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364 lines
14 KiB
364 lines
14 KiB
// Ceres Solver - A fast non-linear least squares minimizer
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// Copyright 2023 Google Inc. All rights reserved.
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// http://ceres-solver.org/
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are met:
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//
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// * Redistributions of source code must retain the above copyright notice,
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// this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above copyright notice,
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// this list of conditions and the following disclaimer in the documentation
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// and/or other materials provided with the distribution.
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// * Neither the name of Google Inc. nor the names of its contributors may be
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// used to endorse or promote products derived from this software without
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// specific prior written permission.
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//
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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// POSSIBILITY OF SUCH DAMAGE.
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//
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// Author: keir@google.com (Keir Mierle)
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//
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// Computation of the Jacobian matrix for vector-valued functions of multiple
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// variables, using automatic differentiation based on the implementation of
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// dual numbers in jet.h. Before reading the rest of this file, it is advisable
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// to read jet.h's header comment in detail.
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//
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// The helper wrapper AutoDifferentiate() computes the jacobian of
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// functors with templated operator() taking this form:
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//
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// struct F {
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// template<typename T>
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// bool operator()(const T *x, const T *y, ..., T *z) {
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// // Compute z[] based on x[], y[], ...
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// // return true if computation succeeded, false otherwise.
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// }
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// };
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//
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// All inputs and outputs may be vector-valued.
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//
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// To understand how jets are used to compute the jacobian, a
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// picture may help. Consider a vector-valued function, F, returning 3
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// dimensions and taking a vector-valued parameter of 4 dimensions:
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//
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// y x
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// [ * ] F [ * ]
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// [ * ] <--- [ * ]
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// [ * ] [ * ]
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// [ * ]
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//
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// Similar to the 2-parameter example for f described in jet.h, computing the
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// jacobian dy/dx is done by substituting a suitable jet object for x and all
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// intermediate steps of the computation of F. Since x is has 4 dimensions, use
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// a Jet<double, 4>.
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//
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// Before substituting a jet object for x, the dual components are set
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// appropriately for each dimension of x:
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//
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// y x
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// [ * | * * * * ] f [ * | 1 0 0 0 ] x0
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// [ * | * * * * ] <--- [ * | 0 1 0 0 ] x1
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// [ * | * * * * ] [ * | 0 0 1 0 ] x2
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// ---+--- [ * | 0 0 0 1 ] x3
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// | ^ ^ ^ ^
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// dy/dx | | | +----- infinitesimal for x3
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// | | +------- infinitesimal for x2
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// | +--------- infinitesimal for x1
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// +----------- infinitesimal for x0
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//
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// The reason to set the internal 4x4 submatrix to the identity is that we wish
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// to take the derivative of y separately with respect to each dimension of x.
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// Each column of the 4x4 identity is therefore for a single component of the
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// independent variable x.
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//
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// Then the jacobian of the mapping, dy/dx, is the 3x4 sub-matrix of the
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// extended y vector, indicated in the above diagram.
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//
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// Functors with multiple parameters
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// ---------------------------------
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// In practice, it is often convenient to use a function f of two or more
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// vector-valued parameters, for example, x[3] and z[6]. Unfortunately, the jet
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// framework is designed for a single-parameter vector-valued input. The wrapper
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// in this file addresses this issue adding support for functions with one or
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// more parameter vectors.
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//
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// To support multiple parameters, all the parameter vectors are concatenated
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// into one and treated as a single parameter vector, except that since the
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// functor expects different inputs, we need to construct the jets as if they
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// were part of a single parameter vector. The extended jets are passed
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// separately for each parameter.
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//
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// For example, consider a functor F taking two vector parameters, p[2] and
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// q[3], and producing an output y[4]:
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//
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// struct F {
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// template<typename T>
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// bool operator()(const T *p, const T *q, T *z) {
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// // ...
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// }
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// };
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//
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// In this case, the necessary jet type is Jet<double, 5>. Here is a
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// visualization of the jet objects in this case:
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//
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// Dual components for p ----+
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// |
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// -+-
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// y [ * | 1 0 | 0 0 0 ] --- p[0]
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// [ * | 0 1 | 0 0 0 ] --- p[1]
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// [ * | . . | + + + ] |
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// [ * | . . | + + + ] v
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// [ * | . . | + + + ] <--- F(p, q)
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// [ * | . . | + + + ] ^
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// ^^^ ^^^^^ |
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// dy/dp dy/dq [ * | 0 0 | 1 0 0 ] --- q[0]
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// [ * | 0 0 | 0 1 0 ] --- q[1]
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// [ * | 0 0 | 0 0 1 ] --- q[2]
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// --+--
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// |
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// Dual components for q --------------+
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//
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// where the 4x2 submatrix (marked with ".") and 4x3 submatrix (marked with "+"
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// of y in the above diagram are the derivatives of y with respect to p and q
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// respectively. This is how autodiff works for functors taking multiple vector
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// valued arguments (up to 6).
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//
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// Jacobian null pointers (nullptr)
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// --------------------------------
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// In general, the functions below will accept nullptr for all or some of the
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// Jacobian parameters, meaning that those Jacobians will not be computed.
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#ifndef CERES_PUBLIC_INTERNAL_AUTODIFF_H_
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#define CERES_PUBLIC_INTERNAL_AUTODIFF_H_
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#include <array>
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#include <cstddef>
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#include <utility>
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#include "ceres/internal/array_selector.h"
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#include "ceres/internal/eigen.h"
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#include "ceres/internal/fixed_array.h"
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#include "ceres/internal/parameter_dims.h"
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#include "ceres/internal/variadic_evaluate.h"
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#include "ceres/jet.h"
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#include "ceres/types.h"
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#include "glog/logging.h"
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// If the number of parameters exceeds this values, the corresponding jets are
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// placed on the heap. This will reduce performance by a factor of 2-5 on
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// current compilers.
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#ifndef CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK
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#define CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK 50
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#endif
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#ifndef CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK
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#define CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK 20
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#endif
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namespace ceres::internal {
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// Extends src by a 1st order perturbation for every dimension and puts it in
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// dst. The size of src is N. Since this is also used for perturbations in
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// blocked arrays, offset is used to shift which part of the jet the
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// perturbation occurs. This is used to set up the extended x augmented by an
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// identity matrix. The JetT type should be a Jet type, and T should be a
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// numeric type (e.g. double). For example,
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//
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// 0 1 2 3 4 5 6 7 8
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// dst[0] [ * | . . | 1 0 0 | . . . ]
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// dst[1] [ * | . . | 0 1 0 | . . . ]
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// dst[2] [ * | . . | 0 0 1 | . . . ]
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//
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// is what would get put in dst if N was 3, offset was 3, and the jet type JetT
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// was 8-dimensional.
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template <int j, int N, int Offset, typename T, typename JetT>
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struct Make1stOrderPerturbation {
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public:
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inline static void Apply(const T* src, JetT* dst) {
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if (j == 0) {
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DCHECK(src);
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DCHECK(dst);
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}
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dst[j] = JetT(src[j], j + Offset);
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Make1stOrderPerturbation<j + 1, N, Offset, T, JetT>::Apply(src, dst);
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}
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};
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template <int N, int Offset, typename T, typename JetT>
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struct Make1stOrderPerturbation<N, N, Offset, T, JetT> {
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public:
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static void Apply(const T* /* NOT USED */, JetT* /* NOT USED */) {}
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};
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// Calls Make1stOrderPerturbation for every parameter block.
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//
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// Example:
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// If one having three parameter blocks with dimensions (3, 2, 4), the call
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// Make1stOrderPerturbations<integer_sequence<3, 2, 4>::Apply(params, x);
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// will result in the following calls to Make1stOrderPerturbation:
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// Make1stOrderPerturbation<0, 3, 0>::Apply(params[0], x + 0);
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// Make1stOrderPerturbation<0, 2, 3>::Apply(params[1], x + 3);
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// Make1stOrderPerturbation<0, 4, 5>::Apply(params[2], x + 5);
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template <typename Seq, int ParameterIdx = 0, int Offset = 0>
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struct Make1stOrderPerturbations;
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template <int N, int... Ns, int ParameterIdx, int Offset>
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struct Make1stOrderPerturbations<std::integer_sequence<int, N, Ns...>,
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ParameterIdx,
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Offset> {
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template <typename T, typename JetT>
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inline static void Apply(T const* const* parameters, JetT* x) {
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Make1stOrderPerturbation<0, N, Offset, T, JetT>::Apply(
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parameters[ParameterIdx], x + Offset);
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Make1stOrderPerturbations<std::integer_sequence<int, Ns...>,
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ParameterIdx + 1,
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Offset + N>::Apply(parameters, x);
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}
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};
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// End of 'recursion'. Nothing more to do.
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template <int ParameterIdx, int Total>
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struct Make1stOrderPerturbations<std::integer_sequence<int>,
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ParameterIdx,
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Total> {
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template <typename T, typename JetT>
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static void Apply(T const* const* /* NOT USED */, JetT* /* NOT USED */) {}
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};
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// Takes the 0th order part of src, assumed to be a Jet type, and puts it in
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// dst. This is used to pick out the "vector" part of the extended y.
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template <typename JetT, typename T>
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inline void Take0thOrderPart(int M, const JetT* src, T dst) {
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DCHECK(src);
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for (int i = 0; i < M; ++i) {
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dst[i] = src[i].a;
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}
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}
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// Takes N 1st order parts, starting at index N0, and puts them in the M x N
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// matrix 'dst'. This is used to pick out the "matrix" parts of the extended y.
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template <int N0, int N, typename JetT, typename T>
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inline void Take1stOrderPart(const int M, const JetT* src, T* dst) {
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DCHECK(src);
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DCHECK(dst);
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for (int i = 0; i < M; ++i) {
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Eigen::Map<Eigen::Matrix<T, N, 1>>(dst + N * i, N) =
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src[i].v.template segment<N>(N0);
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}
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}
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// Calls Take1stOrderPart for every parameter block.
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//
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// Example:
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// If one having three parameter blocks with dimensions (3, 2, 4), the call
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// Take1stOrderParts<integer_sequence<3, 2, 4>::Apply(num_outputs,
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// output,
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// jacobians);
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// will result in the following calls to Take1stOrderPart:
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// if (jacobians[0]) {
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// Take1stOrderPart<0, 3>(num_outputs, output, jacobians[0]);
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// }
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// if (jacobians[1]) {
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// Take1stOrderPart<3, 2>(num_outputs, output, jacobians[1]);
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// }
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// if (jacobians[2]) {
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// Take1stOrderPart<5, 4>(num_outputs, output, jacobians[2]);
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// }
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template <typename Seq, int ParameterIdx = 0, int Offset = 0>
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struct Take1stOrderParts;
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template <int N, int... Ns, int ParameterIdx, int Offset>
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struct Take1stOrderParts<std::integer_sequence<int, N, Ns...>,
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ParameterIdx,
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Offset> {
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template <typename JetT, typename T>
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inline static void Apply(int num_outputs, JetT* output, T** jacobians) {
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if (jacobians[ParameterIdx]) {
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Take1stOrderPart<Offset, N>(num_outputs, output, jacobians[ParameterIdx]);
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}
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Take1stOrderParts<std::integer_sequence<int, Ns...>,
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ParameterIdx + 1,
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Offset + N>::Apply(num_outputs, output, jacobians);
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}
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};
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// End of 'recursion'. Nothing more to do.
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template <int ParameterIdx, int Offset>
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struct Take1stOrderParts<std::integer_sequence<int>, ParameterIdx, Offset> {
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template <typename T, typename JetT>
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static void Apply(int /* NOT USED*/,
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JetT* /* NOT USED*/,
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T** /* NOT USED */) {}
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};
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template <int kNumResiduals,
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typename ParameterDims,
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typename Functor,
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typename T>
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inline bool AutoDifferentiate(const Functor& functor,
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T const* const* parameters,
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int dynamic_num_outputs,
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T* function_value,
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T** jacobians) {
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using JetT = Jet<T, ParameterDims::kNumParameters>;
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using Parameters = typename ParameterDims::Parameters;
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if (kNumResiduals != DYNAMIC) {
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DCHECK_EQ(kNumResiduals, dynamic_num_outputs);
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}
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ArraySelector<JetT,
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ParameterDims::kNumParameters,
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CERES_AUTODIFF_MAX_PARAMETERS_ON_STACK>
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parameters_as_jets(ParameterDims::kNumParameters);
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// Pointers to the beginning of each parameter block
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std::array<JetT*, ParameterDims::kNumParameterBlocks> unpacked_parameters =
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ParameterDims::GetUnpackedParameters(parameters_as_jets.data());
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// If the number of residuals is fixed, we use the template argument as the
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// number of outputs. Otherwise we use the num_outputs parameter. Note: The
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// ?-operator here is compile-time evaluated, therefore num_outputs is also
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// a compile-time constant for functors with fixed residuals.
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const int num_outputs =
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kNumResiduals == DYNAMIC ? dynamic_num_outputs : kNumResiduals;
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DCHECK_GT(num_outputs, 0);
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ArraySelector<JetT, kNumResiduals, CERES_AUTODIFF_MAX_RESIDUALS_ON_STACK>
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residuals_as_jets(num_outputs);
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// Invalidate the output Jets, so that we can detect if the user
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// did not assign values to all of them.
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for (int i = 0; i < num_outputs; ++i) {
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residuals_as_jets[i].a = kImpossibleValue;
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residuals_as_jets[i].v.setConstant(kImpossibleValue);
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}
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Make1stOrderPerturbations<Parameters>::Apply(parameters,
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parameters_as_jets.data());
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if (!VariadicEvaluate<ParameterDims>(
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functor, unpacked_parameters.data(), residuals_as_jets.data())) {
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return false;
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}
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Take0thOrderPart(num_outputs, residuals_as_jets.data(), function_value);
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Take1stOrderParts<Parameters>::Apply(
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num_outputs, residuals_as_jets.data(), jacobians);
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return true;
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}
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} // namespace ceres::internal
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#endif // CERES_PUBLIC_INTERNAL_AUTODIFF_H_
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