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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2017 Google Inc. All rights reserved.
+// http://ceres-solver.org/
+//
+// 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)
+//
+// Preconditioners for linear systems that arise in Structure from
+// Motion problems. VisibilityBasedPreconditioner implements:
+//
+//  CLUSTER_JACOBI
+//  CLUSTER_TRIDIAGONAL
+//
+// Detailed descriptions of these preconditions beyond what is
+// documented here can be found in
+//
+// Visibility Based Preconditioning for Bundle Adjustment
+// A. Kushal & S. Agarwal, CVPR 2012.
+//
+// http://www.cs.washington.edu/homes/sagarwal/vbp.pdf
+//
+// The two preconditioners share enough code that its most efficient
+// to implement them as part of the same code base.
+
+#ifndef CERES_INTERNAL_VISIBILITY_BASED_PRECONDITIONER_H_
+#define CERES_INTERNAL_VISIBILITY_BASED_PRECONDITIONER_H_
+
+#include <memory>
+#include <set>
+#include <unordered_map>
+#include <unordered_set>
+#include <utility>
+#include <vector>
+
+#include "ceres/graph.h"
+#include "ceres/linear_solver.h"
+#include "ceres/pair_hash.h"
+#include "ceres/preconditioner.h"
+#include "ceres/sparse_cholesky.h"
+
+namespace ceres {
+namespace internal {
+
+class BlockRandomAccessSparseMatrix;
+class BlockSparseMatrix;
+struct CompressedRowBlockStructure;
+class SchurEliminatorBase;
+
+// This class implements visibility based preconditioners for
+// Structure from Motion/Bundle Adjustment problems. The name
+// VisibilityBasedPreconditioner comes from the fact that the sparsity
+// structure of the preconditioner matrix is determined by analyzing
+// the visibility structure of the scene, i.e. which cameras see which
+// points.
+//
+// The key idea of visibility based preconditioning is to identify
+// cameras that we expect have strong interactions, and then using the
+// entries in the Schur complement matrix corresponding to these
+// camera pairs as an approximation to the full Schur complement.
+//
+// CLUSTER_JACOBI identifies these camera pairs by clustering cameras,
+// and considering all non-zero camera pairs within each cluster. The
+// clustering in the current implementation is done using the
+// Canonical Views algorithm of Simon et al. (see
+// canonical_views_clustering.h). For the purposes of clustering, the
+// similarity or the degree of interaction between a pair of cameras
+// is measured by counting the number of points visible in both the
+// cameras. Thus the name VisibilityBasedPreconditioner. Further, if we
+// were to permute the parameter blocks such that all the cameras in
+// the same cluster occur contiguously, the preconditioner matrix will
+// be a block diagonal matrix with blocks corresponding to the
+// clusters. Thus in analogy with the Jacobi preconditioner we refer
+// to this as the CLUSTER_JACOBI preconditioner.
+//
+// CLUSTER_TRIDIAGONAL adds more mass to the CLUSTER_JACOBI
+// preconditioner by considering the interaction between clusters and
+// identifying strong interactions between cluster pairs. This is done
+// by constructing a weighted graph on the clusters, with the weight
+// on the edges connecting two clusters proportional to the number of
+// 3D points visible to cameras in both the clusters. A degree-2
+// maximum spanning forest is identified in this graph and the camera
+// pairs contained in the edges of this forest are added to the
+// preconditioner. The detailed reasoning for this construction is
+// explained in the paper mentioned above.
+//
+// Degree-2 spanning trees and forests have the property that they
+// correspond to tri-diagonal matrices. Thus there exist a permutation
+// of the camera blocks under which the CLUSTER_TRIDIAGONAL
+// preconditioner matrix is a block tridiagonal matrix, and thus the
+// name for the preconditioner.
+//
+// Thread Safety: This class is NOT thread safe.
+//
+// Example usage:
+//
+//   LinearSolver::Options options;
+//   options.preconditioner_type = CLUSTER_JACOBI;
+//   options.elimination_groups.push_back(num_points);
+//   options.elimination_groups.push_back(num_cameras);
+//   VisibilityBasedPreconditioner preconditioner(
+//      *A.block_structure(), options);
+//   preconditioner.Update(A, NULL);
+//   preconditioner.RightMultiply(x, y);
+class VisibilityBasedPreconditioner : public BlockSparseMatrixPreconditioner {
+ public:
+  // Initialize the symbolic structure of the preconditioner. bs is
+  // the block structure of the linear system to be solved. It is used
+  // to determine the sparsity structure of the preconditioner matrix.
+  //
+  // It has the same structural requirement as other Schur complement
+  // based solvers. Please see schur_eliminator.h for more details.
+  VisibilityBasedPreconditioner(const CompressedRowBlockStructure& bs,
+                                const Preconditioner::Options& options);
+  VisibilityBasedPreconditioner(const VisibilityBasedPreconditioner&) = delete;
+  void operator=(const VisibilityBasedPreconditioner&) = delete;
+
+  virtual ~VisibilityBasedPreconditioner();
+
+  // Preconditioner interface
+  virtual void RightMultiply(const double* x, double* y) const;
+  virtual int num_rows() const;
+
+  friend class VisibilityBasedPreconditionerTest;
+
+ private:
+  virtual bool UpdateImpl(const BlockSparseMatrix& A, const double* D);
+  void ComputeClusterJacobiSparsity(const CompressedRowBlockStructure& bs);
+  void ComputeClusterTridiagonalSparsity(const CompressedRowBlockStructure& bs);
+  void InitStorage(const CompressedRowBlockStructure& bs);
+  void InitEliminator(const CompressedRowBlockStructure& bs);
+  LinearSolverTerminationType Factorize();
+  void ScaleOffDiagonalCells();
+
+  void ClusterCameras(const std::vector<std::set<int>>& visibility);
+  void FlattenMembershipMap(const std::unordered_map<int, int>& membership_map,
+                            std::vector<int>* membership_vector) const;
+  void ComputeClusterVisibility(
+      const std::vector<std::set<int>>& visibility,
+      std::vector<std::set<int>>* cluster_visibility) const;
+  WeightedGraph<int>* CreateClusterGraph(
+      const std::vector<std::set<int>>& visibility) const;
+  void ForestToClusterPairs(const WeightedGraph<int>& forest,
+                            std::unordered_set<std::pair<int, int>, pair_hash>* cluster_pairs) const;
+  void ComputeBlockPairsInPreconditioner(const CompressedRowBlockStructure& bs);
+  bool IsBlockPairInPreconditioner(int block1, int block2) const;
+  bool IsBlockPairOffDiagonal(int block1, int block2) const;
+
+  Preconditioner::Options options_;
+
+  // Number of parameter blocks in the schur complement.
+  int num_blocks_;
+  int num_clusters_;
+
+  // Sizes of the blocks in the schur complement.
+  std::vector<int> block_size_;
+
+  // Mapping from cameras to clusters.
+  std::vector<int> cluster_membership_;
+
+  // Non-zero camera pairs from the schur complement matrix that are
+  // present in the preconditioner, sorted by row (first element of
+  // each pair), then column (second).
+  std::set<std::pair<int, int>> block_pairs_;
+
+  // Set of cluster pairs (including self pairs (i,i)) in the
+  // preconditioner.
+  std::unordered_set<std::pair<int, int>, pair_hash> cluster_pairs_;
+  std::unique_ptr<SchurEliminatorBase> eliminator_;
+
+  // Preconditioner matrix.
+  std::unique_ptr<BlockRandomAccessSparseMatrix> m_;
+  std::unique_ptr<SparseCholesky> sparse_cholesky_;
+};
+
+}  // namespace internal
+}  // namespace ceres
+
+#endif  // CERES_INTERNAL_VISIBILITY_BASED_PRECONDITIONER_H_