1 // Ceres Solver - A fast non-linear least squares minimizer 2 // Copyright 2010, 2011, 2012 Google Inc. All rights reserved. 3 // http://code.google.com/p/ceres-solver/ 4 // 5 // Redistribution and use in source and binary forms, with or without 6 // modification, are permitted provided that the following conditions are met: 7 // 8 // * Redistributions of source code must retain the above copyright notice, 9 // this list of conditions and the following disclaimer. 10 // * Redistributions in binary form must reproduce the above copyright notice, 11 // this list of conditions and the following disclaimer in the documentation 12 // and/or other materials provided with the distribution. 13 // * Neither the name of Google Inc. nor the names of its contributors may be 14 // used to endorse or promote products derived from this software without 15 // specific prior written permission. 16 // 17 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 18 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 21 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 22 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 23 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 24 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 25 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 26 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 27 // POSSIBILITY OF SUCH DAMAGE. 28 // 29 // Author: keir (at) google.com (Keir Mierle) 30 // 31 // Computation of the Jacobian matrix for vector-valued functions of multiple 32 // variables, using automatic differentiation based on the implementation of 33 // dual numbers in jet.h. Before reading the rest of this file, it is adivsable 34 // to read jet.h's header comment in detail. 35 // 36 // The helper wrapper AutoDiff::Differentiate() computes the jacobian of 37 // functors with templated operator() taking this form: 38 // 39 // struct F { 40 // template<typename T> 41 // bool operator()(const T *x, const T *y, ..., T *z) { 42 // // Compute z[] based on x[], y[], ... 43 // // return true if computation succeeded, false otherwise. 44 // } 45 // }; 46 // 47 // All inputs and outputs may be vector-valued. 48 // 49 // To understand how jets are used to compute the jacobian, a 50 // picture may help. Consider a vector-valued function, F, returning 3 51 // dimensions and taking a vector-valued parameter of 4 dimensions: 52 // 53 // y x 54 // [ * ] F [ * ] 55 // [ * ] <--- [ * ] 56 // [ * ] [ * ] 57 // [ * ] 58 // 59 // Similar to the 2-parameter example for f described in jet.h, computing the 60 // jacobian dy/dx is done by substutiting a suitable jet object for x and all 61 // intermediate steps of the computation of F. Since x is has 4 dimensions, use 62 // a Jet<double, 4>. 63 // 64 // Before substituting a jet object for x, the dual components are set 65 // appropriately for each dimension of x: 66 // 67 // y x 68 // [ * | * * * * ] f [ * | 1 0 0 0 ] x0 69 // [ * | * * * * ] <--- [ * | 0 1 0 0 ] x1 70 // [ * | * * * * ] [ * | 0 0 1 0 ] x2 71 // ---+--- [ * | 0 0 0 1 ] x3 72 // | ^ ^ ^ ^ 73 // dy/dx | | | +----- infinitesimal for x3 74 // | | +------- infinitesimal for x2 75 // | +--------- infinitesimal for x1 76 // +----------- infinitesimal for x0 77 // 78 // The reason to set the internal 4x4 submatrix to the identity is that we wish 79 // to take the derivative of y separately with respect to each dimension of x. 80 // Each column of the 4x4 identity is therefore for a single component of the 81 // independent variable x. 82 // 83 // Then the jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the 84 // extended y vector, indicated in the above diagram. 85 // 86 // Functors with multiple parameters 87 // --------------------------------- 88 // In practice, it is often convenient to use a function f of two or more 89 // vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the jet 90 // framework is designed for a single-parameter vector-valued input. The wrapper 91 // in this file addresses this issue adding support for functions with one or 92 // more parameter vectors. 93 // 94 // To support multiple parameters, all the parameter vectors are concatenated 95 // into one and treated as a single parameter vector, except that since the 96 // functor expects different inputs, we need to construct the jets as if they 97 // were part of a single parameter vector. The extended jets are passed 98 // separately for each parameter. 99 // 100 // For example, consider a functor F taking two vector parameters, p[2] and 101 // q[3], and producing an output y[4]: 102 // 103 // struct F { 104 // template<typename T> 105 // bool operator()(const T *p, const T *q, T *z) { 106 // // ... 107 // } 108 // }; 109 // 110 // In this case, the necessary jet type is Jet<double, 5>. Here is a 111 // visualization of the jet objects in this case: 112 // 113 // Dual components for p ----+ 114 // | 115 // -+- 116 // y [ * | 1 0 | 0 0 0 ] --- p[0] 117 // [ * | 0 1 | 0 0 0 ] --- p[1] 118 // [ * | . . | + + + ] | 119 // [ * | . . | + + + ] v 120 // [ * | . . | + + + ] <--- F(p, q) 121 // [ * | . . | + + + ] ^ 122 // ^^^ ^^^^^ | 123 // dy/dp dy/dq [ * | 0 0 | 1 0 0 ] --- q[0] 124 // [ * | 0 0 | 0 1 0 ] --- q[1] 125 // [ * | 0 0 | 0 0 1 ] --- q[2] 126 // --+-- 127 // | 128 // Dual components for q --------------+ 129 // 130 // where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+" 131 // of y in the above diagram are the derivatives of y with respect to p and q 132 // respectively. This is how autodiff works for functors taking multiple vector 133 // valued arguments (up to 6). 134 // 135 // Jacobian NULL pointers 136 // ---------------------- 137 // In general, the functions below will accept NULL pointers for all or some of 138 // the Jacobian parameters, meaning that those Jacobians will not be computed. 139 140 #ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_ 141 #define CERES_PUBLIC_INTERNAL_AUTODIFF_H_ 142 143 #include <stddef.h> 144 145 #include "ceres/jet.h" 146 #include "ceres/internal/eigen.h" 147 #include "ceres/internal/fixed_array.h" 148 #include "ceres/internal/variadic_evaluate.h" 149 #include "glog/logging.h" 150 151 namespace ceres { 152 namespace internal { 153 154 // Extends src by a 1st order pertubation for every dimension and puts it in 155 // dst. The size of src is N. Since this is also used for perturbations in 156 // blocked arrays, offset is used to shift which part of the jet the 157 // perturbation occurs. This is used to set up the extended x augmented by an 158 // identity matrix. The JetT type should be a Jet type, and T should be a 159 // numeric type (e.g. double). For example, 160 // 161 // 0 1 2 3 4 5 6 7 8 162 // dst[0] [ * | . . | 1 0 0 | . . . ] 163 // dst[1] [ * | . . | 0 1 0 | . . . ] 164 // dst[2] [ * | . . | 0 0 1 | . . . ] 165 // 166 // is what would get put in dst if N was 3, offset was 3, and the jet type JetT 167 // was 8-dimensional. 168 template <typename JetT, typename T, int N> 169 inline void Make1stOrderPerturbation(int offset, const T* src, JetT* dst) { 170 DCHECK(src); 171 DCHECK(dst); 172 for (int j = 0; j < N; ++j) { 173 dst[j].a = src[j]; 174 dst[j].v.setZero(); 175 dst[j].v[offset + j] = 1.0; 176 } 177 } 178 179 // Takes the 0th order part of src, assumed to be a Jet type, and puts it in 180 // dst. This is used to pick out the "vector" part of the extended y. 181 template <typename JetT, typename T> 182 inline void Take0thOrderPart(int M, const JetT *src, T dst) { 183 DCHECK(src); 184 for (int i = 0; i < M; ++i) { 185 dst[i] = src[i].a; 186 } 187 } 188 189 // Takes N 1st order parts, starting at index N0, and puts them in the M x N 190 // matrix 'dst'. This is used to pick out the "matrix" parts of the extended y. 191 template <typename JetT, typename T, int N0, int N> 192 inline void Take1stOrderPart(const int M, const JetT *src, T *dst) { 193 DCHECK(src); 194 DCHECK(dst); 195 for (int i = 0; i < M; ++i) { 196 Eigen::Map<Eigen::Matrix<T, N, 1> >(dst + N * i, N) = 197 src[i].v.template segment<N>(N0); 198 } 199 } 200 201 // This is in a struct because default template parameters on a 202 // function are not supported in C++03 (though it is available in 203 // C++0x). N0 through N5 are the dimension of the input arguments to 204 // the user supplied functor. 205 template <typename Functor, typename T, 206 int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0, 207 int N5 = 0, int N6 = 0, int N7 = 0, int N8 = 0, int N9 = 0> 208 struct AutoDiff { 209 static bool Differentiate(const Functor& functor, 210 T const *const *parameters, 211 int num_outputs, 212 T *function_value, 213 T **jacobians) { 214 // This block breaks the 80 column rule to keep it somewhat readable. 215 DCHECK_GT(num_outputs, 0); 216 DCHECK((!N1 && !N2 && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || 217 ((N1 > 0) && !N2 && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || 218 ((N1 > 0) && (N2 > 0) && !N3 && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || 219 ((N1 > 0) && (N2 > 0) && (N3 > 0) && !N4 && !N5 && !N6 && !N7 && !N8 && !N9) || 220 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && !N5 && !N6 && !N7 && !N8 && !N9) || 221 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && !N6 && !N7 && !N8 && !N9) || 222 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && !N7 && !N8 && !N9) || 223 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && !N8 && !N9) || 224 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && (N8 > 0) && !N9) || 225 ((N1 > 0) && (N2 > 0) && (N3 > 0) && (N4 > 0) && (N5 > 0) && (N6 > 0) && (N7 > 0) && (N8 > 0) && (N9 > 0))) 226 << "Zero block cannot precede a non-zero block. Block sizes are " 227 << "(ignore trailing 0s): " << N0 << ", " << N1 << ", " << N2 << ", " 228 << N3 << ", " << N4 << ", " << N5 << ", " << N6 << ", " << N7 << ", " 229 << N8 << ", " << N9; 230 231 typedef Jet<T, N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9> JetT; 232 FixedArray<JetT, (256 * 7) / sizeof(JetT)> x( 233 N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9 + num_outputs); 234 235 // These are the positions of the respective jets in the fixed array x. 236 const int jet0 = 0; 237 const int jet1 = N0; 238 const int jet2 = N0 + N1; 239 const int jet3 = N0 + N1 + N2; 240 const int jet4 = N0 + N1 + N2 + N3; 241 const int jet5 = N0 + N1 + N2 + N3 + N4; 242 const int jet6 = N0 + N1 + N2 + N3 + N4 + N5; 243 const int jet7 = N0 + N1 + N2 + N3 + N4 + N5 + N6; 244 const int jet8 = N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7; 245 const int jet9 = N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8; 246 247 const JetT *unpacked_parameters[10] = { 248 x.get() + jet0, 249 x.get() + jet1, 250 x.get() + jet2, 251 x.get() + jet3, 252 x.get() + jet4, 253 x.get() + jet5, 254 x.get() + jet6, 255 x.get() + jet7, 256 x.get() + jet8, 257 x.get() + jet9, 258 }; 259 260 JetT* output = x.get() + N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9; 261 262 #define CERES_MAKE_1ST_ORDER_PERTURBATION(i) \ 263 if (N ## i) { \ 264 internal::Make1stOrderPerturbation<JetT, T, N ## i>( \ 265 jet ## i, \ 266 parameters[i], \ 267 x.get() + jet ## i); \ 268 } 269 CERES_MAKE_1ST_ORDER_PERTURBATION(0); 270 CERES_MAKE_1ST_ORDER_PERTURBATION(1); 271 CERES_MAKE_1ST_ORDER_PERTURBATION(2); 272 CERES_MAKE_1ST_ORDER_PERTURBATION(3); 273 CERES_MAKE_1ST_ORDER_PERTURBATION(4); 274 CERES_MAKE_1ST_ORDER_PERTURBATION(5); 275 CERES_MAKE_1ST_ORDER_PERTURBATION(6); 276 CERES_MAKE_1ST_ORDER_PERTURBATION(7); 277 CERES_MAKE_1ST_ORDER_PERTURBATION(8); 278 CERES_MAKE_1ST_ORDER_PERTURBATION(9); 279 #undef CERES_MAKE_1ST_ORDER_PERTURBATION 280 281 if (!VariadicEvaluate<Functor, JetT, 282 N0, N1, N2, N3, N4, N5, N6, N7, N8, N9>::Call( 283 functor, unpacked_parameters, output)) { 284 return false; 285 } 286 287 internal::Take0thOrderPart(num_outputs, output, function_value); 288 289 #define CERES_TAKE_1ST_ORDER_PERTURBATION(i) \ 290 if (N ## i) { \ 291 if (jacobians[i]) { \ 292 internal::Take1stOrderPart<JetT, T, \ 293 jet ## i, \ 294 N ## i>(num_outputs, \ 295 output, \ 296 jacobians[i]); \ 297 } \ 298 } 299 CERES_TAKE_1ST_ORDER_PERTURBATION(0); 300 CERES_TAKE_1ST_ORDER_PERTURBATION(1); 301 CERES_TAKE_1ST_ORDER_PERTURBATION(2); 302 CERES_TAKE_1ST_ORDER_PERTURBATION(3); 303 CERES_TAKE_1ST_ORDER_PERTURBATION(4); 304 CERES_TAKE_1ST_ORDER_PERTURBATION(5); 305 CERES_TAKE_1ST_ORDER_PERTURBATION(6); 306 CERES_TAKE_1ST_ORDER_PERTURBATION(7); 307 CERES_TAKE_1ST_ORDER_PERTURBATION(8); 308 CERES_TAKE_1ST_ORDER_PERTURBATION(9); 309 #undef CERES_TAKE_1ST_ORDER_PERTURBATION 310 return true; 311 } 312 }; 313 314 } // namespace internal 315 } // namespace ceres 316 317 #endif // CERES_PUBLIC_INTERNAL_AUTODIFF_H_ 318
