// Ceres Solver - A fast non-linear least squares minimizer // Copyright 2013 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: sameeragarwal@google.com (Sameer Agarwal) // // CostFunctionToFunctor is an adapter class that allows users to use // CostFunction objects in templated functors which are to be used for // automatic differentiation. This allows the user to seamlessly mix // analytic, numeric and automatic differentiation. // // For example, let us assume that // // class IntrinsicProjection : public SizedCostFunction<2, 5, 3> { // public: // IntrinsicProjection(const double* observations); // virtual bool Evaluate(double const* const* parameters, // double* residuals, // double** jacobians) const; // }; // // is a cost function that implements the projection of a point in its // local coordinate system onto its image plane and subtracts it from // the observed point projection. It can compute its residual and // either via analytic or numerical differentiation can compute its // jacobians. // // Now we would like to compose the action of this CostFunction with // the action of camera extrinsics, i.e., rotation and // translation. Say we have a templated function // // template // void RotateAndTranslatePoint(const T* rotation, // const T* translation, // const T* point, // T* result); // // Then we can now do the following, // // struct CameraProjection { // CameraProjection(double* observation) { // intrinsic_projection_.reset( // new CostFunctionToFunctor<2, 5, 3>( // new IntrinsicProjection(observation_))); // } // template // bool operator()(const T* rotation, // const T* translation, // const T* intrinsics, // const T* point, // T* residual) const { // T transformed_point[3]; // RotateAndTranslatePoint(rotation, translation, point, transformed_point); // // // Note that we call intrinsic_projection_, just like it was // // any other templated functor. // // return (*intrinsic_projection_)(intrinsics, transformed_point, residual); // } // // private: // scoped_ptr > intrinsic_projection_; // }; #ifndef CERES_PUBLIC_COST_FUNCTION_TO_FUNCTOR_H_ #define CERES_PUBLIC_COST_FUNCTION_TO_FUNCTOR_H_ #include #include #include "ceres/cost_function.h" #include "ceres/internal/fixed_array.h" #include "ceres/internal/port.h" #include "ceres/internal/scoped_ptr.h" namespace ceres { template class CostFunctionToFunctor { public: explicit CostFunctionToFunctor(CostFunction* cost_function) : cost_function_(cost_function) { CHECK_NOTNULL(cost_function); CHECK_GE(kNumResiduals, 0); CHECK_EQ(cost_function->num_residuals(), kNumResiduals); // This block breaks the 80 column rule to keep it somewhat readable. CHECK((!N1 && !N2 && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && !N2 && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && !N5 && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && !N6 && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && !N7 && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && !N8 && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && (N8 > 0) && !N9) || ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && (N8 > 0) && (N9 > 0))) << "Zero block cannot precede a non-zero block. Block sizes are " << "(ignore trailing 0s): " << N0 << ", " << N1 << ", " << N2 << ", " << N3 << ", " << N4 << ", " << N5 << ", " << N6 << ", " << N7 << ", " << N8 << ", " << N9; const vector& parameter_block_sizes = cost_function->parameter_block_sizes(); const int num_parameter_blocks = (N0 > 0) + (N1 > 0) + (N2 > 0) + (N3 > 0) + (N4 > 0) + (N5 > 0) + (N6 > 0) + (N7 > 0) + (N8 > 0) + (N9 > 0); CHECK_EQ(parameter_block_sizes.size(), num_parameter_blocks); CHECK_EQ(N0, parameter_block_sizes[0]); if (parameter_block_sizes.size() > 1) CHECK_EQ(N1, parameter_block_sizes[1]); // NOLINT if (parameter_block_sizes.size() > 2) CHECK_EQ(N2, parameter_block_sizes[2]); // NOLINT if (parameter_block_sizes.size() > 3) CHECK_EQ(N3, parameter_block_sizes[3]); // NOLINT if (parameter_block_sizes.size() > 4) CHECK_EQ(N4, parameter_block_sizes[4]); // NOLINT if (parameter_block_sizes.size() > 5) CHECK_EQ(N5, parameter_block_sizes[5]); // NOLINT if (parameter_block_sizes.size() > 6) CHECK_EQ(N6, parameter_block_sizes[6]); // NOLINT if (parameter_block_sizes.size() > 7) CHECK_EQ(N7, parameter_block_sizes[7]); // NOLINT if (parameter_block_sizes.size() > 8) CHECK_EQ(N8, parameter_block_sizes[8]); // NOLINT if (parameter_block_sizes.size() > 9) CHECK_EQ(N9, parameter_block_sizes[9]); // NOLINT CHECK_EQ(accumulate(parameter_block_sizes.begin(), parameter_block_sizes.end(), 0), N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9); } bool operator()(const double* x0, double* residuals) const { CHECK_NE(N0, 0); CHECK_EQ(N1, 0); CHECK_EQ(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); return cost_function_->Evaluate(&x0, residuals, NULL); } bool operator()(const double* x0, const double* x1, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_EQ(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(2); parameter_blocks[0] = x0; parameter_blocks[1] = x1; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(3); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(4); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(5); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, const double* x5, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(6); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; parameter_blocks[5] = x5; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, const double* x5, const double* x6, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(7); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; parameter_blocks[5] = x5; parameter_blocks[6] = x6; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, const double* x5, const double* x6, const double* x7, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(8); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; parameter_blocks[5] = x5; parameter_blocks[6] = x6; parameter_blocks[7] = x7; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, const double* x5, const double* x6, const double* x7, const double* x8, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_NE(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray parameter_blocks(9); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; parameter_blocks[5] = x5; parameter_blocks[6] = x6; parameter_blocks[7] = x7; parameter_blocks[8] = x8; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } bool operator()(const double* x0, const double* x1, const double* x2, const double* x3, const double* x4, const double* x5, const double* x6, const double* x7, const double* x8, const double* x9, double* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_NE(N8, 0); CHECK_NE(N9, 0); internal::FixedArray parameter_blocks(10); parameter_blocks[0] = x0; parameter_blocks[1] = x1; parameter_blocks[2] = x2; parameter_blocks[3] = x3; parameter_blocks[4] = x4; parameter_blocks[5] = x5; parameter_blocks[6] = x6; parameter_blocks[7] = x7; parameter_blocks[8] = x8; parameter_blocks[9] = x9; return cost_function_->Evaluate(parameter_blocks.get(), residuals, NULL); } template bool operator()(const JetT* x0, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_EQ(N1, 0); CHECK_EQ(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); return EvaluateWithJets(&x0, residuals); } template bool operator()(const JetT* x0, const JetT* x1, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_EQ(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(2); jets[0] = x0; jets[1] = x1; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_EQ(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(3); jets[0] = x0; jets[1] = x1; jets[2] = x2; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_EQ(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(4); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_EQ(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(5); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, const JetT* x5, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_EQ(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(6); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; jets[5] = x5; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, const JetT* x5, const JetT* x6, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_EQ(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(7); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; jets[5] = x5; jets[6] = x6; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, const JetT* x5, const JetT* x6, const JetT* x7, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_EQ(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(8); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; jets[5] = x5; jets[6] = x6; jets[7] = x7; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, const JetT* x5, const JetT* x6, const JetT* x7, const JetT* x8, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_NE(N8, 0); CHECK_EQ(N9, 0); internal::FixedArray jets(9); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; jets[5] = x5; jets[6] = x6; jets[7] = x7; jets[8] = x8; return EvaluateWithJets(jets.get(), residuals); } template bool operator()(const JetT* x0, const JetT* x1, const JetT* x2, const JetT* x3, const JetT* x4, const JetT* x5, const JetT* x6, const JetT* x7, const JetT* x8, const JetT* x9, JetT* residuals) const { CHECK_NE(N0, 0); CHECK_NE(N1, 0); CHECK_NE(N2, 0); CHECK_NE(N3, 0); CHECK_NE(N4, 0); CHECK_NE(N5, 0); CHECK_NE(N6, 0); CHECK_NE(N7, 0); CHECK_NE(N8, 0); CHECK_NE(N9, 0); internal::FixedArray jets(10); jets[0] = x0; jets[1] = x1; jets[2] = x2; jets[3] = x3; jets[4] = x4; jets[5] = x5; jets[6] = x6; jets[7] = x7; jets[8] = x8; jets[9] = x9; return EvaluateWithJets(jets.get(), residuals); } private: template bool EvaluateWithJets(const JetT** inputs, JetT* output) const { const int kNumParameters = N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9; const vector& parameter_block_sizes = cost_function_->parameter_block_sizes(); const int num_parameter_blocks = parameter_block_sizes.size(); const int num_residuals = cost_function_->num_residuals(); internal::FixedArray parameters(kNumParameters); internal::FixedArray parameter_blocks(num_parameter_blocks); internal::FixedArray jacobians(num_residuals * kNumParameters); internal::FixedArray jacobian_blocks(num_parameter_blocks); internal::FixedArray residuals(num_residuals); // Build a set of arrays to get the residuals and jacobians from // the CostFunction wrapped by this functor. double* parameter_ptr = parameters.get(); double* jacobian_ptr = jacobians.get(); for (int i = 0; i < num_parameter_blocks; ++i) { parameter_blocks[i] = parameter_ptr; jacobian_blocks[i] = jacobian_ptr; for (int j = 0; j < parameter_block_sizes[i]; ++j) { *parameter_ptr++ = inputs[i][j].a; } jacobian_ptr += num_residuals * parameter_block_sizes[i]; } if (!cost_function_->Evaluate(parameter_blocks.get(), residuals.get(), jacobian_blocks.get())) { return false; } // Now that we have the incoming Jets, which are carrying the // partial derivatives of each of the inputs w.r.t to some other // underlying parameters. The derivative of the outputs of the // cost function w.r.t to the same underlying parameters can now // be computed by applying the chain rule. // // d output[i] d output[i] d input[j] // -------------- = sum_j ----------- * ------------ // d parameter[k] d input[j] d parameter[k] // // d input[j] // -------------- = inputs[j], so // d parameter[k] // // outputJet[i] = sum_k jacobian[i][k] * inputJet[k] // // The following loop, iterates over the residuals, computing one // output jet at a time. for (int i = 0; i < num_residuals; ++i) { output[i].a = residuals[i]; output[i].v.setZero(); for (int j = 0; j < num_parameter_blocks; ++j) { const int32 block_size = parameter_block_sizes[j]; for (int k = 0; k < parameter_block_sizes[j]; ++k) { output[i].v += jacobian_blocks[j][i * block_size + k] * inputs[j][k].v; } } } return true; } private: internal::scoped_ptr cost_function_; }; } // namespace ceres #endif // CERES_PUBLIC_COST_FUNCTION_TO_FUNCTOR_H_