Initial commit of Ceres Solver.
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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
+// http://code.google.com/p/ceres-solver/
+//
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are met:
+//
+// * Redistributions of source code must retain the above copyright notice,
+// this list of conditions and the following disclaimer.
+// * Redistributions in binary form must reproduce the above copyright notice,
+// this list of conditions and the following disclaimer in the documentation
+// and/or other materials provided with the distribution.
+// * Neither the name of Google Inc. nor the names of its contributors may be
+// used to endorse or promote products derived from this software without
+// specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+// POSSIBILITY OF SUCH DAMAGE.
+//
+// Author: sameeragarwal@google.com (Sameer Agarwal)
+//
+// An iterative solver for solving the Schur complement/reduced camera
+// linear system that arise in SfM problems.
+
+#ifndef CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_
+#define CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_
+
+#include "ceres/linear_operator.h"
+#include "ceres/partitioned_matrix_view.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/internal/scoped_ptr.h"
+#include "ceres/types.h"
+
+namespace ceres {
+namespace internal {
+
+class BlockSparseMatrix;
+class BlockSparseMatrixBase;
+
+// This class implements various linear algebraic operations related
+// to the Schur complement without explicitly forming it.
+//
+//
+// Given a reactangular linear system Ax = b, where
+//
+// A = [E F]
+//
+// The normal equations are given by
+//
+// A'Ax = A'b
+//
+// |E'E E'F||y| = |E'b|
+// |F'E F'F||z| |F'b|
+//
+// and the Schur complement system is given by
+//
+// [F'F - F'E (E'E)^-1 E'F] z = F'b - F'E (E'E)^-1 E'b
+//
+// Now if we wish to solve Ax = b in the least squares sense, one way
+// is to form this Schur complement system and solve it using
+// Preconditioned Conjugate Gradients.
+//
+// The key operation in a conjugate gradient solver is the evaluation of the
+// matrix vector product with the Schur complement
+//
+// S = F'F - F'E (E'E)^-1 E'F
+//
+// It is straightforward to see that matrix vector products with S can
+// be evaluated without storing S in memory. Instead, given (E'E)^-1
+// (which for our purposes is an easily inverted block diagonal
+// matrix), it can be done in terms of matrix vector products with E,
+// F and (E'E)^-1. This class implements this functionality and other
+// auxilliary bits needed to implement a CG solver on the Schur
+// complement using the PartitionedMatrixView object.
+//
+// THREAD SAFETY: This class is nqot thread safe. In particular, the
+// RightMultiply (and the LeftMultiply) methods are not thread safe as
+// they depend on mutable arrays used for the temporaries needed to
+// compute the product y += Sx;
+class ImplicitSchurComplement : public LinearOperator {
+ public:
+ // num_eliminate_blocks is the number of E blocks in the matrix
+ // A.
+ //
+ // constant_sparsity indicates if across calls to Init, the sparsity
+ // structure of the matrix A remains constant or not. This makes for
+ // significant savings across multiple matrices A, e.g. when used in
+ // conjunction with an optimization algorithm.
+ //
+ // preconditioner indicates whether the inverse of the matrix F'F
+ // should be computed or not as a preconditioner for the Schur
+ // Complement.
+ //
+ // TODO(sameeragarwal): Get rid of the two bools below and replace
+ // them with enums.
+ ImplicitSchurComplement(int num_eliminate_blocks,
+ bool constant_sparsity,
+ bool preconditioner);
+ virtual ~ImplicitSchurComplement();
+
+ // Initialize the Schur complement for a linear least squares
+ // problem of the form
+ //
+ // |A | x = |b|
+ // |diag(D)| |0|
+ //
+ // If D is null, then it is treated as a zero dimensional matrix. It
+ // is important that the matrix A have a BlockStructure object
+ // associated with it and has a block structure that is compatible
+ // with the SchurComplement solver.
+ void Init(const BlockSparseMatrixBase& A, const double* D, const double* b);
+
+ // y += Sx, where S is the Schur complement.
+ virtual void RightMultiply(const double* x, double* y) const;
+
+ // The Schur complement is a symmetric positive definite matrix,
+ // thus the left and right multiply operators are the same.
+ virtual void LeftMultiply(const double* x, double* y) const {
+ RightMultiply(x, y);
+ }
+
+ // y = (E'E)^-1 (E'b - E'F x). Given an estimate of the solution to
+ // the Schur complement system, this method computes the value of
+ // the e_block variables that were eliminated to form the Schur
+ // complement.
+ void BackSubstitute(const double* x, double* y);
+
+ virtual int num_rows() const { return A_->num_cols_f(); }
+ virtual int num_cols() const { return A_->num_cols_f(); }
+ const Vector& rhs() const { return rhs_; }
+
+ const BlockSparseMatrix* block_diagonal_EtE_inverse() const {
+ return block_diagonal_EtE_inverse_.get();
+ }
+
+ const BlockSparseMatrix* block_diagonal_FtF_inverse() const {
+ return block_diagonal_FtF_inverse_.get();
+ }
+
+ private:
+ void AddDiagonalAndInvert(const double* D, BlockSparseMatrix* matrix);
+ void UpdateRhs();
+
+ int num_eliminate_blocks_;
+ bool constant_sparsity_;
+ bool preconditioner_;
+
+ scoped_ptr<PartitionedMatrixView> A_;
+ const double* D_;
+ const double* b_;
+
+ scoped_ptr<BlockSparseMatrix> block_diagonal_EtE_inverse_;
+ scoped_ptr<BlockSparseMatrix> block_diagonal_FtF_inverse_;
+
+ Vector rhs_;
+
+ // Temporary storage vectors used to implement RightMultiply.
+ mutable Vector tmp_rows_;
+ mutable Vector tmp_e_cols_;
+ mutable Vector tmp_e_cols_2_;
+ mutable Vector tmp_f_cols_;
+};
+
+} // namespace internal
+} // namespace ceres
+
+#endif // CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_
