| // Ceres Solver - A fast non-linear least squares minimizer |
| // Copyright 2019 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) |
| // |
| // Computation of the Jacobian matrix for vector-valued functions of multiple |
| // variables, using automatic differentiation based on the implementation of |
| // dual numbers in jet.h. Before reading the rest of this file, it is advisable |
| // to read jet.h's header comment in detail. |
| // |
| // The helper wrapper AutoDifferentiate() computes the jacobian of |
| // functors with templated operator() taking this form: |
| // |
| // struct F { |
| // template<typename T> |
| // bool operator()(const T *x, const T *y, ..., T *z) { |
| // // Compute z[] based on x[], y[], ... |
| // // return true if computation succeeded, false otherwise. |
| // } |
| // }; |
| // |
| // All inputs and outputs may be vector-valued. |
| // |
| // To understand how jets are used to compute the jacobian, a |
| // picture may help. Consider a vector-valued function, F, returning 3 |
| // dimensions and taking a vector-valued parameter of 4 dimensions: |
| // |
| // y x |
| // [ * ] F [ * ] |
| // [ * ] <--- [ * ] |
| // [ * ] [ * ] |
| // [ * ] |
| // |
| // Similar to the 2-parameter example for f described in jet.h, computing the |
| // jacobian dy/dx is done by substituting a suitable jet object for x and all |
| // intermediate steps of the computation of F. Since x is has 4 dimensions, use |
| // a Jet<double, 4>. |
| // |
| // Before substituting a jet object for x, the dual components are set |
| // appropriately for each dimension of x: |
| // |
| // y x |
| // [ * | * * * * ] f [ * | 1 0 0 0 ] x0 |
| // [ * | * * * * ] <--- [ * | 0 1 0 0 ] x1 |
| // [ * | * * * * ] [ * | 0 0 1 0 ] x2 |
| // ---+--- [ * | 0 0 0 1 ] x3 |
| // | ^ ^ ^ ^ |
| // dy/dx | | | +----- infinitesimal for x3 |
| // | | +------- infinitesimal for x2 |
| // | +--------- infinitesimal for x1 |
| // +----------- infinitesimal for x0 |
| // |
| // The reason to set the internal 4x4 submatrix to the identity is that we wish |
| // to take the derivative of y separately with respect to each dimension of x. |
| // Each column of the 4x4 identity is therefore for a single component of the |
| // independent variable x. |
| // |
| // Then the jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the |
| // extended y vector, indicated in the above diagram. |
| // |
| // Functors with multiple parameters |
| // --------------------------------- |
| // In practice, it is often convenient to use a function f of two or more |
| // vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the jet |
| // framework is designed for a single-parameter vector-valued input. The wrapper |
| // in this file addresses this issue adding support for functions with one or |
| // more parameter vectors. |
| // |
| // To support multiple parameters, all the parameter vectors are concatenated |
| // into one and treated as a single parameter vector, except that since the |
| // functor expects different inputs, we need to construct the jets as if they |
| // were part of a single parameter vector. The extended jets are passed |
| // separately for each parameter. |
| // |
| // For example, consider a functor F taking two vector parameters, p[2] and |
| // q[3], and producing an output y[4]: |
| // |
| // struct F { |
| // template<typename T> |
| // bool operator()(const T *p, const T *q, T *z) { |
| // // ... |
| // } |
| // }; |
| // |
| // In this case, the necessary jet type is Jet<double, 5>. Here is a |
| // visualization of the jet objects in this case: |
| // |
| // Dual components for p ----+ |
| // | |
| // -+- |
| // y [ * | 1 0 | 0 0 0 ] --- p[0] |
| // [ * | 0 1 | 0 0 0 ] --- p[1] |
| // [ * | . . | + + + ] | |
| // [ * | . . | + + + ] v |
| // [ * | . . | + + + ] <--- F(p, q) |
| // [ * | . . | + + + ] ^ |
| // ^^^ ^^^^^ | |
| // dy/dp dy/dq [ * | 0 0 | 1 0 0 ] --- q[0] |
| // [ * | 0 0 | 0 1 0 ] --- q[1] |
| // [ * | 0 0 | 0 0 1 ] --- q[2] |
| // --+-- |
| // | |
| // Dual components for q --------------+ |
| // |
| // where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+" |
| // of y in the above diagram are the derivatives of y with respect to p and q |
| // respectively. This is how autodiff works for functors taking multiple vector |
| // valued arguments (up to 6). |
| // |
| // Jacobian NULL pointers |
| // ---------------------- |
| // In general, the functions below will accept NULL pointers for all or some of |
| // the Jacobian parameters, meaning that those Jacobians will not be computed. |
| |
| #ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_ |
| #define CERES_PUBLIC_INTERNAL_AUTODIFF_H_ |
| |
| #include <stddef.h> |
| |
| #include <array> |
| |
| #include "ceres/internal/array_selector.h" |
| #include "ceres/internal/eigen.h" |
| #include "ceres/internal/fixed_array.h" |
| #include "ceres/internal/parameter_dims.h" |
| #include "ceres/internal/variadic_evaluate.h" |
| #include "ceres/jet.h" |
| #include "ceres/types.h" |
| #include "glog/logging.h" |
| |
| // If the number of parameters exceeds this values, the corresponding jets are |
| // placed on the heap. This will reduce performance by a factor of 2-5 on |
| // current compilers. |
| #ifndef CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK |
| #define CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK 50 |
| #endif |
| |
| #ifndef CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK |
| #define CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK 20 |
| #endif |
| |
| namespace ceres { |
| namespace internal { |
| |
| // Extends src by a 1st order perturbation for every dimension and puts it in |
| // dst. The size of src is N. Since this is also used for perturbations in |
| // blocked arrays, offset is used to shift which part of the jet the |
| // perturbation occurs. This is used to set up the extended x augmented by an |
| // identity matrix. The JetT type should be a Jet type, and T should be a |
| // numeric type (e.g. double). For example, |
| // |
| // 0 1 2 3 4 5 6 7 8 |
| // dst[0] [ * | . . | 1 0 0 | . . . ] |
| // dst[1] [ * | . . | 0 1 0 | . . . ] |
| // dst[2] [ * | . . | 0 0 1 | . . . ] |
| // |
| // is what would get put in dst if N was 3, offset was 3, and the jet type JetT |
| // was 8-dimensional. |
| template <int j, int N, int Offset, typename T, typename JetT> |
| struct Make1stOrderPerturbation { |
| public: |
| inline static void Apply(const T* src, JetT* dst) { |
| if (j == 0) { |
| DCHECK(src); |
| DCHECK(dst); |
| } |
| dst[j] = JetT(src[j], j + Offset); |
| Make1stOrderPerturbation<j + 1, N, Offset, T, JetT>::Apply(src, dst); |
| } |
| }; |
| |
| template <int N, int Offset, typename T, typename JetT> |
| struct Make1stOrderPerturbation<N, N, Offset, T, JetT> { |
| public: |
| static void Apply(const T* src, JetT* dst) {} |
| }; |
| |
| // Calls Make1stOrderPerturbation for every parameter block. |
| // |
| // Example: |
| // If one having three parameter blocks with dimensions (3, 2, 4), the call |
| // Make1stOrderPerturbations<integer_sequence<3, 2, 4>::Apply(params, x); |
| // will result in the following calls to Make1stOrderPerturbation: |
| // Make1stOrderPerturbation<0, 3, 0>::Apply(params[0], x + 0); |
| // Make1stOrderPerturbation<0, 2, 3>::Apply(params[1], x + 3); |
| // Make1stOrderPerturbation<0, 4, 5>::Apply(params[2], x + 5); |
| template <typename Seq, int ParameterIdx = 0, int Offset = 0> |
| struct Make1stOrderPerturbations; |
| |
| template <int N, int... Ns, int ParameterIdx, int Offset> |
| struct Make1stOrderPerturbations<integer_sequence<int, N, Ns...>, |
| ParameterIdx, |
| Offset> { |
| template <typename T, typename JetT> |
| inline static void Apply(T const* const* parameters, JetT* x) { |
| Make1stOrderPerturbation<0, N, Offset, T, JetT>::Apply( |
| parameters[ParameterIdx], x + Offset); |
| Make1stOrderPerturbations<integer_sequence<int, Ns...>, |
| ParameterIdx + 1, |
| Offset + N>::Apply(parameters, x); |
| } |
| }; |
| |
| // End of 'recursion'. Nothing more to do. |
| template <int ParameterIdx, int Total> |
| struct Make1stOrderPerturbations<integer_sequence<int>, ParameterIdx, Total> { |
| template <typename T, typename JetT> |
| static void Apply(T const* const* /* NOT USED */, JetT* /* NOT USED */) {} |
| }; |
| |
| // Takes the 0th order part of src, assumed to be a Jet type, and puts it in |
| // dst. This is used to pick out the "vector" part of the extended y. |
| template <typename JetT, typename T> |
| inline void Take0thOrderPart(int M, const JetT* src, T dst) { |
| DCHECK(src); |
| for (int i = 0; i < M; ++i) { |
| dst[i] = src[i].a; |
| } |
| } |
| |
| // Takes N 1st order parts, starting at index N0, and puts them in the M x N |
| // matrix 'dst'. This is used to pick out the "matrix" parts of the extended y. |
| template <int N0, int N, typename JetT, typename T> |
| inline void Take1stOrderPart(const int M, const JetT* src, T* dst) { |
| DCHECK(src); |
| DCHECK(dst); |
| for (int i = 0; i < M; ++i) { |
| Eigen::Map<Eigen::Matrix<T, N, 1>>(dst + N * i, N) = |
| src[i].v.template segment<N>(N0); |
| } |
| } |
| |
| // Calls Take1stOrderPart for every parameter block. |
| // |
| // Example: |
| // If one having three parameter blocks with dimensions (3, 2, 4), the call |
| // Take1stOrderParts<integer_sequence<3, 2, 4>::Apply(num_outputs, |
| // output, |
| // jacobians); |
| // will result in the following calls to Take1stOrderPart: |
| // if (jacobians[0]) { |
| // Take1stOrderPart<0, 3>(num_outputs, output, jacobians[0]); |
| // } |
| // if (jacobians[1]) { |
| // Take1stOrderPart<3, 2>(num_outputs, output, jacobians[1]); |
| // } |
| // if (jacobians[2]) { |
| // Take1stOrderPart<5, 4>(num_outputs, output, jacobians[2]); |
| // } |
| template <typename Seq, int ParameterIdx = 0, int Offset = 0> |
| struct Take1stOrderParts; |
| |
| template <int N, int... Ns, int ParameterIdx, int Offset> |
| struct Take1stOrderParts<integer_sequence<int, N, Ns...>, |
| ParameterIdx, |
| Offset> { |
| template <typename JetT, typename T> |
| inline static void Apply(int num_outputs, JetT* output, T** jacobians) { |
| if (jacobians[ParameterIdx]) { |
| Take1stOrderPart<Offset, N>(num_outputs, output, jacobians[ParameterIdx]); |
| } |
| Take1stOrderParts<integer_sequence<int, Ns...>, |
| ParameterIdx + 1, |
| Offset + N>::Apply(num_outputs, output, jacobians); |
| } |
| }; |
| |
| // End of 'recursion'. Nothing more to do. |
| template <int ParameterIdx, int Offset> |
| struct Take1stOrderParts<integer_sequence<int>, ParameterIdx, Offset> { |
| template <typename T, typename JetT> |
| static void Apply(int /* NOT USED*/, |
| JetT* /* NOT USED*/, |
| T** /* NOT USED */) {} |
| }; |
| |
| template <int kNumResiduals, |
| typename ParameterDims, |
| typename Functor, |
| typename T> |
| inline bool AutoDifferentiate(const Functor& functor, |
| T const* const* parameters, |
| int dynamic_num_outputs, |
| T* function_value, |
| T** jacobians) { |
| typedef Jet<T, ParameterDims::kNumParameters> JetT; |
| using Parameters = typename ParameterDims::Parameters; |
| |
| if (kNumResiduals != DYNAMIC) { |
| DCHECK_EQ(kNumResiduals, dynamic_num_outputs); |
| } |
| |
| ArraySelector<JetT, |
| ParameterDims::kNumParameters, |
| CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK> |
| parameters_as_jets(ParameterDims::kNumParameters); |
| |
| // Pointers to the beginning of each parameter block |
| std::array<JetT*, ParameterDims::kNumParameterBlocks> unpacked_parameters = |
| ParameterDims::GetUnpackedParameters(parameters_as_jets.data()); |
| |
| // If the number of residuals is fixed, we use the template argument as the |
| // number of outputs. Otherwise we use the num_outputs parameter. Note: The |
| // ?-operator here is compile-time evaluated, therefore num_outputs is also |
| // a compile-time constant for functors with fixed residuals. |
| const int num_outputs = |
| kNumResiduals == DYNAMIC ? dynamic_num_outputs : kNumResiduals; |
| DCHECK_GT(num_outputs, 0); |
| |
| ArraySelector<JetT, kNumResiduals, CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK> |
| residuals_as_jets(num_outputs); |
| |
| // Invalidate the output Jets, so that we can detect if the user |
| // did not assign values to all of them. |
| for (int i = 0; i < num_outputs; ++i) { |
| residuals_as_jets[i].a = kImpossibleValue; |
| residuals_as_jets[i].v.setConstant(kImpossibleValue); |
| } |
| |
| Make1stOrderPerturbations<Parameters>::Apply(parameters, |
| parameters_as_jets.data()); |
| |
| if (!VariadicEvaluate<ParameterDims>( |
| functor, unpacked_parameters.data(), residuals_as_jets.data())) { |
| return false; |
| } |
| |
| Take0thOrderPart(num_outputs, residuals_as_jets.data(), function_value); |
| Take1stOrderParts<Parameters>::Apply( |
| num_outputs, residuals_as_jets.data(), jacobians); |
| |
| return true; |
| } |
| |
| } // namespace internal |
| } // namespace ceres |
| |
| #endif // CERES_PUBLIC_INTERNAL_AUTODIFF_H_ |
