positive semi-definite => ldlt solver
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SimplicialLDLT<SparseMatrix<Scalar>> ldlt_W_prime;
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SparseMatrix<Scalar> c(I + kappa * L_bar);
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// working copy
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DerivedW W_prime(W);
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ldlt_W_prime.compute(c.transpose());
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for (int iter = 0; iter < p; ++iter)
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{
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W_prime = ldlt_W_prime.solve(W_prime);
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}
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// U_precomputed: #V by 10
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// Cache u_i^T \dot u_i \in R^{4 x 4} to reduce computation time.
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Matrix<Scalar, Dynamic, 10> U_precomputed(n, 10);
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for (int k = 0; k < n; ++k)
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{
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Matrix<Scalar, 4, 4> u_full = V_homogeneous.row(k).transpose() * V_homogeneous.row(k);
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U_precomputed.row(k) = extract_upper_triangle(u_full);
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}
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// U_prime: #V by #T*10 of u_{jx}
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// Each column of U_prime (u_{jx}) is the element-wise product of
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// W_j and U_precomputed_x where j \in {1...m}, x \in {1...10}
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Matrix<Scalar, Dynamic, Dynamic> U_prime(n, m * 10);
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for (int j = 0; j < m; ++j)
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{
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Matrix<Scalar, Dynamic, 1> w_j = W.col(j);
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for (int x = 0; x < 10; ++x)
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{
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Matrix<Scalar, Dynamic, 1> u_x = U_precomputed.col(x);
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U_prime.col(10 * j + x) = w_j.array() * u_x.array();
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}
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}
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// Implicitly and iteratively solve for Psi: #V by #T*10 of \Psi_{ij}s.
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// Note: Using dense matrices to solve for Psi will cause the program to hang.
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// The following won't work
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// Matrix<Scalar, Dynamic, Dynamic> Psi(U_prime);
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// Matrix<Scalar, Dynamic, Dynamic> b((I + lambda * L_bar).transpose());
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// for (int iter = 0; iter < p; ++iter)
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// {
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// Psi = b.ldlt().solve(Psi); // hangs here
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// }
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// Convert to sparse matrices and compute
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Matrix<Scalar, Dynamic, Dynamic> Psi = U_prime.sparseView();
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SparseMatrix<Scalar> b = (I + lambda * L_bar).transpose();
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SimplicialLDLT<SparseMatrix<Scalar>> ldlt_Psi;
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ldlt_Psi.compute(b);
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for (int iter = 0; iter < p; ++iter)
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{
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Psi = ldlt_Psi.solve(Psi);
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}
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// P: #V by 10 precomputed upper triangle of
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// p_i p_i^T , p_i
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// p_i^T , 1
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// where p_i = (\sum_{j=1}^{n} Psi_{ij})'s top right 3 by 1 column
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Matrix<Scalar, Dynamic, 10> P(n, 10);
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for (int i = 0; i < n; ++i)
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{
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Matrix<Scalar, 3, 1> p_i = Matrix<Scalar, 3, 1>::Zero(3);
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Scalar last = 0;
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for (int j = 0; j < m; ++j)
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{
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Matrix<Scalar, 3, 1> p_i_curr(3);
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p_i_curr << Psi(i, j * 10 + 3), Psi(i, j * 10 + 6), Psi(i, j * 10 + 8);
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p_i += p_i_curr;
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last += Psi(i, j * 10 + 9);
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}
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p_i /= last; // normalize
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Matrix<Scalar, 4, 4> p_matrix(4, 4);
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p_matrix.block(0, 0, 3, 3) = p_i * p_i.transpose();
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p_matrix.block(0, 3, 3, 1) = p_i;
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p_matrix.block(3, 0, 1, 3) = p_i.transpose();
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p_matrix(3, 3) = 1;
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P.row(i) = extract_upper_triangle(p_matrix);
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}
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// Omega
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Omega.resize(n, m * 10);
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for (int i = 0; i < n; ++i)
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{
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Matrix<Scalar, 10, 1> p_vector = P.row(i);
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for (int j = 0; j < m; ++j)
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{
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Matrix<Scalar, 10, 1> Omega_curr(10);
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Matrix<Scalar, 10, 1> Psi_curr = Psi.block(i, j * 10, 1, 10).transpose();
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Omega_curr = (1. - alpha) * Psi_curr + alpha * W_prime(i, j) * p_vector;
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Omega.block(i, j * 10, 1, 10) = Omega_curr.transpose();
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}
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}
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}
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#ifdef IGL_STATIC_LIBRARY
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// Explicit template instantiation
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template void igl::direct_delta_mush<Eigen::Matrix<double, -1, -1, 0, -1, -1>, Eigen::Matrix<double, -1, -1, 0, -1, -1>, Eigen::Matrix<double, -1, -1, 0, -1, -1> >(Eigen::MatrixBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> > const&, std::vector<Eigen::Transform<double, 3, 2, 0>, Eigen::aligned_allocator<Eigen::Transform<double, 3, 2, 0> > > const&, Eigen::MatrixBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >&); template void igl::direct_delta_mush_precomputation<Eigen::Matrix<double, -1, -1, 0, -1, -1>, Eigen::Matrix<int, -1, -1, 0, -1, -1>, Eigen::Matrix<double, -1, -1, 0, -1, -1>, Eigen::Matrix<double, -1, -1, 0, -1, -1> >(Eigen::MatrixBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> > const&, Eigen::MatrixBase<Eigen::Matrix<int, -1, -1, 0, -1, -1> > const&, Eigen::MatrixBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> > const&, int, Eigen::Matrix<double, -1, -1, 0, -1, -1>::Scalar, Eigen::Matrix<double, -1, -1, 0, -1, -1>::Scalar, Eigen::Matrix<double, -1, -1, 0, -1, -1>::Scalar, Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >&);
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#endif
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