1 // Ceres Solver - A fast non-linear least squares minimizer 2 // Copyright 2010, 2011, 2012 Google Inc. All rights reserved. 3 // http://code.google.com/p/ceres-solver/ 4 // 5 // Redistribution and use in source and binary forms, with or without 6 // modification, are permitted provided that the following conditions are met: 7 // 8 // * Redistributions of source code must retain the above copyright notice, 9 // this list of conditions and the following disclaimer. 10 // * Redistributions in binary form must reproduce the above copyright notice, 11 // this list of conditions and the following disclaimer in the documentation 12 // and/or other materials provided with the distribution. 13 // * Neither the name of Google Inc. nor the names of its contributors may be 14 // used to endorse or promote products derived from this software without 15 // specific prior written permission. 16 // 17 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 18 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 21 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 22 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 23 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 24 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 25 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 26 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 27 // POSSIBILITY OF SUCH DAMAGE. 28 // 29 // Author: sameeragarwal@google.com (Sameer Agarwal) 30 // 31 // An iterative solver for solving the Schur complement/reduced camera 32 // linear system that arise in SfM problems. 33 34 #ifndef CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_ 35 #define CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_ 36 37 #include "ceres/linear_operator.h" 38 #include "ceres/linear_solver.h" 39 #include "ceres/partitioned_matrix_view.h" 40 #include "ceres/internal/eigen.h" 41 #include "ceres/internal/scoped_ptr.h" 42 #include "ceres/types.h" 43 44 namespace ceres { 45 namespace internal { 46 47 class BlockSparseMatrix; 48 49 // This class implements various linear algebraic operations related 50 // to the Schur complement without explicitly forming it. 51 // 52 // 53 // Given a reactangular linear system Ax = b, where 54 // 55 // A = [E F] 56 // 57 // The normal equations are given by 58 // 59 // A'Ax = A'b 60 // 61 // |E'E E'F||y| = |E'b| 62 // |F'E F'F||z| |F'b| 63 // 64 // and the Schur complement system is given by 65 // 66 // [F'F - F'E (E'E)^-1 E'F] z = F'b - F'E (E'E)^-1 E'b 67 // 68 // Now if we wish to solve Ax = b in the least squares sense, one way 69 // is to form this Schur complement system and solve it using 70 // Preconditioned Conjugate Gradients. 71 // 72 // The key operation in a conjugate gradient solver is the evaluation of the 73 // matrix vector product with the Schur complement 74 // 75 // S = F'F - F'E (E'E)^-1 E'F 76 // 77 // It is straightforward to see that matrix vector products with S can 78 // be evaluated without storing S in memory. Instead, given (E'E)^-1 79 // (which for our purposes is an easily inverted block diagonal 80 // matrix), it can be done in terms of matrix vector products with E, 81 // F and (E'E)^-1. This class implements this functionality and other 82 // auxilliary bits needed to implement a CG solver on the Schur 83 // complement using the PartitionedMatrixView object. 84 // 85 // THREAD SAFETY: This class is nqot thread safe. In particular, the 86 // RightMultiply (and the LeftMultiply) methods are not thread safe as 87 // they depend on mutable arrays used for the temporaries needed to 88 // compute the product y += Sx; 89 class ImplicitSchurComplement : public LinearOperator { 90 public: 91 // num_eliminate_blocks is the number of E blocks in the matrix 92 // A. 93 // 94 // preconditioner indicates whether the inverse of the matrix F'F 95 // should be computed or not as a preconditioner for the Schur 96 // Complement. 97 // 98 // TODO(sameeragarwal): Get rid of the two bools below and replace 99 // them with enums. 100 ImplicitSchurComplement(const LinearSolver::Options& options); 101 virtual ~ImplicitSchurComplement(); 102 103 // Initialize the Schur complement for a linear least squares 104 // problem of the form 105 // 106 // |A | x = |b| 107 // |diag(D)| |0| 108 // 109 // If D is null, then it is treated as a zero dimensional matrix. It 110 // is important that the matrix A have a BlockStructure object 111 // associated with it and has a block structure that is compatible 112 // with the SchurComplement solver. 113 void Init(const BlockSparseMatrix& A, const double* D, const double* b); 114 115 // y += Sx, where S is the Schur complement. 116 virtual void RightMultiply(const double* x, double* y) const; 117 118 // The Schur complement is a symmetric positive definite matrix, 119 // thus the left and right multiply operators are the same. LeftMultiply(const double * x,double * y)120 virtual void LeftMultiply(const double* x, double* y) const { 121 RightMultiply(x, y); 122 } 123 124 // y = (E'E)^-1 (E'b - E'F x). Given an estimate of the solution to 125 // the Schur complement system, this method computes the value of 126 // the e_block variables that were eliminated to form the Schur 127 // complement. 128 void BackSubstitute(const double* x, double* y); 129 num_rows()130 virtual int num_rows() const { return A_->num_cols_f(); } num_cols()131 virtual int num_cols() const { return A_->num_cols_f(); } rhs()132 const Vector& rhs() const { return rhs_; } 133 block_diagonal_EtE_inverse()134 const BlockSparseMatrix* block_diagonal_EtE_inverse() const { 135 return block_diagonal_EtE_inverse_.get(); 136 } 137 block_diagonal_FtF_inverse()138 const BlockSparseMatrix* block_diagonal_FtF_inverse() const { 139 return block_diagonal_FtF_inverse_.get(); 140 } 141 142 private: 143 void AddDiagonalAndInvert(const double* D, BlockSparseMatrix* matrix); 144 void UpdateRhs(); 145 146 const LinearSolver::Options& options_; 147 148 scoped_ptr<PartitionedMatrixViewBase> A_; 149 const double* D_; 150 const double* b_; 151 152 scoped_ptr<BlockSparseMatrix> block_diagonal_EtE_inverse_; 153 scoped_ptr<BlockSparseMatrix> block_diagonal_FtF_inverse_; 154 155 Vector rhs_; 156 157 // Temporary storage vectors used to implement RightMultiply. 158 mutable Vector tmp_rows_; 159 mutable Vector tmp_e_cols_; 160 mutable Vector tmp_e_cols_2_; 161 mutable Vector tmp_f_cols_; 162 }; 163 164 } // namespace internal 165 } // namespace ceres 166 167 #endif // CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_ 168