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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2023 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:
//
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// this list of conditions and the following disclaimer.
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// 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.
//
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//
// Author: sameeragarwal@google.com (Sameer Agarwal)
#ifndef CERES_PUBLIC_MANIFOLD_H_
#define CERES_PUBLIC_MANIFOLD_H_
#include <Eigen/Core>
#include <algorithm>
#include <array>
#include <memory>
#include <utility>
#include <vector>
#include "absl/log/check.h"
#include "ceres/internal/disable_warnings.h"
#include "ceres/internal/export.h"
#include "ceres/types.h"
namespace ceres {
// In sensor fusion problems, often we have to model quantities that live in
// spaces known as Manifolds, for example the rotation/orientation of a sensor
// that is represented by a quaternion.
//
// Manifolds are spaces which locally look like Euclidean spaces. More
// precisely, at each point on the manifold there is a linear space that is
// tangent to the manifold. It has dimension equal to the intrinsic dimension of
// the manifold itself, which is less than or equal to the ambient space in
// which the manifold is embedded.
//
// For example, the tangent space to a point on a sphere in three dimensions is
// the two dimensional plane that is tangent to the sphere at that point. There
// are two reasons tangent spaces are interesting:
//
// 1. They are Eucliean spaces so the usual vector space operations apply there,
// which makes numerical operations easy.
// 2. Movement in the tangent space translate into movements along the manifold.
// Movements perpendicular to the tangent space do not translate into
// movements on the manifold.
//
// Returning to our sphere example, moving in the 2 dimensional plane
// tangent to the sphere and projecting back onto the sphere will move you away
// from the point you started from but moving along the normal at the same point
// and the projecting back onto the sphere brings you back to the point.
//
// The Manifold interface defines two operations (and their derivatives)
// involving the tangent space, allowing filtering and optimization to be
// performed on said manifold:
//
// 1. x_plus_delta = Plus(x, delta)
// 2. delta = Minus(x_plus_delta, x)
//
// "Plus" computes the result of moving along delta in the tangent space at x,
// and then projecting back onto the manifold that x belongs to. In Differential
// Geometry this is known as a "Retraction". It is a generalization of vector
// addition in Euclidean spaces.
//
// Given two points on the manifold, "Minus" computes the change delta to x in
// the tangent space at x, that will take it to x_plus_delta.
//
// Let us now consider two examples.
//
// The Euclidean space R^n is the simplest example of a manifold. It has
// dimension n (and so does its tangent space) and Plus and Minus are the
// familiar vector sum and difference operations.
//
// Plus(x, delta) = x + delta = y,
// Minus(y, x) = y - x = delta.
//
// A more interesting case is SO(3), the special orthogonal group in three
// dimensions - the space of 3x3 rotation matrices. SO(3) is a three dimensional
// manifold embedded in R^9 or R^(3x3). So points on SO(3) are represented using
// 9 dimensional vectors or 3x3 matrices, and points in its tangent spaces are
// represented by 3 dimensional vectors.
//
// Defining Plus and Minus are defined in terms of the matrix Exp and Log
// operations as follows:
//
// Let Exp(p, q, r) = [cos(theta) + cp^2, -sr + cpq , sq + cpr ]
// [sr + cpq , cos(theta) + cq^2, -sp + cqr ]
// [-sq + cpr , sp + cqr , cos(theta) + cr^2]
//
// where: theta = sqrt(p^2 + q^2 + r^2)
// s = sinc(theta)
// c = (1 - cos(theta))/theta^2
//
// and Log(x) = 1/(2 sinc(theta))[x_32 - x_23, x_13 - x_31, x_21 - x_12]
//
// where: theta = acos((Trace(x) - 1)/2)
//
// Then,
//
// Plus(x, delta) = x Exp(delta)
// Minus(y, x) = Log(x^T y)
//
// For Plus and Minus to be mathematically consistent, the following identities
// must be satisfied at all points x on the manifold:
//
// 1. Plus(x, 0) = x.
// 2. For all y, Plus(x, Minus(y, x)) = y.
// 3. For all delta, Minus(Plus(x, delta), x) = delta.
// 4. For all delta_1, delta_2
// |Minus(Plus(x, delta_1), Plus(x, delta_2)) <= |delta_1 - delta_2|
//
// Briefly:
// (1) Ensures that the tangent space is "centered" at x, and the zero vector is
// the identity element.
// (2) Ensures that any y can be reached from x.
// (3) Ensures that Plus is an injective (one-to-one) map.
// (4) Allows us to define a metric on the manifold.
//
// Additionally we require that Plus and Minus be sufficiently smooth. In
// particular they need to be differentiable everywhere on the manifold.
//
// For more details, please see
//
// "Integrating Generic Sensor Fusion Algorithms with Sound State
// Representations through Encapsulation of Manifolds"
// By C. Hertzberg, R. Wagner, U. Frese and L. Schroder
// https://arxiv.org/pdf/1107.1119.pdf
class CERES_EXPORT Manifold {
public:
virtual ~Manifold();
// Dimension of the ambient space in which the manifold is embedded.
virtual int AmbientSize() const = 0;
// Dimension of the manifold/tangent space.
virtual int TangentSize() const = 0;
// x_plus_delta = Plus(x, delta),
//
// A generalization of vector addition in Euclidean space, Plus computes the
// result of moving along delta in the tangent space at x, and then projecting
// back onto the manifold that x belongs to.
//
// x and x_plus_delta are AmbientSize() vectors.
// delta is a TangentSize() vector.
//
// Return value indicates if the operation was successful or not.
virtual bool Plus(const double* x,
const double* delta,
double* x_plus_delta) const = 0;
// Compute the derivative of Plus(x, delta) w.r.t delta at delta = 0, i.e.
//
// (D_2 Plus)(x, 0)
//
// jacobian is a row-major AmbientSize() x TangentSize() matrix.
//
// Return value indicates whether the operation was successful or not.
virtual bool PlusJacobian(const double* x, double* jacobian) const = 0;
// tangent_matrix = ambient_matrix * (D_2 Plus)(x, 0)
//
// ambient_matrix is a row-major num_rows x AmbientSize() matrix.
// tangent_matrix is a row-major num_rows x TangentSize() matrix.
//
// Return value indicates whether the operation was successful or not.
//
// This function is only used by the GradientProblemSolver, where the
// dimension of the parameter block can be large and it may be more efficient
// to compute this product directly rather than first evaluating the Jacobian
// into a matrix and then doing a matrix vector product.
//
// Because this is not an often used function, we provide a default
// implementation for convenience. If performance becomes an issue then the
// user should consider implementing a specialization.
virtual bool RightMultiplyByPlusJacobian(const double* x,
const int num_rows,
const double* ambient_matrix,
double* tangent_matrix) const;
// y_minus_x = Minus(y, x)
//
// Given two points on the manifold, Minus computes the change to x in the
// tangent space at x, that will take it to y.
//
// x and y are AmbientSize() vectors.
// y_minus_x is a TangentSize() vector.
//
// Return value indicates if the operation was successful or not.
virtual bool Minus(const double* y,
const double* x,
double* y_minus_x) const = 0;
// Compute the derivative of Minus(y, x) w.r.t y at y = x, i.e.
//
// (D_1 Minus) (x, x)
//
// Jacobian is a row-major TangentSize() x AmbientSize() matrix.
//
// Return value indicates whether the operation was successful or not.
virtual bool MinusJacobian(const double* x, double* jacobian) const = 0;
};
// The Euclidean manifold is another name for the ordinary vector space R^size,
// where the plus and minus operations are the usual vector addition and
// subtraction:
// Plus(x, delta) = x + delta
// Minus(y, x) = y - x.
//
// The class works with dynamic and static ambient space dimensions. If the
// ambient space dimensions is know at compile time use
//
// EuclideanManifold<3> manifold;
//
// If the ambient space dimensions is not known at compile time the template
// parameter needs to be set to ceres::DYNAMIC and the actual dimension needs
// to be provided as a constructor argument:
//
// EuclideanManifold<ceres::DYNAMIC> manifold(ambient_dim);
template <int Size>
class EuclideanManifold final : public Manifold {
public:
static_assert(Size == ceres::DYNAMIC || Size >= 0,
"The size of the manifold needs to be non-negative.");
static_assert(ceres::DYNAMIC == Eigen::Dynamic,
"ceres::DYNAMIC needs to be the same as Eigen::Dynamic.");
EuclideanManifold() : size_{Size} {
static_assert(
Size != ceres::DYNAMIC,
"The size is set to dynamic. Please call the constructor with a size.");
}
explicit EuclideanManifold(int size) : size_(size) {
if (Size != ceres::DYNAMIC) {
CHECK_EQ(Size, size)
<< "Specified size by template parameter differs from the supplied "
"one.";
} else {
CHECK_GE(size_, 0)
<< "The size of the manifold needs to be non-negative.";
}
}
int AmbientSize() const override { return size_; }
int TangentSize() const override { return size_; }
bool Plus(const double* x_ptr,
const double* delta_ptr,
double* x_plus_delta_ptr) const override {
Eigen::Map<const AmbientVector> x(x_ptr, size_);
Eigen::Map<const AmbientVector> delta(delta_ptr, size_);
Eigen::Map<AmbientVector> x_plus_delta(x_plus_delta_ptr, size_);
x_plus_delta = x + delta;
return true;
}
bool PlusJacobian(const double* x_ptr, double* jacobian_ptr) const override {
Eigen::Map<MatrixJacobian> jacobian(jacobian_ptr, size_, size_);
jacobian.setIdentity();
return true;
}
bool RightMultiplyByPlusJacobian(const double* x,
const int num_rows,
const double* ambient_matrix,
double* tangent_matrix) const override {
std::copy_n(ambient_matrix, num_rows * size_, tangent_matrix);
return true;
}
bool Minus(const double* y_ptr,
const double* x_ptr,
double* y_minus_x_ptr) const override {
Eigen::Map<const AmbientVector> x(x_ptr, size_);
Eigen::Map<const AmbientVector> y(y_ptr, size_);
Eigen::Map<AmbientVector> y_minus_x(y_minus_x_ptr, size_);
y_minus_x = y - x;
return true;
}
bool MinusJacobian(const double* x_ptr, double* jacobian_ptr) const override {
Eigen::Map<MatrixJacobian> jacobian(jacobian_ptr, size_, size_);
jacobian.setIdentity();
return true;
}
private:
static constexpr bool IsDynamic = (Size == ceres::DYNAMIC);
using AmbientVector = Eigen::Matrix<double, Size, 1>;
using MatrixJacobian = Eigen::Matrix<double, Size, Size, Eigen::RowMajor>;
int size_{};
};
// Hold a subset of the parameters inside a parameter block constant.
class CERES_EXPORT SubsetManifold final : public Manifold {
public:
SubsetManifold(int size, const std::vector<int>& constant_parameters);
int AmbientSize() const override;
int TangentSize() const override;
bool Plus(const double* x,
const double* delta,
double* x_plus_delta) const override;
bool PlusJacobian(const double* x, double* jacobian) const override;
bool RightMultiplyByPlusJacobian(const double* x,
const int num_rows,
const double* ambient_matrix,
double* tangent_matrix) const override;
bool Minus(const double* y,
const double* x,
double* y_minus_x) const override;
bool MinusJacobian(const double* x, double* jacobian) const override;
private:
const int tangent_size_ = 0;
std::vector<bool> constancy_mask_;
};
// Implements the manifold for a Hamilton quaternion as defined in
// https://en.wikipedia.org/wiki/Quaternion. Quaternions are represented as
// unit norm 4-vectors, i.e.
//
// q = [q0; q1; q2; q3], |q| = 1
//
// is the ambient space representation.
//
// q0 scalar part.
// q1 coefficient of i.
// q2 coefficient of j.
// q3 coefficient of k.
//
// where: i*i = j*j = k*k = -1 and i*j = k, j*k = i, k*i = j.
//
// The tangent space is R^3, which relates to the ambient space through the
// Plus and Minus operations defined as:
//
// Plus(x, delta) = [cos(|delta|); sin(|delta|) * delta / |delta|] * x
// Minus(y, x) = to_delta(y * x^{-1})
//
// where "*" is the quaternion product and because q is a unit quaternion
// (|q|=1), q^-1 = [q0; -q1; -q2; -q3]
//
// and to_delta( [q0; u_{3x1}] ) = u / |u| * atan2(|u|, q0)
class CERES_EXPORT QuaternionManifold final : public Manifold {
public:
int AmbientSize() const override { return 4; }
int TangentSize() const override { return 3; }
bool Plus(const double* x,
const double* delta,
double* x_plus_delta) const override;
bool PlusJacobian(const double* x, double* jacobian) const override;
bool Minus(const double* y,
const double* x,
double* y_minus_x) const override;
bool MinusJacobian(const double* x, double* jacobian) const override;
};
// Implements the quaternion manifold for Eigen's representation of the
// Hamilton quaternion. Geometrically it is exactly the same as the
// QuaternionManifold defined above. However, Eigen uses a different internal
// memory layout for the elements of the quaternion than what is commonly
// used. It stores the quaternion in memory as [q1, q2, q3, q0] or
// [x, y, z, w] where the real (scalar) part is last.
//
// Since Ceres operates on parameter blocks which are raw double pointers this
// difference is important and requires a different manifold.
class CERES_EXPORT EigenQuaternionManifold final : public Manifold {
public:
int AmbientSize() const override { return 4; }
int TangentSize() const override { return 3; }
bool Plus(const double* x,
const double* delta,
double* x_plus_delta) const override;
bool PlusJacobian(const double* x, double* jacobian) const override;
bool Minus(const double* y,
const double* x,
double* y_minus_x) const override;
bool MinusJacobian(const double* x, double* jacobian) const override;
};
} // namespace ceres
// clang-format off
#include "ceres/internal/reenable_warnings.h"
// clang-format on
#endif // CERES_PUBLIC_MANIFOLD_H_