text: // Ceres Solver - A fast non-linear least squares minimizer
  // Copyright 2015 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: kushalav@google.com (Avanish Kushal)
  
  #include "ceres/visibility.h"
  
  #include <algorithm>
  #include <cmath>
  #include <ctime>
  #include <set>
  #include <unordered_map>
  #include <utility>
  #include <vector>
  
  #include "ceres/block_structure.h"
  #include "ceres/graph.h"
  #include "ceres/pair_hash.h"
  #include "glog/logging.h"
  
  namespace ceres {
  namespace internal {
  
  using std::make_pair;
  using std::max;
  using std::pair;
  using std::set;
  using std::vector;
  
  void ComputeVisibility(const CompressedRowBlockStructure& block_structure,
                         const int num_eliminate_blocks,
                         vector<set<int>>* visibility) {
    CHECK(visibility != nullptr);
  
    // Clear the visibility vector and resize it to hold a
    // vector for each camera.
    visibility->resize(0);
    visibility->resize(block_structure.cols.size() - num_eliminate_blocks);
  
    for (int i = 0; i < block_structure.rows.size(); ++i) {
      const vector<Cell>& cells = block_structure.rows[i].cells;
      int block_id = cells[0].block_id;
      // If the first block is not an e_block, then skip this row block.
      if (block_id >= num_eliminate_blocks) {
        continue;
      }
  
      for (int j = 1; j < cells.size(); ++j) {
        int camera_block_id = cells[j].block_id - num_eliminate_blocks;
        DCHECK_GE(camera_block_id, 0);
        DCHECK_LT(camera_block_id, visibility->size());
        (*visibility)[camera_block_id].insert(block_id);
      }
    }
  }
  
  WeightedGraph<int>* CreateSchurComplementGraph(
      const vector<set<int>>& visibility) {
    const time_t start_time = time(NULL);
    // Compute the number of e_blocks/point blocks. Since the visibility
    // set for each e_block/camera contains the set of e_blocks/points
    // visible to it, we find the maximum across all visibility sets.
    int num_points = 0;
    for (int i = 0; i < visibility.size(); i++) {
      if (visibility[i].size() > 0) {
        num_points = max(num_points, (*visibility[i].rbegin()) + 1);
      }
    }
  
    // Invert the visibility. The input is a camera->point mapping,
    // which tells us which points are visible in which
    // cameras. However, to compute the sparsity structure of the Schur
    // Complement efficiently, its better to have the point->camera
    // mapping.
    vector<set<int>> inverse_visibility(num_points);
    for (int i = 0; i < visibility.size(); i++) {
      const set<int>& visibility_set = visibility[i];
      for (const int v : visibility_set) {
        inverse_visibility[v].insert(i);
      }
    }
  
    // Map from camera pairs to number of points visible to both cameras
    // in the pair.
    std::unordered_map<pair<int, int>, int, pair_hash> camera_pairs;
  
    // Count the number of points visible to each camera/f_block pair.
    for (const auto& inverse_visibility_set : inverse_visibility) {
      for (set<int>::const_iterator camera1 = inverse_visibility_set.begin();
           camera1 != inverse_visibility_set.end();
           ++camera1) {
        set<int>::const_iterator camera2 = camera1;
        for (++camera2; camera2 != inverse_visibility_set.end(); ++camera2) {
          ++(camera_pairs[make_pair(*camera1, *camera2)]);
        }
      }
    }
  
    WeightedGraph<int>* graph = new WeightedGraph<int>;
  
    // Add vertices and initialize the pairs for self edges so that self
    // edges are guaranteed. This is needed for the Canonical views
    // algorithm to work correctly.
    static constexpr double kSelfEdgeWeight = 1.0;
    for (int i = 0; i < visibility.size(); ++i) {
      graph->AddVertex(i);
      graph->AddEdge(i, i, kSelfEdgeWeight);
    }
  
    // Add an edge for each camera pair.
    for (const auto& camera_pair_count : camera_pairs) {
      const int camera1 = camera_pair_count.first.first;
      const int camera2 = camera_pair_count.first.second;
      const int count = camera_pair_count.second;
      DCHECK_NE(camera1, camera2);
      // Static cast necessary for Windows.
      const double weight =
          static_cast<double>(count) /
          (sqrt(static_cast<double>(visibility[camera1].size() *
                                    visibility[camera2].size())));
      graph->AddEdge(camera1, camera2, weight);
    }
  
    VLOG(2) << "Schur complement graph time: " << (time(NULL) - start_time);
    return graph;
  }
  
  }  // namespace internal
  }  // namespace ceres
  