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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2010, 2011, 2012 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: keir@google.com (Keir Mierle)
//
// This fits circles to a collection of points, where the error is related to
// the distance of a point from the circle. This uses auto-differentiation to
// take the derivatives.
//
// The input format is simple text. Feed on standard in:
//
// x_initial y_initial r_initial
// x1 y1
// x2 y2
// y3 y3
// ...
//
// And the result after solving will be printed to stdout:
//
// x y r
//
// There are closed form solutions [1] to this problem which you may want to
// consider instead of using this one. If you already have a decent guess, Ceres
// can squeeze down the last bit of error.
//
// [1] http://www.mathworks.com/matlabcentral/fileexchange/5557-circle-fit/content/circfit.m
#include <cstdio>
#include <vector>
#include "ceres/ceres.h"
#include "gflags/gflags.h"
#include "glog/logging.h"
using ceres::AutoDiffCostFunction;
using ceres::CauchyLoss;
using ceres::CostFunction;
using ceres::LossFunction;
using ceres::Problem;
using ceres::Solve;
using ceres::Solver;
DEFINE_double(robust_threshold, 0.0, "Robust loss parameter. Set to 0 for "
"normal squared error (no robustification).");
// The cost for a single sample. The returned residual is related to the
// distance of the point from the circle (passed in as x, y, m parameters).
//
// Note that the radius is parameterized as r = m^2 to constrain the radius to
// positive values.
class DistanceFromCircleCost {
public:
DistanceFromCircleCost(double xx, double yy) : xx_(xx), yy_(yy) {}
template <typename T> bool operator()(const T* const x,
const T* const y,
const T* const m, // r = m^2
T* residual) const {
// Since the radius is parameterized as m^2, unpack m to get r.
T r = *m * *m;
// Get the position of the sample in the circle's coordinate system.
T xp = xx_ - *x;
T yp = yy_ - *y;
// It is tempting to use the following cost:
//
// residual[0] = r - sqrt(xp*xp + yp*yp);
//
// which is the distance of the sample from the circle. This works
// reasonably well, but the sqrt() adds strong nonlinearities to the cost
// function. Instead, a different cost is used, which while not strictly a
// distance in the metric sense (it has units distance^2) it produces more
// robust fits when there are outliers. This is because the cost surface is
// more convex.
residual[0] = r*r - xp*xp - yp*yp;
return true;
}
private:
// The measured x,y coordinate that should be on the circle.
double xx_, yy_;
};
int main(int argc, char** argv) {
google::ParseCommandLineFlags(&argc, &argv, true);
google::InitGoogleLogging(argv[0]);
double x, y, r;
if (scanf("%lg %lg %lg", &x, &y, &r) != 3) {
fprintf(stderr, "Couldn't read first line.\n");
return 1;
}
fprintf(stderr, "Got x, y, r %lg, %lg, %lg\n", x, y, r);
// Save initial values for comparison.
double initial_x = x;
double initial_y = y;
double initial_r = r;
// Parameterize r as m^2 so that it can't be negative.
double m = sqrt(r);
Problem problem;
// Configure the loss function.
LossFunction* loss = NULL;
if (FLAGS_robust_threshold) {
loss = new CauchyLoss(FLAGS_robust_threshold);
}
// Add the residuals.
double xx, yy;
int num_points = 0;
while (scanf("%lf %lf\n", &xx, &yy) == 2) {
CostFunction *cost =
new AutoDiffCostFunction<DistanceFromCircleCost, 1, 1, 1, 1>(
new DistanceFromCircleCost(xx, yy));
problem.AddResidualBlock(cost, loss, &x, &y, &m);
num_points++;
}
std::cout << "Got " << num_points << " points.\n";
// Build and solve the problem.
Solver::Options options;
options.max_num_iterations = 500;
options.linear_solver_type = ceres::DENSE_QR;
Solver::Summary summary;
Solve(options, &problem, &summary);
// Recover r from m.
r = m * m;
std::cout << summary.BriefReport() << "\n";
std::cout << "x : " << initial_x << " -> " << x << "\n";
std::cout << "y : " << initial_y << " -> " << y << "\n";
std::cout << "r : " << initial_r << " -> " << r << "\n";
return 0;
}
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