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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2022 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
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// 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.
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// used to endorse or promote products derived from this software without
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// Author: sameeragarwal@google.com (Sameer Agarwal)
#include "ceres/autodiff_manifold.h"
#include <cmath>
#include "ceres/manifold.h"
#include "ceres/manifold_test_utils.h"
#include "ceres/rotation.h"
#include "gtest/gtest.h"
namespace ceres::internal {
namespace {
constexpr int kNumTrials = 1000;
constexpr double kTolerance = 1e-9;
Vector RandomQuaternion() {
Vector x = Vector::Random(4);
x.normalize();
return x;
}
} // namespace
struct EuclideanFunctor {
template <typename T>
bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
for (int i = 0; i < 3; ++i) {
x_plus_delta[i] = x[i] + delta[i];
}
return true;
}
template <typename T>
bool Minus(const T* y, const T* x, T* y_minus_x) const {
for (int i = 0; i < 3; ++i) {
y_minus_x[i] = y[i] - x[i];
}
return true;
}
};
TEST(AutoDiffLManifoldTest, EuclideanManifold) {
AutoDiffManifold<EuclideanFunctor, 3, 3> manifold;
EXPECT_EQ(manifold.AmbientSize(), 3);
EXPECT_EQ(manifold.TangentSize(), 3);
for (int trial = 0; trial < kNumTrials; ++trial) {
const Vector x = Vector::Random(manifold.AmbientSize());
const Vector y = Vector::Random(manifold.AmbientSize());
Vector delta = Vector::Random(manifold.TangentSize());
Vector x_plus_delta = Vector::Zero(manifold.AmbientSize());
manifold.Plus(x.data(), delta.data(), x_plus_delta.data());
EXPECT_NEAR((x_plus_delta - x - delta).norm() / (x + delta).norm(),
0.0,
kTolerance);
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(manifold, x, delta, y, kTolerance);
}
}
struct ScaledFunctor {
explicit ScaledFunctor(const double s) : s(s) {}
template <typename T>
bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
for (int i = 0; i < 3; ++i) {
x_plus_delta[i] = x[i] + s * delta[i];
}
return true;
}
template <typename T>
bool Minus(const T* y, const T* x, T* y_minus_x) const {
for (int i = 0; i < 3; ++i) {
y_minus_x[i] = (y[i] - x[i]) / s;
}
return true;
}
const double s;
};
TEST(AutoDiffManifoldTest, ScaledManifold) {
constexpr double kScale = 1.2342;
AutoDiffManifold<ScaledFunctor, 3, 3> manifold(new ScaledFunctor(kScale));
EXPECT_EQ(manifold.AmbientSize(), 3);
EXPECT_EQ(manifold.TangentSize(), 3);
for (int trial = 0; trial < kNumTrials; ++trial) {
const Vector x = Vector::Random(manifold.AmbientSize());
const Vector y = Vector::Random(manifold.AmbientSize());
Vector delta = Vector::Random(manifold.TangentSize());
Vector x_plus_delta = Vector::Zero(manifold.AmbientSize());
manifold.Plus(x.data(), delta.data(), x_plus_delta.data());
EXPECT_NEAR((x_plus_delta - x - delta * kScale).norm() /
(x + delta * kScale).norm(),
0.0,
kTolerance);
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(manifold, x, delta, y, kTolerance);
}
}
// Templated functor that implements the Plus and Minus operations on the
// Quaternion manifold.
struct QuaternionFunctor {
template <typename T>
bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
const T squared_norm_delta =
delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2];
T q_delta[4];
if (squared_norm_delta > T(0.0)) {
T norm_delta = sqrt(squared_norm_delta);
const T sin_delta_by_delta = sin(norm_delta) / norm_delta;
q_delta[0] = cos(norm_delta);
q_delta[1] = sin_delta_by_delta * delta[0];
q_delta[2] = sin_delta_by_delta * delta[1];
q_delta[3] = sin_delta_by_delta * delta[2];
} else {
// We do not just use q_delta = [1,0,0,0] here because that is a
// constant and when used for automatic differentiation will
// lead to a zero derivative. Instead we take a first order
// approximation and evaluate it at zero.
q_delta[0] = T(1.0);
q_delta[1] = delta[0];
q_delta[2] = delta[1];
q_delta[3] = delta[2];
}
QuaternionProduct(q_delta, x, x_plus_delta);
return true;
}
template <typename T>
bool Minus(const T* y, const T* x, T* y_minus_x) const {
T minus_x[4] = {x[0], -x[1], -x[2], -x[3]};
T ambient_y_minus_x[4];
QuaternionProduct(y, minus_x, ambient_y_minus_x);
T u_norm = sqrt(ambient_y_minus_x[1] * ambient_y_minus_x[1] +
ambient_y_minus_x[2] * ambient_y_minus_x[2] +
ambient_y_minus_x[3] * ambient_y_minus_x[3]);
if (u_norm > 0.0) {
T theta = atan2(u_norm, ambient_y_minus_x[0]);
y_minus_x[0] = theta * ambient_y_minus_x[1] / u_norm;
y_minus_x[1] = theta * ambient_y_minus_x[2] / u_norm;
y_minus_x[2] = theta * ambient_y_minus_x[3] / u_norm;
} else {
// We do not use [0,0,0] here because even though the value part is
// a constant, the derivative part is not.
y_minus_x[0] = ambient_y_minus_x[1];
y_minus_x[1] = ambient_y_minus_x[2];
y_minus_x[2] = ambient_y_minus_x[3];
}
return true;
}
};
TEST(AutoDiffManifoldTest, QuaternionPlusPiBy2) {
AutoDiffManifold<QuaternionFunctor, 4, 3> manifold;
Vector x = Vector::Zero(4);
x[0] = 1.0;
for (int i = 0; i < 3; ++i) {
Vector delta = Vector::Zero(3);
delta[i] = M_PI / 2;
Vector x_plus_delta = Vector::Zero(4);
EXPECT_TRUE(manifold.Plus(x.data(), delta.data(), x_plus_delta.data()));
// Expect that the element corresponding to pi/2 is +/- 1. All other
// elements should be zero.
for (int j = 0; j < 4; ++j) {
if (i == (j - 1)) {
EXPECT_LT(std::abs(x_plus_delta[j]) - 1,
std::numeric_limits<double>::epsilon())
<< "\ndelta = " << delta.transpose()
<< "\nx_plus_delta = " << x_plus_delta.transpose()
<< "\n expected the " << j
<< "th element of x_plus_delta to be +/- 1.";
} else {
EXPECT_LT(std::abs(x_plus_delta[j]),
std::numeric_limits<double>::epsilon())
<< "\ndelta = " << delta.transpose()
<< "\nx_plus_delta = " << x_plus_delta.transpose()
<< "\n expected the " << j << "th element of x_plus_delta to be 0.";
}
}
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(
manifold, x, delta, x_plus_delta, kTolerance);
}
}
// Compute the expected value of Quaternion::Plus via functions in rotation.h
// and compares it to the one computed by Quaternion::Plus.
MATCHER_P2(QuaternionPlusIsCorrectAt, x, delta, "") {
// This multiplication by 2 is needed because AngleAxisToQuaternion uses
// |delta|/2 as the angle of rotation where as in the implementation of
// Quaternion for historical reasons we use |delta|.
const Vector two_delta = delta * 2;
Vector delta_q(4);
AngleAxisToQuaternion(two_delta.data(), delta_q.data());
Vector expected(4);
QuaternionProduct(delta_q.data(), x.data(), expected.data());
Vector actual(4);
EXPECT_TRUE(arg.Plus(x.data(), delta.data(), actual.data()));
const double n = (actual - expected).norm();
const double d = expected.norm();
const double diffnorm = n / d;
if (diffnorm > kTolerance) {
*result_listener << "\nx: " << x.transpose()
<< "\ndelta: " << delta.transpose()
<< "\nexpected: " << expected.transpose()
<< "\nactual: " << actual.transpose()
<< "\ndiff: " << (expected - actual).transpose()
<< "\ndiffnorm : " << diffnorm;
return false;
}
return true;
}
TEST(AutoDiffManifoldTest, QuaternionGenericDelta) {
AutoDiffManifold<QuaternionFunctor, 4, 3> manifold;
for (int trial = 0; trial < kNumTrials; ++trial) {
const Vector x = RandomQuaternion();
const Vector y = RandomQuaternion();
Vector delta = Vector::Random(3);
EXPECT_THAT(manifold, QuaternionPlusIsCorrectAt(x, delta));
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(manifold, x, delta, y, kTolerance);
}
}
TEST(AutoDiffManifoldTest, QuaternionSmallDelta) {
AutoDiffManifold<QuaternionFunctor, 4, 3> manifold;
for (int trial = 0; trial < kNumTrials; ++trial) {
const Vector x = RandomQuaternion();
const Vector y = RandomQuaternion();
Vector delta = Vector::Random(3);
delta.normalize();
delta *= 1e-6;
EXPECT_THAT(manifold, QuaternionPlusIsCorrectAt(x, delta));
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(manifold, x, delta, y, kTolerance);
}
}
TEST(AutoDiffManifold, QuaternionDeltaJustBelowPi) {
AutoDiffManifold<QuaternionFunctor, 4, 3> manifold;
for (int trial = 0; trial < kNumTrials; ++trial) {
const Vector x = RandomQuaternion();
const Vector y = RandomQuaternion();
Vector delta = Vector::Random(3);
delta.normalize();
delta *= (M_PI - 1e-6);
EXPECT_THAT(manifold, QuaternionPlusIsCorrectAt(x, delta));
EXPECT_THAT_MANIFOLD_INVARIANTS_HOLD(manifold, x, delta, y, kTolerance);
}
}
} // namespace ceres::internal