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// Ceres Solver - A fast non-linear least squares minimizer
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// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
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// http://code.google.com/p/ceres-solver/
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are met:
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// * Redistributions of source code must retain the above copyright notice,
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// this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above copyright notice,
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// this list of conditions and the following disclaimer in the documentation
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// and/or other materials provided with the distribution.
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// * Neither the name of Google Inc. nor the names of its contributors may be
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// used to endorse or promote products derived from this software without
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// specific prior written permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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// POSSIBILITY OF SUCH DAMAGE.
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// Author: sameeragarwal@google.com (Sameer Agarwal)
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#include "ceres/visibility_based_preconditioner.h"
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#include <glog/logging.h>
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#include "Eigen/Dense"
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#include "ceres/block_random_access_sparse_matrix.h"
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#include "ceres/block_sparse_matrix.h"
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#include "ceres/canonical_views_clustering.h"
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#include "ceres/collections_port.h"
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#include "ceres/detect_structure.h"
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#include "ceres/graph.h"
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#include "ceres/graph_algorithms.h"
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#include "ceres/linear_solver.h"
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#include "ceres/schur_eliminator.h"
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#include "ceres/visibility.h"
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#include "ceres/internal/scoped_ptr.h"
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// TODO(sameeragarwal): Currently these are magic weights for the
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// preconditioner construction. Move these higher up into the Options
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// struct and provide some guidelines for choosing them.
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// This will require some more work on the clustering algorithm and
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// possibly some more refactoring of the code.
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static const double kSizePenaltyWeight = 3.0;
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static const double kSimilarityPenaltyWeight = 0.0;
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#ifndef CERES_NO_SUITESPARSE
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VisibilityBasedPreconditioner::VisibilityBasedPreconditioner(
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const CompressedRowBlockStructure& bs,
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const LinearSolver::Options& options)
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CHECK_GT(options_.num_eliminate_blocks, 0);
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CHECK(options_.preconditioner_type == SCHUR_JACOBI ||
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options_.preconditioner_type == CLUSTER_JACOBI ||
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options_.preconditioner_type == CLUSTER_TRIDIAGONAL)
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<< "Unknown preconditioner type: " << options_.preconditioner_type;
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num_blocks_ = bs.cols.size() - options_.num_eliminate_blocks;
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CHECK_GT(num_blocks_, 0)
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<< "Jacobian should have atleast 1 f_block for "
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<< "visibility based preconditioning.";
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// Vector of camera block sizes
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block_size_.resize(num_blocks_);
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for (int i = 0; i < num_blocks_; ++i) {
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block_size_[i] = bs.cols[i + options_.num_eliminate_blocks].size;
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const time_t start_time = time(NULL);
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switch (options_.preconditioner_type) {
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ComputeSchurJacobiSparsity(bs);
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ComputeClusterJacobiSparsity(bs);
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case CLUSTER_TRIDIAGONAL:
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ComputeClusterTridiagonalSparsity(bs);
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LOG(FATAL) << "Unknown preconditioner type";
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const time_t structure_time = time(NULL);
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const time_t storage_time = time(NULL);
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const time_t eliminator_time = time(NULL);
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// Allocate temporary storage for a vector used during
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tmp_rhs_ = CHECK_NOTNULL(ss_.CreateDenseVector(NULL,
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const time_t init_time = time(NULL);
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VLOG(2) << "init time: "
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<< init_time - start_time
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<< " structure time: " << structure_time - start_time
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<< " storage time:" << storage_time - structure_time
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<< " eliminator time: " << eliminator_time - storage_time;
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VisibilityBasedPreconditioner::~VisibilityBasedPreconditioner() {
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if (factor_ != NULL) {
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if (tmp_rhs_ != NULL) {
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// Determine the sparsity structure of the SCHUR_JACOBI
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// preconditioner. SCHUR_JACOBI is an extreme case of a visibility
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// based preconditioner where each camera block corresponds to a
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// cluster and there is no interaction between clusters.
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void VisibilityBasedPreconditioner::ComputeSchurJacobiSparsity(
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const CompressedRowBlockStructure& bs) {
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num_clusters_ = num_blocks_;
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cluster_membership_.resize(num_blocks_);
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cluster_pairs_.clear();
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// Each camea block is a member of its own cluster and the only
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// cluster pairs are the self edges (i,i).
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for (int i = 0; i < num_clusters_; ++i) {
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cluster_membership_[i] = i;
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cluster_pairs_.insert(make_pair(i, i));
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// Determine the sparsity structure of the CLUSTER_JACOBI
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// preconditioner. It clusters cameras using their scene
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// visibility. The clusters form the diagonal blocks of the
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// preconditioner matrix.
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void VisibilityBasedPreconditioner::ComputeClusterJacobiSparsity(
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const CompressedRowBlockStructure& bs) {
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vector<set<int> > visibility;
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ComputeVisibility(bs, options_.num_eliminate_blocks, &visibility);
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CHECK_EQ(num_blocks_, visibility.size());
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ClusterCameras(visibility);
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cluster_pairs_.clear();
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for (int i = 0; i < num_clusters_; ++i) {
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cluster_pairs_.insert(make_pair(i, i));
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// Determine the sparsity structure of the CLUSTER_TRIDIAGONAL
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// preconditioner. It clusters cameras using using the scene
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// visibility and then finds the strongly interacting pairs of
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// clusters by constructing another graph with the clusters as
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// vertices and approximating it with a degree-2 maximum spanning
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// forest. The set of edges in this forest are the cluster pairs.
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void VisibilityBasedPreconditioner::ComputeClusterTridiagonalSparsity(
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const CompressedRowBlockStructure& bs) {
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vector<set<int> > visibility;
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ComputeVisibility(bs, options_.num_eliminate_blocks, &visibility);
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CHECK_EQ(num_blocks_, visibility.size());
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ClusterCameras(visibility);
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// Construct a weighted graph on the set of clusters, where the
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// edges are the number of 3D points/e_blocks visible in both the
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// clusters at the ends of the edge. Return an approximate degree-2
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// maximum spanning forest of this graph.
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vector<set<int> > cluster_visibility;
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ComputeClusterVisibility(visibility, &cluster_visibility);
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scoped_ptr<Graph<int> > cluster_graph(
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CHECK_NOTNULL(CreateClusterGraph(cluster_visibility)));
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scoped_ptr<Graph<int> > forest(
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CHECK_NOTNULL(Degree2MaximumSpanningForest(*cluster_graph)));
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ForestToClusterPairs(*forest, &cluster_pairs_);
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// Allocate storage for the preconditioner matrix.
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void VisibilityBasedPreconditioner::InitStorage(
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const CompressedRowBlockStructure& bs) {
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ComputeBlockPairsInPreconditioner(bs);
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m_.reset(new BlockRandomAccessSparseMatrix(block_size_, block_pairs_));
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// Call the canonical views algorithm and cluster the cameras based on
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// their visibility sets. The visibility set of a camera is the set of
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// e_blocks/3D points in the scene that are seen by it.
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// The cluster_membership_ vector is updated to indicate cluster
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// memberships for each camera block.
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void VisibilityBasedPreconditioner::ClusterCameras(
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const vector<set<int> >& visibility) {
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scoped_ptr<Graph<int> > schur_complement_graph(
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CHECK_NOTNULL(CreateSchurComplementGraph(visibility)));
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CanonicalViewsClusteringOptions options;
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options.size_penalty_weight = kSizePenaltyWeight;
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options.similarity_penalty_weight = kSimilarityPenaltyWeight;
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HashMap<int, int> membership;
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ComputeCanonicalViewsClustering(*schur_complement_graph,
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num_clusters_ = centers.size();
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CHECK_GT(num_clusters_, 0);
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VLOG(2) << "num_clusters: " << num_clusters_;
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FlattenMembershipMap(membership, &cluster_membership_);
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// Compute the block sparsity structure of the Schur complement
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// matrix. For each pair of cameras contributing a non-zero cell to
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// the schur complement, determine if that cell is present in the
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// preconditioner or not.
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// A pair of cameras contribute a cell to the preconditioner if they
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// are part of the same cluster or if the the two clusters that they
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// belong have an edge connecting them in the degree-2 maximum
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// For example, a camera pair (i,j) where i belonges to cluster1 and
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// j belongs to cluster2 (assume that cluster1 < cluster2).
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// The cell corresponding to (i,j) is present in the preconditioner
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// if cluster1 == cluster2 or the pair (cluster1, cluster2) were
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// connected by an edge in the degree-2 maximum spanning forest.
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// Since we have already expanded the forest into a set of camera
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// pairs/edges, including self edges, the check can be reduced to
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// checking membership of (cluster1, cluster2) in cluster_pairs_.
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void VisibilityBasedPreconditioner::ComputeBlockPairsInPreconditioner(
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const CompressedRowBlockStructure& bs) {
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block_pairs_.clear();
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for (int i = 0; i < num_blocks_; ++i) {
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block_pairs_.insert(make_pair(i, i));
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set<pair<int, int> > skipped_pairs;
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const int num_row_blocks = bs.rows.size();
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const int num_eliminate_blocks = options_.num_eliminate_blocks;
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// Iterate over each row of the matrix. The block structure of the
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// matrix is assumed to be sorted in order of the e_blocks/point
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// blocks. Thus all row blocks containing an e_block/point occur
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// contiguously. Further, if present, an e_block is always the first
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// parameter block in each row block. These structural assumptions
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// are common to all Schur complement based solvers in Ceres.
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// For each e_block/point block we identify the set of cameras
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// seeing it. The cross product of this set with itself is the set
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// of non-zero cells contibuted by this e_block.
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// The time complexity of this is O(nm^2) where, n is the number of
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// 3d points and m is the maximum number of cameras seeing any
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// point, which for most scenes is a fairly small number.
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while (r < num_row_blocks) {
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int e_block_id = bs.rows[r].cells.front().block_id;
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if (e_block_id >= num_eliminate_blocks) {
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// Skip the rows whose first block is an f_block.
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for (; r < num_row_blocks; ++r) {
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const CompressedRow& row = bs.rows[r];
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if (row.cells.front().block_id != e_block_id) {
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// Iterate over the blocks in the row, ignoring the first block
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// since it is the one to be eliminated and adding the rest to
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// the list of f_blocks associated with this e_block.
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for (int c = 1; c < row.cells.size(); ++c) {
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const Cell& cell = row.cells[c];
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const int f_block_id = cell.block_id - num_eliminate_blocks;
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CHECK_GE(f_block_id, 0);
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f_blocks.insert(f_block_id);
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for (set<int>::const_iterator block1 = f_blocks.begin();
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block1 != f_blocks.end();
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set<int>::const_iterator block2 = block1;
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for (; block2 != f_blocks.end(); ++block2) {
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if (IsBlockPairInPreconditioner(*block1, *block2)) {
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block_pairs_.insert(make_pair(*block1, *block2));
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skipped_pairs.insert(make_pair(*block1, *block2));
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// The remaining rows which do not contain any e_blocks.
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for (; r < num_row_blocks; ++r) {
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const CompressedRow& row = bs.rows[r];
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CHECK_GE(row.cells.front().block_id, num_eliminate_blocks);
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for (int i = 0; i < row.cells.size(); ++i) {
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const int block1 = row.cells[i].block_id - num_eliminate_blocks;
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for (int j = 0; j < row.cells.size(); ++j) {
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const int block2 = row.cells[j].block_id - num_eliminate_blocks;
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if (block1 <= block2) {
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if (IsBlockPairInPreconditioner(block1, block2)) {
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block_pairs_.insert(make_pair(block1, block2));
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skipped_pairs.insert(make_pair(block1, block2));
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VLOG(1) << "Block pair stats: "
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<< block_pairs_.size() << " included "
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<< skipped_pairs.size() << " excluded";
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// Initialize the SchurEliminator.
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void VisibilityBasedPreconditioner::InitEliminator(
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const CompressedRowBlockStructure& bs) {
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LinearSolver::Options eliminator_options;
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eliminator_options.num_eliminate_blocks = options_.num_eliminate_blocks;
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eliminator_options.num_threads = options_.num_threads;
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DetectStructure(bs, options_.num_eliminate_blocks,
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&eliminator_options.row_block_size,
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&eliminator_options.e_block_size,
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&eliminator_options.f_block_size);
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eliminator_.reset(SchurEliminatorBase::Create(eliminator_options));
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eliminator_->Init(options_.num_eliminate_blocks, &bs);
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// Update the values of the preconditioner matrix and factorize it.
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bool VisibilityBasedPreconditioner::Update(const BlockSparseMatrixBase& A,
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const time_t start_time = time(NULL);
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const int num_rows = m_->num_rows();
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CHECK_GT(num_rows, 0);
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// We need a dummy rhs vector and a dummy b vector since the Schur
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// eliminator combines the computation of the reduced camera matrix
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// with the computation of the right hand side of that linear
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// TODO(sameeragarwal): Perhaps its worth refactoring the
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// SchurEliminator::Eliminate function to allow NULL for the rhs. As
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// of now it does not seem to be worth the effort.
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Vector rhs = Vector::Zero(m_->num_rows());
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Vector b = Vector::Zero(A.num_rows());
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// Compute a subset of the entries of the Schur complement.
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eliminator_->Eliminate(&A, b.data(), D, m_.get(), rhs.data());
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// Try factorizing the matrix. For SCHUR_JACOBI and CLUSTER_JACOBI,
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// this should always succeed modulo some numerical/conditioning
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// problems. For CLUSTER_TRIDIAGONAL, in general the preconditioner
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// matrix as constructed is not positive definite. However, we will
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// go ahead and try factorizing it. If it works, great, otherwise we
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// scale all the cells in the preconditioner corresponding to the
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// edges in the degree-2 forest and that guarantees positive
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// definiteness. The proof of this fact can be found in Lemma 1 in
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// "Visibility Based Preconditioning for Bundle Adjustment".
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// Doing the factorization like this saves us matrix mass when
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// scaling is not needed, which is quite often in our experience.
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bool status = Factorize();
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// The scaling only affects the tri-diagonal case, since
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// ScaleOffDiagonalBlocks only pays attenion to the cells that
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// belong to the edges of the degree-2 forest. In the SCHUR_JACOBI
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// and the CLUSTER_JACOBI cases, the preconditioner is guaranteed to
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// be positive semidefinite.
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if (!status && options_.preconditioner_type == CLUSTER_TRIDIAGONAL) {
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VLOG(1) << "Unscaled factorization failed. Retrying with off-diagonal "
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ScaleOffDiagonalCells();
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status = Factorize();
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VLOG(2) << "Compute time: " << time(NULL) - start_time;
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// Consider the preconditioner matrix as meta-block matrix, whose
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// blocks correspond to the clusters. Then cluster pairs corresponding
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// to edges in the degree-2 forest are off diagonal entries of this
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// matrix. Scaling these off-diagonal entries by 1/2 forces this
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// matrix to be positive definite.
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void VisibilityBasedPreconditioner::ScaleOffDiagonalCells() {
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for (set< pair<int, int> >::const_iterator it = block_pairs_.begin();
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it != block_pairs_.end();
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const int block1 = it->first;
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const int block2 = it->second;
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if (!IsBlockPairOffDiagonal(block1, block2)) {
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int r, c, row_stride, col_stride;
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CellInfo* cell_info = m_->GetCell(block1, block2,
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&row_stride, &col_stride);
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CHECK(cell_info != NULL)
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<< "Cell missing for block pair (" << block1 << "," << block2 << ")"
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<< " cluster pair (" << cluster_membership_[block1]
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<< " " << cluster_membership_[block2] << ")";
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// Ah the magic of tri-diagonal matrices and diagonal
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// dominance. See Lemma 1 in "Visibility Based Preconditioning
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// For Bundle Adjustment".
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MatrixRef m(cell_info->values, row_stride, col_stride);
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m.block(r, c, block_size_[block1], block_size_[block2]) *= 0.5;
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// Compute the sparse Cholesky factorization of the preconditioner
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bool VisibilityBasedPreconditioner::Factorize() {
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// Extract the TripletSparseMatrix that is used for actually storing
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// S and convert it into a cholmod_sparse object.
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cholmod_sparse* lhs = ss_.CreateSparseMatrix(
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down_cast<BlockRandomAccessSparseMatrix*>(
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m_.get())->mutable_matrix());
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// The matrix is symmetric, and the upper triangular part of the
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// matrix contains the values.
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// Symbolic factorization is computed if we don't already have one
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if (factor_ == NULL) {
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factor_ = ss_.AnalyzeCholesky(lhs);
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bool status = ss_.Cholesky(lhs, factor_);
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void VisibilityBasedPreconditioner::RightMultiply(const double* x,
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SuiteSparse* ss = const_cast<SuiteSparse*>(&ss_);
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const int num_rows = m_->num_rows();
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memcpy(CHECK_NOTNULL(tmp_rhs_)->x, x, m_->num_rows() * sizeof(*x));
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cholmod_dense* solution = CHECK_NOTNULL(ss->Solve(factor_, tmp_rhs_));
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memcpy(y, solution->x, sizeof(*y) * num_rows);
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int VisibilityBasedPreconditioner::num_rows() const {
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return m_->num_rows();
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// Classify camera/f_block pairs as in and out of the preconditioner,
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// based on whether the cluster pair that they belong to is in the
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// preconditioner or not.
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bool VisibilityBasedPreconditioner::IsBlockPairInPreconditioner(
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const int block2) const {
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int cluster1 = cluster_membership_[block1];
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int cluster2 = cluster_membership_[block2];
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if (cluster1 > cluster2) {
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std::swap(cluster1, cluster2);
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return (cluster_pairs_.count(make_pair(cluster1, cluster2)) > 0);
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bool VisibilityBasedPreconditioner::IsBlockPairOffDiagonal(
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const int block2) const {
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return (cluster_membership_[block1] != cluster_membership_[block2]);
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// Convert a graph into a list of edges that includes self edges for
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void VisibilityBasedPreconditioner::ForestToClusterPairs(
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const Graph<int>& forest,
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HashSet<pair<int, int> >* cluster_pairs) const {
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CHECK_NOTNULL(cluster_pairs)->clear();
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const HashSet<int>& vertices = forest.vertices();
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CHECK_EQ(vertices.size(), num_clusters_);
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// Add all the cluster pairs corresponding to the edges in the
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for (HashSet<int>::const_iterator it1 = vertices.begin();
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it1 != vertices.end();
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const int cluster1 = *it1;
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cluster_pairs->insert(make_pair(cluster1, cluster1));
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const HashSet<int>& neighbors = forest.Neighbors(cluster1);
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for (HashSet<int>::const_iterator it2 = neighbors.begin();
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it2 != neighbors.end();
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const int cluster2 = *it2;
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if (cluster1 < cluster2) {
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cluster_pairs->insert(make_pair(cluster1, cluster2));
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// The visibilty set of a cluster is the union of the visibilty sets
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// of all its cameras. In other words, the set of points visible to
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// any camera in the cluster.
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void VisibilityBasedPreconditioner::ComputeClusterVisibility(
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const vector<set<int> >& visibility,
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vector<set<int> >* cluster_visibility) const {
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CHECK_NOTNULL(cluster_visibility)->resize(0);
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cluster_visibility->resize(num_clusters_);
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for (int i = 0; i < num_blocks_; ++i) {
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const int cluster_id = cluster_membership_[i];
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(*cluster_visibility)[cluster_id].insert(visibility[i].begin(),
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visibility[i].end());
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// Construct a graph whose vertices are the clusters, and the edge
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// weights are the number of 3D points visible to cameras in both the
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Graph<int>* VisibilityBasedPreconditioner::CreateClusterGraph(
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const vector<set<int> >& cluster_visibility) const {
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Graph<int>* cluster_graph = new Graph<int>;
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for (int i = 0; i < num_clusters_; ++i) {
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cluster_graph->AddVertex(i);
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for (int i = 0; i < num_clusters_; ++i) {
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const set<int>& cluster_i = cluster_visibility[i];
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for (int j = i+1; j < num_clusters_; ++j) {
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vector<int> intersection;
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const set<int>& cluster_j = cluster_visibility[j];
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set_intersection(cluster_i.begin(), cluster_i.end(),
558
cluster_j.begin(), cluster_j.end(),
559
back_inserter(intersection));
561
if (intersection.size() > 0) {
562
// Clusters interact strongly when they share a large number
563
// of 3D points. The degree-2 maximum spanning forest
564
// alorithm, iterates on the edges in decreasing order of
565
// their weight, which is the number of points shared by the
566
// two cameras that it connects.
567
cluster_graph->AddEdge(i, j, intersection.size());
571
return cluster_graph;
574
// Canonical views clustering returns a HashMap from vertices to
575
// cluster ids. Convert this into a flat array for quick lookup. It is
576
// possible that some of the vertices may not be associated with any
577
// cluster. In that case, randomly assign them to one of the clusters.
578
void VisibilityBasedPreconditioner::FlattenMembershipMap(
579
const HashMap<int, int>& membership_map,
580
vector<int>* membership_vector) const {
581
CHECK_NOTNULL(membership_vector)->resize(0);
582
membership_vector->resize(num_blocks_, -1);
583
// Iterate over the cluster membership map and update the
584
// cluster_membership_ vector assigning arbitrary cluster ids to
585
// the few cameras that have not been clustered.
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for (HashMap<int, int>::const_iterator it = membership_map.begin();
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it != membership_map.end();
589
const int camera_id = it->first;
590
int cluster_id = it->second;
592
// If the view was not clustered, randomly assign it to one of the
593
// clusters. This preserves the mathematical correctness of the
594
// preconditioner. If there are too many views which are not
595
// clustered, it may lead to some quality degradation though.
597
// TODO(sameeragarwal): Check if a large number of views have not
598
// been clustered and deal with it?
599
if (cluster_id == -1) {
600
cluster_id = camera_id % num_clusters_;
603
membership_vector->at(camera_id) = cluster_id;
607
#endif // CERES_NO_SUITESPARSE
609
} // namespace internal
added
removed
internal/ceres/visibility_based_preconditioner.cc
