// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2010, 2011, 2012 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
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// 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 adivsable
// to read jet.h's header comment in detail.
//
// The helper wrapper AutoDiff::Differentiate() 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 substutiting 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 "ceres/jet.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/fixed_array.h"
#include "ceres/internal/variadic_evaluate.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
// Extends src by a 1st order pertubation 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 <typename JetT, typename T, int N>
inline void Make1stOrderPerturbation(int offset, const T* src, JetT* dst) {
DCHECK(src);
DCHECK(dst);
for (int j = 0; j < N; ++j) {
dst[j].a = src[j];
dst[j].v.setZero();
dst[j].v[offset + j] = 1.0;
}
}
// 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 <typename JetT, typename T, int N0, int N>
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);
}
}
// This is in a struct because default template parameters on a
// function are not supported in C++03 (though it is available in
// C++0x). N0 through N5 are the dimension of the input arguments to
// the user supplied functor.
template <typename Functor, typename T,
int N0 = 0, int N1 = 0, int N2 = 0, int N3 = 0, int N4 = 0,
int N5 = 0, int N6 = 0, int N7 = 0, int N8 = 0, int N9 = 0>
struct AutoDiff {
static bool Differentiate(const Functor& functor,
T const *const *parameters,
int num_outputs,
T *function_value,
T **jacobians) {
// This block breaks the 80 column rule to keep it somewhat readable.
DCHECK_GT(num_outputs, 0);
DCHECK((!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;
typedef Jet<T, N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9> JetT;
FixedArray<JetT, (256 * 7) / sizeof(JetT)> x(
N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9 + num_outputs);
// These are the positions of the respective jets in the fixed array x.
const int jet0 = 0;
const int jet1 = N0;
const int jet2 = N0 + N1;
const int jet3 = N0 + N1 + N2;
const int jet4 = N0 + N1 + N2 + N3;
const int jet5 = N0 + N1 + N2 + N3 + N4;
const int jet6 = N0 + N1 + N2 + N3 + N4 + N5;
const int jet7 = N0 + N1 + N2 + N3 + N4 + N5 + N6;
const int jet8 = N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7;
const int jet9 = N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8;
const JetT *unpacked_parameters[10] = {
x.get() + jet0,
x.get() + jet1,
x.get() + jet2,
x.get() + jet3,
x.get() + jet4,
x.get() + jet5,
x.get() + jet6,
x.get() + jet7,
x.get() + jet8,
x.get() + jet9,
};
JetT* output = x.get() + N0 + N1 + N2 + N3 + N4 + N5 + N6 + N7 + N8 + N9;
#define CERES_MAKE_1ST_ORDER_PERTURBATION(i) \
if (N ## i) { \
internal::Make1stOrderPerturbation<JetT, T, N ## i>( \
jet ## i, \
parameters[i], \
x.get() + jet ## i); \
}
CERES_MAKE_1ST_ORDER_PERTURBATION(0);
CERES_MAKE_1ST_ORDER_PERTURBATION(1);
CERES_MAKE_1ST_ORDER_PERTURBATION(2);
CERES_MAKE_1ST_ORDER_PERTURBATION(3);
CERES_MAKE_1ST_ORDER_PERTURBATION(4);
CERES_MAKE_1ST_ORDER_PERTURBATION(5);
CERES_MAKE_1ST_ORDER_PERTURBATION(6);
CERES_MAKE_1ST_ORDER_PERTURBATION(7);
CERES_MAKE_1ST_ORDER_PERTURBATION(8);
CERES_MAKE_1ST_ORDER_PERTURBATION(9);
#undef CERES_MAKE_1ST_ORDER_PERTURBATION
if (!VariadicEvaluate<Functor, JetT,
N0, N1, N2, N3, N4, N5, N6, N7, N8, N9>::Call(
functor, unpacked_parameters, output)) {
return false;
}
internal::Take0thOrderPart(num_outputs, output, function_value);
#define CERES_TAKE_1ST_ORDER_PERTURBATION(i) \
if (N ## i) { \
if (jacobians[i]) { \
internal::Take1stOrderPart<JetT, T, \
jet ## i, \
N ## i>(num_outputs, \
output, \
jacobians[i]); \
} \
}
CERES_TAKE_1ST_ORDER_PERTURBATION(0);
CERES_TAKE_1ST_ORDER_PERTURBATION(1);
CERES_TAKE_1ST_ORDER_PERTURBATION(2);
CERES_TAKE_1ST_ORDER_PERTURBATION(3);
CERES_TAKE_1ST_ORDER_PERTURBATION(4);
CERES_TAKE_1ST_ORDER_PERTURBATION(5);
CERES_TAKE_1ST_ORDER_PERTURBATION(6);
CERES_TAKE_1ST_ORDER_PERTURBATION(7);
CERES_TAKE_1ST_ORDER_PERTURBATION(8);
CERES_TAKE_1ST_ORDER_PERTURBATION(9);
#undef CERES_TAKE_1ST_ORDER_PERTURBATION
return true;
}
};
} // namespace internal
} // namespace ceres
#endif // CERES_PUBLIC_INTERNAL_AUTODIFF_H_