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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2022 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
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// Author: sameeragarwal@google.com (Sameer Agarwal)
#include "ceres/visibility_based_preconditioner.h"
#include <algorithm>
#include <functional>
#include <iterator>
#include <memory>
#include <set>
#include <utility>
#include <vector>
#include "Eigen/Dense"
#include "ceres/block_random_access_sparse_matrix.h"
#include "ceres/block_sparse_matrix.h"
#include "ceres/canonical_views_clustering.h"
#include "ceres/graph.h"
#include "ceres/graph_algorithms.h"
#include "ceres/linear_solver.h"
#include "ceres/schur_eliminator.h"
#include "ceres/single_linkage_clustering.h"
#include "ceres/visibility.h"
#include "glog/logging.h"
namespace ceres::internal {
using std::make_pair;
using std::pair;
using std::set;
using std::swap;
using std::vector;
// TODO(sameeragarwal): Currently these are magic weights for the
// preconditioner construction. Move these higher up into the Options
// struct and provide some guidelines for choosing them.
//
// This will require some more work on the clustering algorithm and
// possibly some more refactoring of the code.
static constexpr double kCanonicalViewsSizePenaltyWeight = 3.0;
static constexpr double kCanonicalViewsSimilarityPenaltyWeight = 0.0;
static constexpr double kSingleLinkageMinSimilarity = 0.9;
VisibilityBasedPreconditioner::VisibilityBasedPreconditioner(
const CompressedRowBlockStructure& bs, Preconditioner::Options options)
: options_(std::move(options)), num_blocks_(0), num_clusters_(0) {
CHECK_GT(options_.elimination_groups.size(), 1);
CHECK_GT(options_.elimination_groups[0], 0);
CHECK(options_.type == CLUSTER_JACOBI || options_.type == CLUSTER_TRIDIAGONAL)
<< "Unknown preconditioner type: " << options_.type;
num_blocks_ = bs.cols.size() - options_.elimination_groups[0];
CHECK_GT(num_blocks_, 0) << "Jacobian should have at least 1 f_block for "
<< "visibility based preconditioning.";
CHECK(options_.context != nullptr);
// Vector of camera block sizes
block_size_.resize(num_blocks_);
for (int i = 0; i < num_blocks_; ++i) {
block_size_[i] = bs.cols[i + options_.elimination_groups[0]].size;
}
const time_t start_time = time(nullptr);
switch (options_.type) {
case CLUSTER_JACOBI:
ComputeClusterJacobiSparsity(bs);
break;
case CLUSTER_TRIDIAGONAL:
ComputeClusterTridiagonalSparsity(bs);
break;
default:
LOG(FATAL) << "Unknown preconditioner type";
}
const time_t structure_time = time(nullptr);
InitStorage(bs);
const time_t storage_time = time(nullptr);
InitEliminator(bs);
const time_t eliminator_time = time(nullptr);
LinearSolver::Options sparse_cholesky_options;
sparse_cholesky_options.sparse_linear_algebra_library_type =
options_.sparse_linear_algebra_library_type;
sparse_cholesky_options.ordering_type = options_.ordering_type;
sparse_cholesky_ = SparseCholesky::Create(sparse_cholesky_options);
const time_t init_time = time(nullptr);
VLOG(2) << "init time: " << init_time - start_time
<< " structure time: " << structure_time - start_time
<< " storage time:" << storage_time - structure_time
<< " eliminator time: " << eliminator_time - storage_time;
}
VisibilityBasedPreconditioner::~VisibilityBasedPreconditioner() = default;
// Determine the sparsity structure of the CLUSTER_JACOBI
// preconditioner. It clusters cameras using their scene
// visibility. The clusters form the diagonal blocks of the
// preconditioner matrix.
void VisibilityBasedPreconditioner::ComputeClusterJacobiSparsity(
const CompressedRowBlockStructure& bs) {
vector<set<int>> visibility;
ComputeVisibility(bs, options_.elimination_groups[0], &visibility);
CHECK_EQ(num_blocks_, visibility.size());
ClusterCameras(visibility);
cluster_pairs_.clear();
for (int i = 0; i < num_clusters_; ++i) {
cluster_pairs_.insert(make_pair(i, i));
}
}
// Determine the sparsity structure of the CLUSTER_TRIDIAGONAL
// preconditioner. It clusters cameras using the scene visibility and
// then finds the strongly interacting pairs of clusters by
// constructing another graph with the clusters as vertices and
// approximating it with a degree-2 maximum spanning forest. The set
// of edges in this forest are the cluster pairs.
void VisibilityBasedPreconditioner::ComputeClusterTridiagonalSparsity(
const CompressedRowBlockStructure& bs) {
vector<set<int>> visibility;
ComputeVisibility(bs, options_.elimination_groups[0], &visibility);
CHECK_EQ(num_blocks_, visibility.size());
ClusterCameras(visibility);
// Construct a weighted graph on the set of clusters, where the
// edges are the number of 3D points/e_blocks visible in both the
// clusters at the ends of the edge. Return an approximate degree-2
// maximum spanning forest of this graph.
vector<set<int>> cluster_visibility;
ComputeClusterVisibility(visibility, &cluster_visibility);
auto cluster_graph = CreateClusterGraph(cluster_visibility);
CHECK(cluster_graph != nullptr);
auto forest = Degree2MaximumSpanningForest(*cluster_graph);
CHECK(forest != nullptr);
ForestToClusterPairs(*forest, &cluster_pairs_);
}
// Allocate storage for the preconditioner matrix.
void VisibilityBasedPreconditioner::InitStorage(
const CompressedRowBlockStructure& bs) {
ComputeBlockPairsInPreconditioner(bs);
m_ = std::make_unique<BlockRandomAccessSparseMatrix>(block_size_,
block_pairs_);
}
// Call the canonical views algorithm and cluster the cameras based on
// their visibility sets. The visibility set of a camera is the set of
// e_blocks/3D points in the scene that are seen by it.
//
// The cluster_membership_ vector is updated to indicate cluster
// memberships for each camera block.
void VisibilityBasedPreconditioner::ClusterCameras(
const vector<set<int>>& visibility) {
auto schur_complement_graph = CreateSchurComplementGraph(visibility);
CHECK(schur_complement_graph != nullptr);
std::unordered_map<int, int> membership;
if (options_.visibility_clustering_type == CANONICAL_VIEWS) {
vector<int> centers;
CanonicalViewsClusteringOptions clustering_options;
clustering_options.size_penalty_weight = kCanonicalViewsSizePenaltyWeight;
clustering_options.similarity_penalty_weight =
kCanonicalViewsSimilarityPenaltyWeight;
ComputeCanonicalViewsClustering(
clustering_options, *schur_complement_graph, &centers, &membership);
num_clusters_ = centers.size();
} else if (options_.visibility_clustering_type == SINGLE_LINKAGE) {
SingleLinkageClusteringOptions clustering_options;
clustering_options.min_similarity = kSingleLinkageMinSimilarity;
num_clusters_ = ComputeSingleLinkageClustering(
clustering_options, *schur_complement_graph, &membership);
} else {
LOG(FATAL) << "Unknown visibility clustering algorithm.";
}
CHECK_GT(num_clusters_, 0);
VLOG(2) << "num_clusters: " << num_clusters_;
FlattenMembershipMap(membership, &cluster_membership_);
}
// Compute the block sparsity structure of the Schur complement
// matrix. For each pair of cameras contributing a non-zero cell to
// the schur complement, determine if that cell is present in the
// preconditioner or not.
//
// A pair of cameras contribute a cell to the preconditioner if they
// are part of the same cluster or if the two clusters that they
// belong have an edge connecting them in the degree-2 maximum
// spanning forest.
//
// For example, a camera pair (i,j) where i belongs to cluster1 and
// j belongs to cluster2 (assume that cluster1 < cluster2).
//
// The cell corresponding to (i,j) is present in the preconditioner
// if cluster1 == cluster2 or the pair (cluster1, cluster2) were
// connected by an edge in the degree-2 maximum spanning forest.
//
// Since we have already expanded the forest into a set of camera
// pairs/edges, including self edges, the check can be reduced to
// checking membership of (cluster1, cluster2) in cluster_pairs_.
void VisibilityBasedPreconditioner::ComputeBlockPairsInPreconditioner(
const CompressedRowBlockStructure& bs) {
block_pairs_.clear();
for (int i = 0; i < num_blocks_; ++i) {
block_pairs_.insert(make_pair(i, i));
}
int r = 0;
const int num_row_blocks = bs.rows.size();
const int num_eliminate_blocks = options_.elimination_groups[0];
// Iterate over each row of the matrix. The block structure of the
// matrix is assumed to be sorted in order of the e_blocks/point
// blocks. Thus all row blocks containing an e_block/point occur
// contiguously. Further, if present, an e_block is always the first
// parameter block in each row block. These structural assumptions
// are common to all Schur complement based solvers in Ceres.
//
// For each e_block/point block we identify the set of cameras
// seeing it. The cross product of this set with itself is the set
// of non-zero cells contributed by this e_block.
//
// The time complexity of this is O(nm^2) where, n is the number of
// 3d points and m is the maximum number of cameras seeing any
// point, which for most scenes is a fairly small number.
while (r < num_row_blocks) {
int e_block_id = bs.rows[r].cells.front().block_id;
if (e_block_id >= num_eliminate_blocks) {
// Skip the rows whose first block is an f_block.
break;
}
set<int> f_blocks;
for (; r < num_row_blocks; ++r) {
const CompressedRow& row = bs.rows[r];
if (row.cells.front().block_id != e_block_id) {
break;
}
// Iterate over the blocks in the row, ignoring the first block
// since it is the one to be eliminated and adding the rest to
// the list of f_blocks associated with this e_block.
for (int c = 1; c < row.cells.size(); ++c) {
const Cell& cell = row.cells[c];
const int f_block_id = cell.block_id - num_eliminate_blocks;
CHECK_GE(f_block_id, 0);
f_blocks.insert(f_block_id);
}
}
for (auto block1 = f_blocks.begin(); block1 != f_blocks.end(); ++block1) {
auto block2 = block1;
++block2;
for (; block2 != f_blocks.end(); ++block2) {
if (IsBlockPairInPreconditioner(*block1, *block2)) {
block_pairs_.emplace(*block1, *block2);
}
}
}
}
// The remaining rows which do not contain any e_blocks.
for (; r < num_row_blocks; ++r) {
const CompressedRow& row = bs.rows[r];
CHECK_GE(row.cells.front().block_id, num_eliminate_blocks);
for (int i = 0; i < row.cells.size(); ++i) {
const int block1 = row.cells[i].block_id - num_eliminate_blocks;
for (const auto& cell : row.cells) {
const int block2 = cell.block_id - num_eliminate_blocks;
if (block1 <= block2) {
if (IsBlockPairInPreconditioner(block1, block2)) {
block_pairs_.insert(make_pair(block1, block2));
}
}
}
}
}
VLOG(1) << "Block pair stats: " << block_pairs_.size();
}
// Initialize the SchurEliminator.
void VisibilityBasedPreconditioner::InitEliminator(
const CompressedRowBlockStructure& bs) {
LinearSolver::Options eliminator_options;
eliminator_options.elimination_groups = options_.elimination_groups;
eliminator_options.num_threads = options_.num_threads;
eliminator_options.e_block_size = options_.e_block_size;
eliminator_options.f_block_size = options_.f_block_size;
eliminator_options.row_block_size = options_.row_block_size;
eliminator_options.context = options_.context;
eliminator_ = SchurEliminatorBase::Create(eliminator_options);
const bool kFullRankETE = true;
eliminator_->Init(
eliminator_options.elimination_groups[0], kFullRankETE, &bs);
}
// Update the values of the preconditioner matrix and factorize it.
bool VisibilityBasedPreconditioner::UpdateImpl(const BlockSparseMatrix& A,
const double* D) {
const time_t start_time = time(nullptr);
const int num_rows = m_->num_rows();
CHECK_GT(num_rows, 0);
// Compute a subset of the entries of the Schur complement.
eliminator_->Eliminate(
BlockSparseMatrixData(A), nullptr, D, m_.get(), nullptr);
// Try factorizing the matrix. For CLUSTER_JACOBI, this should
// always succeed modulo some numerical/conditioning problems. For
// CLUSTER_TRIDIAGONAL, in general the preconditioner matrix as
// constructed is not positive definite. However, we will go ahead
// and try factorizing it. If it works, great, otherwise we scale
// all the cells in the preconditioner corresponding to the edges in
// the degree-2 forest and that guarantees positive
// definiteness. The proof of this fact can be found in Lemma 1 in
// "Visibility Based Preconditioning for Bundle Adjustment".
//
// Doing the factorization like this saves us matrix mass when
// scaling is not needed, which is quite often in our experience.
LinearSolverTerminationType status = Factorize();
if (status == LinearSolverTerminationType::FATAL_ERROR) {
return false;
}
// The scaling only affects the tri-diagonal case, since
// ScaleOffDiagonalBlocks only pays attention to the cells that
// belong to the edges of the degree-2 forest. In the CLUSTER_JACOBI
// case, the preconditioner is guaranteed to be positive
// semidefinite.
if (status == LinearSolverTerminationType::FAILURE &&
options_.type == CLUSTER_TRIDIAGONAL) {
VLOG(1) << "Unscaled factorization failed. Retrying with off-diagonal "
<< "scaling";
ScaleOffDiagonalCells();
status = Factorize();
}
VLOG(2) << "Compute time: " << time(nullptr) - start_time;
return (status == LinearSolverTerminationType::SUCCESS);
}
// Consider the preconditioner matrix as meta-block matrix, whose
// blocks correspond to the clusters. Then cluster pairs corresponding
// to edges in the degree-2 forest are off diagonal entries of this
// matrix. Scaling these off-diagonal entries by 1/2 forces this
// matrix to be positive definite.
void VisibilityBasedPreconditioner::ScaleOffDiagonalCells() {
for (const auto& block_pair : block_pairs_) {
const int block1 = block_pair.first;
const int block2 = block_pair.second;
if (!IsBlockPairOffDiagonal(block1, block2)) {
continue;
}
int r, c, row_stride, col_stride;
CellInfo* cell_info =
m_->GetCell(block1, block2, &r, &c, &row_stride, &col_stride);
CHECK(cell_info != nullptr)
<< "Cell missing for block pair (" << block1 << "," << block2 << ")"
<< " cluster pair (" << cluster_membership_[block1] << " "
<< cluster_membership_[block2] << ")";
// Ah the magic of tri-diagonal matrices and diagonal
// dominance. See Lemma 1 in "Visibility Based Preconditioning
// For Bundle Adjustment".
MatrixRef m(cell_info->values, row_stride, col_stride);
m.block(r, c, block_size_[block1], block_size_[block2]) *= 0.5;
}
}
// Compute the sparse Cholesky factorization of the preconditioner
// matrix.
LinearSolverTerminationType VisibilityBasedPreconditioner::Factorize() {
// Extract the TripletSparseMatrix that is used for actually storing
// S and convert it into a CompressedRowSparseMatrix.
const TripletSparseMatrix* tsm =
down_cast<BlockRandomAccessSparseMatrix*>(m_.get())->mutable_matrix();
std::unique_ptr<CompressedRowSparseMatrix> lhs;
const CompressedRowSparseMatrix::StorageType storage_type =
sparse_cholesky_->StorageType();
if (storage_type ==
CompressedRowSparseMatrix::StorageType::UPPER_TRIANGULAR) {
lhs = CompressedRowSparseMatrix::FromTripletSparseMatrix(*tsm);
lhs->set_storage_type(
CompressedRowSparseMatrix::StorageType::UPPER_TRIANGULAR);
} else {
lhs = CompressedRowSparseMatrix::FromTripletSparseMatrixTransposed(*tsm);
lhs->set_storage_type(
CompressedRowSparseMatrix::StorageType::LOWER_TRIANGULAR);
}
std::string message;
return sparse_cholesky_->Factorize(lhs.get(), &message);
}
void VisibilityBasedPreconditioner::RightMultiply(const double* x,
double* y) const {
CHECK(x != nullptr);
CHECK(y != nullptr);
CHECK(sparse_cholesky_ != nullptr);
std::string message;
sparse_cholesky_->Solve(x, y, &message);
}
int VisibilityBasedPreconditioner::num_rows() const { return m_->num_rows(); }
// Classify camera/f_block pairs as in and out of the preconditioner,
// based on whether the cluster pair that they belong to is in the
// preconditioner or not.
bool VisibilityBasedPreconditioner::IsBlockPairInPreconditioner(
const int block1, const int block2) const {
int cluster1 = cluster_membership_[block1];
int cluster2 = cluster_membership_[block2];
if (cluster1 > cluster2) {
swap(cluster1, cluster2);
}
return (cluster_pairs_.count(make_pair(cluster1, cluster2)) > 0);
}
bool VisibilityBasedPreconditioner::IsBlockPairOffDiagonal(
const int block1, const int block2) const {
return (cluster_membership_[block1] != cluster_membership_[block2]);
}
// Convert a graph into a list of edges that includes self edges for
// each vertex.
void VisibilityBasedPreconditioner::ForestToClusterPairs(
const WeightedGraph<int>& forest,
std::unordered_set<pair<int, int>, pair_hash>* cluster_pairs) const {
CHECK(cluster_pairs != nullptr);
cluster_pairs->clear();
const std::unordered_set<int>& vertices = forest.vertices();
CHECK_EQ(vertices.size(), num_clusters_);
// Add all the cluster pairs corresponding to the edges in the
// forest.
for (const int cluster1 : vertices) {
cluster_pairs->insert(make_pair(cluster1, cluster1));
const std::unordered_set<int>& neighbors = forest.Neighbors(cluster1);
for (const int cluster2 : neighbors) {
if (cluster1 < cluster2) {
cluster_pairs->insert(make_pair(cluster1, cluster2));
}
}
}
}
// The visibility set of a cluster is the union of the visibility sets
// of all its cameras. In other words, the set of points visible to
// any camera in the cluster.
void VisibilityBasedPreconditioner::ComputeClusterVisibility(
const vector<set<int>>& visibility,
vector<set<int>>* cluster_visibility) const {
CHECK(cluster_visibility != nullptr);
cluster_visibility->resize(0);
cluster_visibility->resize(num_clusters_);
for (int i = 0; i < num_blocks_; ++i) {
const int cluster_id = cluster_membership_[i];
(*cluster_visibility)[cluster_id].insert(visibility[i].begin(),
visibility[i].end());
}
}
// Construct a graph whose vertices are the clusters, and the edge
// weights are the number of 3D points visible to cameras in both the
// vertices.
std::unique_ptr<WeightedGraph<int>>
VisibilityBasedPreconditioner::CreateClusterGraph(
const vector<set<int>>& cluster_visibility) const {
auto cluster_graph = std::make_unique<WeightedGraph<int>>();
for (int i = 0; i < num_clusters_; ++i) {
cluster_graph->AddVertex(i);
}
for (int i = 0; i < num_clusters_; ++i) {
const set<int>& cluster_i = cluster_visibility[i];
for (int j = i + 1; j < num_clusters_; ++j) {
vector<int> intersection;
const set<int>& cluster_j = cluster_visibility[j];
set_intersection(cluster_i.begin(),
cluster_i.end(),
cluster_j.begin(),
cluster_j.end(),
back_inserter(intersection));
if (intersection.size() > 0) {
// Clusters interact strongly when they share a large number
// of 3D points. The degree-2 maximum spanning forest
// algorithm, iterates on the edges in decreasing order of
// their weight, which is the number of points shared by the
// two cameras that it connects.
cluster_graph->AddEdge(i, j, intersection.size());
}
}
}
return cluster_graph;
}
// Canonical views clustering returns a std::unordered_map from vertices to
// cluster ids. Convert this into a flat array for quick lookup. It is
// possible that some of the vertices may not be associated with any
// cluster. In that case, randomly assign them to one of the clusters.
//
// The cluster ids can be non-contiguous integers. So as we flatten
// the membership_map, we also map the cluster ids to a contiguous set
// of integers so that the cluster ids are in [0, num_clusters_).
void VisibilityBasedPreconditioner::FlattenMembershipMap(
const std::unordered_map<int, int>& membership_map,
vector<int>* membership_vector) const {
CHECK(membership_vector != nullptr);
membership_vector->resize(0);
membership_vector->resize(num_blocks_, -1);
std::unordered_map<int, int> cluster_id_to_index;
// Iterate over the cluster membership map and update the
// cluster_membership_ vector assigning arbitrary cluster ids to
// the few cameras that have not been clustered.
for (const auto& m : membership_map) {
const int camera_id = m.first;
int cluster_id = m.second;
// If the view was not clustered, randomly assign it to one of the
// clusters. This preserves the mathematical correctness of the
// preconditioner. If there are too many views which are not
// clustered, it may lead to some quality degradation though.
//
// TODO(sameeragarwal): Check if a large number of views have not
// been clustered and deal with it?
if (cluster_id == -1) {
cluster_id = camera_id % num_clusters_;
}
const int index = FindWithDefault(
cluster_id_to_index, cluster_id, cluster_id_to_index.size());
if (index == cluster_id_to_index.size()) {
cluster_id_to_index[cluster_id] = index;
}
CHECK_LT(index, num_clusters_);
membership_vector->at(camera_id) = index;
}
}
} // namespace ceres::internal