Add readme for the sampled_function example.

Change-Id: I9468b6a7b9f2ffdd2bf9f0dd1f4e1d5f894e540c

examples/sampled_function/CMakeLists.txt

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+# 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: vitus@google.com (Michael Vitus)
+
+# Only Ceres itself should be compiled with CERES_BUILDING_SHARED_LIBRARY
+# defined, any users of Ceres will have CERES_USING_SHARED_LIBRARY defined
+# for them in Ceres' config.h if appropriate.
+if (BUILD_SHARED_LIBS)
+  remove_definitions(-DCERES_BUILDING_SHARED_LIBRARY)
+endif()
+
+add_executable(sampled_function sampled_function.cc)
+target_link_libraries(sampled_function ceres)

examples/sampled_function/README.md

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+Sampled Functions
+--
+
+It is common to not have an analytical representation of the optimization
+problem but rather a table of values at specific inputs. This commonly occurs
+when working with images or when the functions in the problem are expensive to
+evaluate. To use this data in an optimization problem we can use interpolation
+to evaluate the function and derivatives at intermediate input values.
+
+There are many libraries that implement a variety of interpolation schemes, but
+it is difficult to use them in Ceres' automatic differentiation framework.
+Instead, Ceres provides the ability to interpolate one and two dimensional data.
+
+The one dimensional interpolation is based on the Cubic Hermite Spline. This
+interpolation method requires knowledge of the function derivatives at the
+control points, however we only know the function values. Consequently, we will
+use the data to estimate derivatives at the control points. The choice of how to
+compute the derivatives is not unique and Ceres uses the Catmull–Rom Spline
+variant which uses `0.5 * (p_{k+1} - p_{k-1})` as the derivative for control
+point `p_k.` This produces a first order differentiable interpolating
+function. The two dimensional interpolation scheme is a generalization of the
+one dimensional scheme where the interpolating function is assumed to be
+separable in the two dimensions.
+
+This example shows how to use interpolation schemes within the Ceres automatic
+differentiation framework. This is a one dimensional example and the objective
+function is to minimize `0.5 * f(x)^2` where `f(x) = (x - 4.5)^2`.
+
+It is also possible to use analytical derivatives with the provided
+interpolation schemes by using a `SizedCostFunction` and defining the
+``Evaluate` function. For this example, the evaluate function would be:
+
+```c++
+bool Evaluate(double const* const* parameters, double* residuals, double** jacobians) const {
+  if (jacobians == NULL || jacobians[0] == NULL)
+    interpolator_.Evaluate(parameters[0][0], residuals);
+  else
+    interpolator_.Evaluate(parameters[0][0], residuals, jacobians[0]);
+
+  return true;
+}
+```

examples/sampled_function/sampled_function.cc

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+// 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: sameeragarwal@google.com (Sameer Agarwal)
+//
+// A simple example of optimizing a sampled function by using cubic
+// interpolation.
+
+#include "ceres/ceres.h"
+#include "ceres/cubic_interpolation.h"
+#include "glog/logging.h"
+
+using ceres::Grid1D;
+using ceres::CubicInterpolator;
+using ceres::AutoDiffCostFunction;
+using ceres::CostFunction;
+using ceres::Problem;
+using ceres::Solver;
+using ceres::Solve;
+
+// A simple cost functor that interfaces an interpolated table of
+// values with automatic differentiation.
+struct InterpolatedCostFunctor {
+  explicit InterpolatedCostFunctor(
+      const CubicInterpolator<Grid1D<double> >& interpolator)
+      : interpolator_(interpolator) {
+  }
+
+  template<typename T> bool operator()(const T* x, T* residuals) const {
+    interpolator_.Evaluate(*x, residuals);
+    return true;
+  }
+
+  static CostFunction* Create(
+      const CubicInterpolator<Grid1D<double> >& interpolator) {
+    return new AutoDiffCostFunction<InterpolatedCostFunctor, 1, 1>(
+        new InterpolatedCostFunctor(interpolator));
+  }
+
+ private:
+  const CubicInterpolator<Grid1D<double> >& interpolator_;
+};
+
+int main(int argc, char** argv) {
+  google::InitGoogleLogging(argv[0]);
+
+  // Evaluate the function f(x) = (x - 4.5)^2;
+  const int kNumSamples = 10;
+  double values[kNumSamples];
+  for (int i = 0; i < kNumSamples; ++i) {
+    values[i] = (i - 4.5) * (i - 4.5);
+  }
+
+  Grid1D<double> array(values, 0, kNumSamples);
+  CubicInterpolator<Grid1D<double> > interpolator(array);
+
+  double x = 1.0;
+  Problem problem;
+  CostFunction* cost_function = InterpolatedCostFunctor::Create(interpolator);
+  problem.AddResidualBlock(cost_function, NULL, &x);
+
+  Solver::Options options;
+  options.minimizer_progress_to_stdout = true;
+  Solver::Summary summary;
+  Solve(options, &problem, &summary);
+  std::cout << summary.BriefReport() << "\n";
+  std::cout << "Expected x: 4.5. Actual x : " << x << std::endl;
+  return 0;
+}