include/ceres/numeric_diff_cost_function.h - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2024 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: keir@google.com (Keir Mierle)
 //         sameeragarwal@google.com (Sameer Agarwal)
 //
 // Create CostFunctions as needed by the least squares framework with jacobians
 // computed via numeric (a.k.a. finite) differentiation. For more details see
 // http://en.wikipedia.org/wiki/Numerical_differentiation.
 //
 // To get an numerically differentiated cost function, you must define
 // a class with a operator() (a functor) that computes the residuals.
 //
 // The function must write the computed value in the last argument
 // (the only non-const one) and return true to indicate success.
 // Please see cost_function.h for details on how the return value
 // maybe used to impose simple constraints on the parameter block.
 //
 // For example, consider a scalar error e = k - x'y, where both x and y are
 // two-dimensional column vector parameters, the prime sign indicates
 // transposition, and k is a constant. The form of this error, which is the
 // difference between a constant and an expression, is a common pattern in least
 // squares problems. For example, the value x'y might be the model expectation
 // for a series of measurements, where there is an instance of the cost function
 // for each measurement k.
 //
 // The actual cost added to the total problem is e^2, or (k - x'k)^2; however,
 // the squaring is implicitly done by the optimization framework.
 //
 // To write an numerically-differentiable cost function for the above model,
 // first define the object
 //
 //   class MyScalarCostFunctor {
 //     explicit MyScalarCostFunctor(double k): k_(k) {}
 //
 //     bool operator()(const double* const x,
 //                     const double* const y,
 //                     double* residuals) const {
 //       residuals[0] = k_ - x[0] * y[0] - x[1] * y[1];
 //       return true;
 //     }
 //
 //    private:
 //     double k_;
 //   };
 //
 // Note that in the declaration of operator() the input parameters x
 // and y come first, and are passed as const pointers to arrays of
 // doubles. If there were three input parameters, then the third input
 // parameter would come after y. The output is always the last
 // parameter, and is also a pointer to an array. In the example above,
 // the residual is a scalar, so only residuals[0] is set.
 //
 // Then given this class definition, the numerically differentiated
 // cost function with central differences used for computing the
 // derivative can be constructed as follows.
 //
 //   CostFunction* cost_function
 //       = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, 1, 2, 2>(
 //           new MyScalarCostFunctor(1.0));                    ^     ^  ^  ^
 //                                                             |     |  |  |
 //                                 Finite Differencing Scheme -+     |  |  |
 //                                 Dimension of residual ------------+  |  |
 //                                 Dimension of x ----------------------+  |
 //                                 Dimension of y -------------------------+
 //
 // In this example, there is usually an instance for each measurement of k.
 //
 // In the instantiation above, the template parameters following
 // "MyScalarCostFunctor", "1, 2, 2", describe the functor as computing
 // a 1-dimensional output from two arguments, both 2-dimensional.
 //
 // NumericDiffCostFunction also supports cost functions with a
 // runtime-determined number of residuals. For example:
 //
 // clang-format off
 //
 //   CostFunction* cost_function
 //       = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, DYNAMIC, 2, 2>(
 //           new CostFunctorWithDynamicNumResiduals(1.0),               ^     ^  ^
 //           TAKE_OWNERSHIP,                                            |     |  |
 //           runtime_number_of_residuals); <----+                       |     |  |
 //                                              |                       |     |  |
 //                                              |                       |     |  |
 //             Actual number of residuals ------+                       |     |  |
 //             Indicate dynamic number of residuals --------------------+     |  |
 //             Dimension of x ------------------------------------------------+  |
 //             Dimension of y ---------------------------------------------------+
 // clang-format on
 //
 //
 // The central difference method is considerably more accurate at the cost of
 // twice as many function evaluations than forward difference. Consider using
 // central differences begin with, and only after that works, trying forward
 // difference to improve performance.
 //
 // WARNING #1: A common beginner's error when first using
 // NumericDiffCostFunction is to get the sizing wrong. In particular,
 // there is a tendency to set the template parameters to (dimension of
 // residual, number of parameters) instead of passing a dimension
 // parameter for *every parameter*. In the example above, that would
 // be <MyScalarCostFunctor, 1, 2>, which is missing the last '2'
 // argument. Please be careful when setting the size parameters.
 //
 ////////////////////////////////////////////////////////////////////////////
 ////////////////////////////////////////////////////////////////////////////
 //
 // ALTERNATE INTERFACE
 //
 // For a variety of reasons, including compatibility with legacy code,
 // NumericDiffCostFunction can also take CostFunction objects as
 // input. The following describes how.
 //
 // To get a numerically differentiated cost function, define a
 // subclass of CostFunction such that the Evaluate() function ignores
 // the jacobian parameter. The numeric differentiation wrapper will
 // fill in the jacobian parameter if necessary by repeatedly calling
 // the Evaluate() function with small changes to the appropriate
 // parameters, and computing the slope. For performance, the numeric
 // differentiation wrapper class is templated on the concrete cost
 // function, even though it could be implemented only in terms of the
 // virtual CostFunction interface.
 //
 // The numerically differentiated version of a cost function for a cost function
 // can be constructed as follows:
 //
 //   auto* cost_function
 //       = new NumericDiffCostFunction<MyCostFunction, CENTRAL, 1, 4, 8>();
 //
 // where MyCostFunction has 1 residual and 2 parameter blocks with sizes 4 and 8
 // respectively. Look at the tests for a more detailed example.
 //
 // TODO(keir): Characterize accuracy; mention pitfalls; provide alternatives.

 #ifndef CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_
 #define CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_

 #include <array>
 #include <memory>
 #include <type_traits>

 #include "Eigen/Dense"
 #include "ceres/cost_function.h"
 #include "ceres/internal/numeric_diff.h"
 #include "ceres/internal/parameter_dims.h"
 #include "ceres/numeric_diff_options.h"
 #include "ceres/sized_cost_function.h"
 #include "ceres/types.h"

 namespace ceres {

 template <typename CostFunctor,
           NumericDiffMethodType kMethod = CENTRAL,
           int kNumResiduals = 0,  // Number of residuals, or ceres::DYNAMIC
           int... Ns>              // Parameters dimensions for each block.
 class NumericDiffCostFunction final
     : public SizedCostFunction<kNumResiduals, Ns...> {
  public:
   explicit NumericDiffCostFunction(
       CostFunctor* functor,
       Ownership ownership = TAKE_OWNERSHIP,
       int num_residuals = kNumResiduals,
       const NumericDiffOptions& options = NumericDiffOptions())
       : NumericDiffCostFunction{std::unique_ptr<CostFunctor>{functor},
                                 ownership,
                                 num_residuals,
                                 options} {}

   explicit NumericDiffCostFunction(
       std::unique_ptr<CostFunctor> functor,
       int num_residuals = kNumResiduals,
       const NumericDiffOptions& options = NumericDiffOptions())
       : NumericDiffCostFunction{
             std::move(functor), TAKE_OWNERSHIP, num_residuals, options} {}

   // Constructs the CostFunctor on the heap and takes the ownership.
   // Invocable only if the number of residuals is known at compile-time.
   template <class... Args,
             bool kIsDynamic = kNumResiduals == DYNAMIC,
             std::enable_if_t<!kIsDynamic &&
                              std::is_constructible_v<CostFunctor, Args&&...>>* =
                 nullptr>
   explicit NumericDiffCostFunction(Args&&... args)
       // NOTE We explicitly use direct initialization using parentheses instead
       // of uniform initialization using braces to avoid narrowing conversion
       // warnings.
       : NumericDiffCostFunction{
             std::make_unique<CostFunctor>(std::forward<Args>(args)...),
             TAKE_OWNERSHIP} {}

   NumericDiffCostFunction(NumericDiffCostFunction&& other) noexcept = default;
   NumericDiffCostFunction& operator=(NumericDiffCostFunction&& other) noexcept =
       default;
   NumericDiffCostFunction(const NumericDiffCostFunction&) = delete;
   NumericDiffCostFunction& operator=(const NumericDiffCostFunction&) = delete;

   ~NumericDiffCostFunction() override {
     if (ownership_ != TAKE_OWNERSHIP) {
       functor_.release();
     }
   }

   bool Evaluate(double const* const* parameters,
                 double* residuals,
                 double** jacobians) const override {
     using internal::FixedArray;
     using internal::NumericDiff;

     using ParameterDims =
         typename SizedCostFunction<kNumResiduals, Ns...>::ParameterDims;

     constexpr int kNumParameters = ParameterDims::kNumParameters;
     constexpr int kNumParameterBlocks = ParameterDims::kNumParameterBlocks;

     // Get the function value (residuals) at the the point to evaluate.
     if (!internal::VariadicEvaluate<ParameterDims>(
             *functor_, parameters, residuals)) {
       return false;
     }

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

     // Create a copy of the parameters which will get mutated.
     FixedArray<double> parameters_copy(kNumParameters);
     std::array<double*, kNumParameterBlocks> parameters_reference_copy =
         ParameterDims::GetUnpackedParameters(parameters_copy.data());

     for (int block = 0; block < kNumParameterBlocks; ++block) {
       memcpy(parameters_reference_copy[block],
              parameters[block],
              sizeof(double) * ParameterDims::GetDim(block));
     }

     internal::EvaluateJacobianForParameterBlocks<ParameterDims>::
         template Apply<kMethod, kNumResiduals>(
             functor_.get(),
             residuals,
             options_,
             SizedCostFunction<kNumResiduals, Ns...>::num_residuals(),
             parameters_reference_copy.data(),
             jacobians);

     return true;
   }

   const CostFunctor& functor() const { return *functor_; }

  private:
   explicit NumericDiffCostFunction(std::unique_ptr<CostFunctor> functor,
                                    Ownership ownership,
                                    [[maybe_unused]] int num_residuals,
                                    const NumericDiffOptions& options)
       : functor_(std::move(functor)), ownership_(ownership), options_(options) {
     if constexpr (kNumResiduals == DYNAMIC) {
       SizedCostFunction<kNumResiduals, Ns...>::set_num_residuals(num_residuals);
     }
   }

   std::unique_ptr<CostFunctor> functor_;
   Ownership ownership_;
   NumericDiffOptions options_;
 };

 }  // namespace ceres

 #endif  // CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2024 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: keir@google.com (Keir Mierle)
	// sameeragarwal@google.com (Sameer Agarwal)
	//
	// Create CostFunctions as needed by the least squares framework with jacobians
	// computed via numeric (a.k.a. finite) differentiation. For more details see
	// http://en.wikipedia.org/wiki/Numerical_differentiation.
	//
	// To get an numerically differentiated cost function, you must define
	// a class with a operator() (a functor) that computes the residuals.
	//
	// The function must write the computed value in the last argument
	// (the only non-const one) and return true to indicate success.
	// Please see cost_function.h for details on how the return value
	// maybe used to impose simple constraints on the parameter block.
	//
	// For example, consider a scalar error e = k - x'y, where both x and y are
	// two-dimensional column vector parameters, the prime sign indicates
	// transposition, and k is a constant. The form of this error, which is the
	// difference between a constant and an expression, is a common pattern in least
	// squares problems. For example, the value x'y might be the model expectation
	// for a series of measurements, where there is an instance of the cost function
	// for each measurement k.
	//
	// The actual cost added to the total problem is e^2, or (k - x'k)^2; however,
	// the squaring is implicitly done by the optimization framework.
	//
	// To write an numerically-differentiable cost function for the above model,
	// first define the object
	//
	// class MyScalarCostFunctor {
	// explicit MyScalarCostFunctor(double k): k_(k) {}
	//
	// bool operator()(const double* const x,
	// const double* const y,
	// double* residuals) const {
	// residuals[0] = k_ - x[0] * y[0] - x[1] * y[1];
	// return true;
	// }
	//
	// private:
	// double k_;
	// };
	//
	// Note that in the declaration of operator() the input parameters x
	// and y come first, and are passed as const pointers to arrays of
	// doubles. If there were three input parameters, then the third input
	// parameter would come after y. The output is always the last
	// parameter, and is also a pointer to an array. In the example above,
	// the residual is a scalar, so only residuals[0] is set.
	//
	// Then given this class definition, the numerically differentiated
	// cost function with central differences used for computing the
	// derivative can be constructed as follows.
	//
	// CostFunction* cost_function
	// = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, 1, 2, 2>(
	// new MyScalarCostFunctor(1.0)); ^ ^ ^ ^
	// \| \| \| \|
	// Finite Differencing Scheme -+ \| \| \|
	// Dimension of residual ------------+ \| \|
	// Dimension of x ----------------------+ \|
	// Dimension of y -------------------------+
	//
	// In this example, there is usually an instance for each measurement of k.
	//
	// In the instantiation above, the template parameters following
	// "MyScalarCostFunctor", "1, 2, 2", describe the functor as computing
	// a 1-dimensional output from two arguments, both 2-dimensional.
	//
	// NumericDiffCostFunction also supports cost functions with a
	// runtime-determined number of residuals. For example:
	//
	// clang-format off
	//
	// CostFunction* cost_function
	// = new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, DYNAMIC, 2, 2>(
	// new CostFunctorWithDynamicNumResiduals(1.0), ^ ^ ^
	// TAKE_OWNERSHIP, \| \| \|
	// runtime_number_of_residuals); <----+ \| \| \|
	// \| \| \| \|
	// \| \| \| \|
	// Actual number of residuals ------+ \| \| \|
	// Indicate dynamic number of residuals --------------------+ \| \|
	// Dimension of x ------------------------------------------------+ \|
	// Dimension of y ---------------------------------------------------+
	// clang-format on
	//
	//
	// The central difference method is considerably more accurate at the cost of
	// twice as many function evaluations than forward difference. Consider using
	// central differences begin with, and only after that works, trying forward
	// difference to improve performance.
	//
	// WARNING #1: A common beginner's error when first using
	// NumericDiffCostFunction is to get the sizing wrong. In particular,
	// there is a tendency to set the template parameters to (dimension of
	// residual, number of parameters) instead of passing a dimension
	// parameter for every parameter. In the example above, that would
	// be <MyScalarCostFunctor, 1, 2>, which is missing the last '2'
	// argument. Please be careful when setting the size parameters.
	//
	////////////////////////////////////////////////////////////////////////////
	////////////////////////////////////////////////////////////////////////////
	//
	// ALTERNATE INTERFACE
	//
	// For a variety of reasons, including compatibility with legacy code,
	// NumericDiffCostFunction can also take CostFunction objects as
	// input. The following describes how.
	//
	// To get a numerically differentiated cost function, define a
	// subclass of CostFunction such that the Evaluate() function ignores
	// the jacobian parameter. The numeric differentiation wrapper will
	// fill in the jacobian parameter if necessary by repeatedly calling
	// the Evaluate() function with small changes to the appropriate
	// parameters, and computing the slope. For performance, the numeric
	// differentiation wrapper class is templated on the concrete cost
	// function, even though it could be implemented only in terms of the
	// virtual CostFunction interface.
	//
	// The numerically differentiated version of a cost function for a cost function
	// can be constructed as follows:
	//
	// auto* cost_function
	// = new NumericDiffCostFunction<MyCostFunction, CENTRAL, 1, 4, 8>();
	//
	// where MyCostFunction has 1 residual and 2 parameter blocks with sizes 4 and 8
	// respectively. Look at the tests for a more detailed example.
	//
	// TODO(keir): Characterize accuracy; mention pitfalls; provide alternatives.

	#ifndef CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_
	#define CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_

	#include <array>
	#include <memory>
	#include <type_traits>

	#include "Eigen/Dense"
	#include "ceres/cost_function.h"
	#include "ceres/internal/numeric_diff.h"
	#include "ceres/internal/parameter_dims.h"
	#include "ceres/numeric_diff_options.h"
	#include "ceres/sized_cost_function.h"
	#include "ceres/types.h"

	namespace ceres {

	template <typename CostFunctor,
	NumericDiffMethodType kMethod = CENTRAL,
	int kNumResiduals = 0, // Number of residuals, or ceres::DYNAMIC
	int... Ns> // Parameters dimensions for each block.
	class NumericDiffCostFunction final
	: public SizedCostFunction<kNumResiduals, Ns...> {
	public:
	explicit NumericDiffCostFunction(
	CostFunctor* functor,
	Ownership ownership = TAKE_OWNERSHIP,
	int num_residuals = kNumResiduals,
	const NumericDiffOptions& options = NumericDiffOptions())
	: NumericDiffCostFunction{std::unique_ptr<CostFunctor>{functor},
	ownership,
	num_residuals,
	options} {}

	explicit NumericDiffCostFunction(
	std::unique_ptr<CostFunctor> functor,
	int num_residuals = kNumResiduals,
	const NumericDiffOptions& options = NumericDiffOptions())
	: NumericDiffCostFunction{
	std::move(functor), TAKE_OWNERSHIP, num_residuals, options} {}

	// Constructs the CostFunctor on the heap and takes the ownership.
	// Invocable only if the number of residuals is known at compile-time.
	template <class... Args,
	bool kIsDynamic = kNumResiduals == DYNAMIC,
	std::enable_if_t<!kIsDynamic &&
	std::is_constructible_v<CostFunctor, Args&&...>>* =
	nullptr>
	explicit NumericDiffCostFunction(Args&&... args)
	// NOTE We explicitly use direct initialization using parentheses instead
	// of uniform initialization using braces to avoid narrowing conversion
	// warnings.
	: NumericDiffCostFunction{
	std::make_unique<CostFunctor>(std::forward<Args>(args)...),
	TAKE_OWNERSHIP} {}

	NumericDiffCostFunction(NumericDiffCostFunction&& other) noexcept = default;
	NumericDiffCostFunction& operator=(NumericDiffCostFunction&& other) noexcept =
	default;
	NumericDiffCostFunction(const NumericDiffCostFunction&) = delete;
	NumericDiffCostFunction& operator=(const NumericDiffCostFunction&) = delete;

	~NumericDiffCostFunction() override {
	if (ownership_ != TAKE_OWNERSHIP) {
	functor_.release();
	}
	}

	bool Evaluate(double const* const* parameters,
	double* residuals,
	double** jacobians) const override {
	using internal::FixedArray;
	using internal::NumericDiff;

	using ParameterDims =
	typename SizedCostFunction<kNumResiduals, Ns...>::ParameterDims;

	constexpr int kNumParameters = ParameterDims::kNumParameters;
	constexpr int kNumParameterBlocks = ParameterDims::kNumParameterBlocks;

	// Get the function value (residuals) at the the point to evaluate.
	if (!internal::VariadicEvaluate<ParameterDims>(
	*functor_, parameters, residuals)) {
	return false;
	}

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

	// Create a copy of the parameters which will get mutated.
	FixedArray<double> parameters_copy(kNumParameters);
	std::array<double*, kNumParameterBlocks> parameters_reference_copy =
	ParameterDims::GetUnpackedParameters(parameters_copy.data());

	for (int block = 0; block < kNumParameterBlocks; ++block) {
	memcpy(parameters_reference_copy[block],
	parameters[block],
	sizeof(double) * ParameterDims::GetDim(block));
	}

	internal::EvaluateJacobianForParameterBlocks<ParameterDims>::
	template Apply<kMethod, kNumResiduals>(
	functor_.get(),
	residuals,
	options_,
	SizedCostFunction<kNumResiduals, Ns...>::num_residuals(),
	parameters_reference_copy.data(),
	jacobians);

	return true;
	}

	const CostFunctor& functor() const { return *functor_; }

	private:
	explicit NumericDiffCostFunction(std::unique_ptr<CostFunctor> functor,
	Ownership ownership,
	[[maybe_unused]] int num_residuals,
	const NumericDiffOptions& options)
	: functor_(std::move(functor)), ownership_(ownership), options_(options) {
	if constexpr (kNumResiduals == DYNAMIC) {
	SizedCostFunction<kNumResiduals, Ns...>::set_num_residuals(num_residuals);
	}
	}

	std::unique_ptr<CostFunctor> functor_;
	Ownership ownership_;
	NumericDiffOptions options_;
	};

	} // namespace ceres

	#endif // CERES_PUBLIC_NUMERIC_DIFF_COST_FUNCTION_H_