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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2014 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: richie.stebbing@gmail.com (Richard Stebbing)
//
// This fits points randomly distributed on an ellipse with an approximate
// line segment contour. This is done by jointly optimizing the control points
// of the line segment contour along with the preimage positions for the data
// points. The purpose of this example is to show an example use case for
// dynamic_sparsity, and how it can benefit problems which are numerically
// dense but dynamically sparse.

#include <cmath>
#include <vector>
#include "ceres/ceres.h"
#include "glog/logging.h"

// Data generated with the following Python code.
//   import numpy as np
//   np.random.seed(1337)
//   t = np.linspace(0.0, 2.0 * np.pi, 212, endpoint=False)
//   t += 2.0 * np.pi * 0.01 * np.random.randn(t.size)
//   theta = np.deg2rad(15)
//   a, b = np.cos(theta), np.sin(theta)
//   R = np.array([[a, -b],
//                 [b, a]])
//   Y = np.dot(np.c_[4.0 * np.cos(t), np.sin(t)], R.T)

const int kYRows = 212;
const int kYCols = 2;
const double kYData[kYRows * kYCols] = {
  +3.871364e+00, +9.916027e-01,
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  +3.870542e+00, +9.996121e-01,
  +3.865424e+00, +1.028474e+00
};
ceres::ConstMatrixRef kY(kYData, kYRows, kYCols);

class PointToLineSegmentContourCostFunction : public ceres::CostFunction {
 public:
  PointToLineSegmentContourCostFunction(const int num_segments,
                                        const Eigen::Vector2d y)
      : num_segments_(num_segments), y_(y) {
    // The first parameter is the preimage position.
    mutable_parameter_block_sizes()->push_back(1);
    // The next parameters are the control points for the line segment contour.
    for (int i = 0; i < num_segments_; ++i) {
      mutable_parameter_block_sizes()->push_back(2);
    }
    set_num_residuals(2);
  }

  virtual bool Evaluate(const double* const* x,
                        double* residuals,
                        double** jacobians) const {
    // Convert the preimage position `t` into a segment index `i0` and the
    // line segment interpolation parameter `u`. `i1` is the index of the next
    // control point.
    const double t = ModuloNumSegments(*x[0]);
    CHECK_GE(t, 0.0);
    CHECK_LT(t, num_segments_);
    const int i0 = floor(t), i1 = (i0 + 1) % num_segments_;
    const double u = t - i0;

    // Linearly interpolate between control points `i0` and `i1`.
    residuals[0] = y_[0] - ((1.0 - u) * x[1 + i0][0] + u * x[1 + i1][0]);
    residuals[1] = y_[1] - ((1.0 - u) * x[1 + i0][1] + u * x[1 + i1][1]);

    if (jacobians == NULL) {
      return true;
    }

    if (jacobians[0] != NULL) {
      jacobians[0][0] = x[1 + i0][0] - x[1 + i1][0];
      jacobians[0][1] = x[1 + i0][1] - x[1 + i1][1];
    }
    for (int i = 0; i < num_segments_; ++i) {
      if (jacobians[i + 1] != NULL) {
        ceres::MatrixRef(jacobians[i + 1], 2, 2).setZero();
        if (i == i0) {
          jacobians[i + 1][0] = -(1.0 - u);
          jacobians[i + 1][3] = -(1.0 - u);
        } else if (i == i1) {
          jacobians[i + 1][0] = -u;
          jacobians[i + 1][3] = -u;
        }
      }
    }
    return true;
  }

  static ceres::CostFunction* Create(const int num_segments,
                                     const Eigen::Vector2d y) {
    return new PointToLineSegmentContourCostFunction(num_segments, y);
  }

 private:
  inline double ModuloNumSegments(const double& t) const {
    return t - num_segments_ * floor(t / num_segments_);
  }

  const int num_segments_;
  const Eigen::Vector2d y_;
};

struct EuclideanDistanceFunctor {
  EuclideanDistanceFunctor(const double& sqrt_weight)
      : sqrt_weight_(sqrt_weight) {}

  template <typename T>
  bool operator()(const T* x0, const T* x1, T* residuals) const {
    residuals[0] = T(sqrt_weight_) * (x0[0] - x1[0]);
    residuals[1] = T(sqrt_weight_) * (x0[1] - x1[1]);
    return true;
  }

  static ceres::CostFunction* Create(const double& sqrt_weight) {
    return new ceres::AutoDiffCostFunction<EuclideanDistanceFunctor, 2, 2, 2>(
        new EuclideanDistanceFunctor(sqrt_weight));
  }

 private:
  const double sqrt_weight_;
};

bool SolveWithFullReport(ceres::Solver::Options options,
                         ceres::Problem* problem,
                         bool dynamic_sparsity) {
  options.dynamic_sparsity = dynamic_sparsity;

  ceres::Solver::Summary summary;
  ceres::Solve(options, problem, &summary);

  std::cout << "####################" << std::endl;
  std::cout << "dynamic_sparsity = " << dynamic_sparsity << std::endl;
  std::cout << "####################" << std::endl;
  std::cout << summary.FullReport() << std::endl;

  return summary.termination_type == ceres::CONVERGENCE;
}

int main(int argc, char** argv) {
  google::InitGoogleLogging(argv[0]);

  // Problem configuration.
  const int num_segments = 151;
  const double regularization_weight = 1e-2;

  // Eigen::MatrixXd is column major so we define our own MatrixXd which is
  // row major. Eigen::VectorXd can be used directly.
  typedef Eigen::Matrix<double,
                        Eigen::Dynamic, Eigen::Dynamic,
                        Eigen::RowMajor> MatrixXd;
  using Eigen::VectorXd;

  // `X` is the matrix of control points which make up the contour of line
  // segments. The number of control points is equal to the number of line
  // segments because the contour is closed.
  //
  // Initialize `X` to points on the unit circle.
  VectorXd w(num_segments + 1);
  w.setLinSpaced(num_segments + 1, 0.0, 2.0 * M_PI);
  w.conservativeResize(num_segments);
  MatrixXd X(num_segments, 2);
  X.col(0) = w.array().cos();
  X.col(1) = w.array().sin();

  // Each data point has an associated preimage position on the line segment
  // contour. For each data point we initialize the preimage positions to
  // the index of the closest control point.
  const int num_observations = kY.rows();
  VectorXd t(num_observations);
  for (int i = 0; i < num_observations; ++i) {
    (X.rowwise() - kY.row(i)).rowwise().squaredNorm().minCoeff(&t[i]);
  }

  ceres::Problem problem;

  // For each data point add a residual which measures its distance to its
  // corresponding position on the line segment contour.
  std::vector<double*> parameter_blocks(1 + num_segments);
  parameter_blocks[0] = NULL;
  for (int i = 0; i < num_segments; ++i) {
    parameter_blocks[i + 1] = X.data() + 2 * i;
  }
  for (int i = 0; i < num_observations; ++i) {
    parameter_blocks[0] = &t[i];
    problem.AddResidualBlock(
      PointToLineSegmentContourCostFunction::Create(num_segments, kY.row(i)),
      NULL,
      parameter_blocks);
  }

  // Add regularization to minimize the length of the line segment contour.
  for (int i = 0; i < num_segments; ++i) {
    problem.AddResidualBlock(
      EuclideanDistanceFunctor::Create(sqrt(regularization_weight)),
      NULL,
      X.data() + 2 * i,
      X.data() + 2 * ((i + 1) % num_segments));
  }

  ceres::Solver::Options options;
  options.max_num_iterations = 100;
  options.linear_solver_type = ceres::SPARSE_NORMAL_CHOLESKY;

  // First, solve `X` and `t` jointly with dynamic_sparsity = true.
  MatrixXd X0 = X;
  VectorXd t0 = t;
  CHECK(SolveWithFullReport(options, &problem, true));

  // Second, solve with dynamic_sparsity = false.
  X = X0;
  t = t0;
  CHECK(SolveWithFullReport(options, &problem, false));

  return 0;
}