Initial commit of Ceres Solver.
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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 <gflags/gflags.h>
+#include "ceres/ceres.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, -1.0, "Robust loss parameter. Set to -1 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;
+}
