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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 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:
//
// * 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
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// 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)
// sameeragarwal@google.com (Sameer Agarwal)
//
// Templated functions for manipulating rotations. The templated
// functions are useful when implementing functors for automatic
// differentiation.
//
// In the following, the Quaternions are laid out as 4-vectors, thus:
//
// q[0] scalar part.
// q[1] coefficient of i.
// q[2] coefficient of j.
// q[3] coefficient of k.
//
// where: i*i = j*j = k*k = -1 and i*j = k, j*k = i, k*i = j.
#ifndef CERES_PUBLIC_ROTATION_H_
#define CERES_PUBLIC_ROTATION_H_
#include <algorithm>
#include <cmath>
#include <limits>
namespace ceres {
// Trivial wrapper to index linear arrays as matrices, given a fixed
// column and row stride. When an array "T* array" is wrapped by a
//
// (const) MatrixAdapter<T, row_stride, col_stride> M"
//
// the expression M(i, j) is equivalent to
//
// arrary[i * row_stride + j * col_stride]
//
// Conversion functions to and from rotation matrices accept
// MatrixAdapters to permit using row-major and column-major layouts,
// and rotation matrices embedded in larger matrices (such as a 3x4
// projection matrix).
template <typename T, int row_stride, int col_stride>
struct MatrixAdapter;
// Convenience functions to create a MatrixAdapter that treats the
// array pointed to by "pointer" as a 3x3 (contiguous) column-major or
// row-major matrix.
template <typename T>
MatrixAdapter<T, 1, 3> ColumnMajorAdapter3x3(T* pointer);
template <typename T>
MatrixAdapter<T, 3, 1> RowMajorAdapter3x3(T* pointer);
// Convert a value in combined axis-angle representation to a quaternion.
// The value angle_axis is a triple whose norm is an angle in radians,
// and whose direction is aligned with the axis of rotation,
// and quaternion is a 4-tuple that will contain the resulting quaternion.
// The implementation may be used with auto-differentiation up to the first
// derivative, higher derivatives may have unexpected results near the origin.
template<typename T>
void AngleAxisToQuaternion(const T* angle_axis, T* quaternion);
// Convert a quaternion to the equivalent combined axis-angle representation.
// The value quaternion must be a unit quaternion - it is not normalized first,
// and angle_axis will be filled with a value whose norm is the angle of
// rotation in radians, and whose direction is the axis of rotation.
// The implemention may be used with auto-differentiation up to the first
// derivative, higher derivatives may have unexpected results near the origin.
template<typename T>
void QuaternionToAngleAxis(const T* quaternion, T* angle_axis);
// Conversions between 3x3 rotation matrix (in column major order) and
// quaternion rotation representations. Templated for use with
// autodifferentiation.
template <typename T>
void RotationMatrixToQuaternion(const T* R, T* quaternion);
template <typename T, int row_stride, int col_stride>
void RotationMatrixToQuaternion(
const MatrixAdapter<const T, row_stride, col_stride>& R,
T* quaternion);
// Conversions between 3x3 rotation matrix (in column major order) and
// axis-angle rotation representations. Templated for use with
// autodifferentiation.
template <typename T>
void RotationMatrixToAngleAxis(const T* R, T* angle_axis);
template <typename T, int row_stride, int col_stride>
void RotationMatrixToAngleAxis(
const MatrixAdapter<const T, row_stride, col_stride>& R,
T* angle_axis);
template <typename T>
void AngleAxisToRotationMatrix(const T* angle_axis, T* R);
template <typename T, int row_stride, int col_stride>
void AngleAxisToRotationMatrix(
const T* angle_axis,
const MatrixAdapter<T, row_stride, col_stride>& R);
// Conversions between 3x3 rotation matrix (in row major order) and
// Euler angle (in degrees) rotation representations.
//
// The {pitch,roll,yaw} Euler angles are rotations around the {x,y,z}
// axes, respectively. They are applied in that same order, so the
// total rotation R is Rz * Ry * Rx.
template <typename T>
void EulerAnglesToRotationMatrix(const T* euler, int row_stride, T* R);
template <typename T, int row_stride, int col_stride>
void EulerAnglesToRotationMatrix(
const T* euler,
const MatrixAdapter<T, row_stride, col_stride>& R);
// Convert a 4-vector to a 3x3 scaled rotation matrix.
//
// The choice of rotation is such that the quaternion [1 0 0 0] goes to an
// identity matrix and for small a, b, c the quaternion [1 a b c] goes to
// the matrix
//
// [ 0 -c b ]
// I + 2 [ c 0 -a ] + higher order terms
// [ -b a 0 ]
//
// which corresponds to a Rodrigues approximation, the last matrix being
// the cross-product matrix of [a b c]. Together with the property that
// R(q1 * q2) = R(q1) * R(q2) this uniquely defines the mapping from q to R.
//
// No normalization of the quaternion is performed, i.e.
// R = ||q||^2 * Q, where Q is an orthonormal matrix
// such that det(Q) = 1 and Q*Q' = I
//
// WARNING: The rotation matrix is ROW MAJOR
template <typename T> inline
void QuaternionToScaledRotation(const T q[4], T R[3 * 3]);
template <typename T, int row_stride, int col_stride> inline
void QuaternionToScaledRotation(
const T q[4],
const MatrixAdapter<T, row_stride, col_stride>& R);
// Same as above except that the rotation matrix is normalized by the
// Frobenius norm, so that R * R' = I (and det(R) = 1).
//
// WARNING: The rotation matrix is ROW MAJOR
template <typename T> inline
void QuaternionToRotation(const T q[4], T R[3 * 3]);
template <typename T, int row_stride, int col_stride> inline
void QuaternionToRotation(
const T q[4],
const MatrixAdapter<T, row_stride, col_stride>& R);
// Rotates a point pt by a quaternion q:
//
// result = R(q) * pt
//
// Assumes the quaternion is unit norm. This assumption allows us to
// write the transform as (something)*pt + pt, as is clear from the
// formula below. If you pass in a quaternion with |q|^2 = 2 then you
// WILL NOT get back 2 times the result you get for a unit quaternion.
template <typename T> inline
void UnitQuaternionRotatePoint(const T q[4], const T pt[3], T result[3]);
// With this function you do not need to assume that q has unit norm.
// It does assume that the norm is non-zero.
template <typename T> inline
void QuaternionRotatePoint(const T q[4], const T pt[3], T result[3]);
// zw = z * w, where * is the Quaternion product between 4 vectors.
template<typename T> inline
void QuaternionProduct(const T z[4], const T w[4], T zw[4]);
// xy = x cross y;
template<typename T> inline
void CrossProduct(const T x[3], const T y[3], T x_cross_y[3]);
template<typename T> inline
T DotProduct(const T x[3], const T y[3]);
// y = R(angle_axis) * x;
template<typename T> inline
void AngleAxisRotatePoint(const T angle_axis[3], const T pt[3], T result[3]);
// --- IMPLEMENTATION
template<typename T, int row_stride, int col_stride>
struct MatrixAdapter {
T* pointer_;
explicit MatrixAdapter(T* pointer)
: pointer_(pointer)
{}
T& operator()(int r, int c) const {
return pointer_[r * row_stride + c * col_stride];
}
};
template <typename T>
MatrixAdapter<T, 1, 3> ColumnMajorAdapter3x3(T* pointer) {
return MatrixAdapter<T, 1, 3>(pointer);
}
template <typename T>
MatrixAdapter<T, 3, 1> RowMajorAdapter3x3(T* pointer) {
return MatrixAdapter<T, 3, 1>(pointer);
}
template<typename T>
inline void AngleAxisToQuaternion(const T* angle_axis, T* quaternion) {
const T& a0 = angle_axis[0];
const T& a1 = angle_axis[1];
const T& a2 = angle_axis[2];
const T theta_squared = a0 * a0 + a1 * a1 + a2 * a2;
// For points not at the origin, the full conversion is numerically stable.
if (theta_squared > T(0.0)) {
const T theta = sqrt(theta_squared);
const T half_theta = theta * T(0.5);
const T k = sin(half_theta) / theta;
quaternion[0] = cos(half_theta);
quaternion[1] = a0 * k;
quaternion[2] = a1 * k;
quaternion[3] = a2 * k;
} else {
// At the origin, sqrt() will produce NaN in the derivative since
// the argument is zero. By approximating with a Taylor series,
// and truncating at one term, the value and first derivatives will be
// computed correctly when Jets are used.
const T k(0.5);
quaternion[0] = T(1.0);
quaternion[1] = a0 * k;
quaternion[2] = a1 * k;
quaternion[3] = a2 * k;
}
}
template<typename T>
inline void QuaternionToAngleAxis(const T* quaternion, T* angle_axis) {
const T& q1 = quaternion[1];
const T& q2 = quaternion[2];
const T& q3 = quaternion[3];
const T sin_squared_theta = q1 * q1 + q2 * q2 + q3 * q3;
// For quaternions representing non-zero rotation, the conversion
// is numerically stable.
if (sin_squared_theta > T(0.0)) {
const T sin_theta = sqrt(sin_squared_theta);
const T& cos_theta = quaternion[0];
// If cos_theta is negative, theta is greater than pi/2, which
// means that angle for the angle_axis vector which is 2 * theta
// would be greater than pi.
//
// While this will result in the correct rotation, it does not
// result in a normalized angle-axis vector.
//
// In that case we observe that 2 * theta ~ 2 * theta - 2 * pi,
// which is equivalent saying
//
// theta - pi = atan(sin(theta - pi), cos(theta - pi))
// = atan(-sin(theta), -cos(theta))
//
const T two_theta =
T(2.0) * ((cos_theta < 0.0)
? atan2(-sin_theta, -cos_theta)
: atan2(sin_theta, cos_theta));
const T k = two_theta / sin_theta;
angle_axis[0] = q1 * k;
angle_axis[1] = q2 * k;
angle_axis[2] = q3 * k;
} else {
// For zero rotation, sqrt() will produce NaN in the derivative since
// the argument is zero. By approximating with a Taylor series,
// and truncating at one term, the value and first derivatives will be
// computed correctly when Jets are used.
const T k(2.0);
angle_axis[0] = q1 * k;
angle_axis[1] = q2 * k;
angle_axis[2] = q3 * k;
}
}
template <typename T>
void RotationMatrixToQuaternion(const T* R, T* angle_axis) {
RotationMatrixToQuaternion(ColumnMajorAdapter3x3(R), angle_axis);
}
// This algorithm comes from "Quaternion Calculus and Fast Animation",
// Ken Shoemake, 1987 SIGGRAPH course notes
template <typename T, int row_stride, int col_stride>
void RotationMatrixToQuaternion(
const MatrixAdapter<const T, row_stride, col_stride>& R,
T* quaternion) {
const T trace = R(0, 0) + R(1, 1) + R(2, 2);
if (trace >= 0.0) {
T t = sqrt(trace + T(1.0));
quaternion[0] = T(0.5) * t;
t = T(0.5) / t;
quaternion[1] = (R(2, 1) - R(1, 2)) * t;
quaternion[2] = (R(0, 2) - R(2, 0)) * t;
quaternion[3] = (R(1, 0) - R(0, 1)) * t;
} else {
int i = 0;
if (R(1, 1) > R(0, 0)) {
i = 1;
}
if (R(2, 2) > R(i, i)) {
i = 2;
}
const int j = (i + 1) % 3;
const int k = (j + 1) % 3;
T t = sqrt(R(i, i) - R(j, j) - R(k, k) + T(1.0));
quaternion[i + 1] = T(0.5) * t;
t = T(0.5) / t;
quaternion[0] = (R(k, j) - R(j, k)) * t;
quaternion[j + 1] = (R(j, i) + R(i, j)) * t;
quaternion[k + 1] = (R(k, i) + R(i, k)) * t;
}
}
// The conversion of a rotation matrix to the angle-axis form is
// numerically problematic when then rotation angle is close to zero
// or to Pi. The following implementation detects when these two cases
// occurs and deals with them by taking code paths that are guaranteed
// to not perform division by a small number.
template <typename T>
inline void RotationMatrixToAngleAxis(const T* R, T* angle_axis) {
RotationMatrixToAngleAxis(ColumnMajorAdapter3x3(R), angle_axis);
}
template <typename T, int row_stride, int col_stride>
void RotationMatrixToAngleAxis(
const MatrixAdapter<const T, row_stride, col_stride>& R,
T* angle_axis) {
T quaternion[4];
RotationMatrixToQuaternion(R, quaternion);
QuaternionToAngleAxis(quaternion, angle_axis);
return;
}
template <typename T>
inline void AngleAxisToRotationMatrix(const T* angle_axis, T* R) {
AngleAxisToRotationMatrix(angle_axis, ColumnMajorAdapter3x3(R));
}
template <typename T, int row_stride, int col_stride>
void AngleAxisToRotationMatrix(
const T* angle_axis,
const MatrixAdapter<T, row_stride, col_stride>& R) {
static const T kOne = T(1.0);
const T theta2 = DotProduct(angle_axis, angle_axis);
if (theta2 > T(std::numeric_limits<double>::epsilon())) {
// We want to be careful to only evaluate the square root if the
// norm of the angle_axis vector is greater than zero. Otherwise
// we get a division by zero.
const T theta = sqrt(theta2);
const T wx = angle_axis[0] / theta;
const T wy = angle_axis[1] / theta;
const T wz = angle_axis[2] / theta;
const T costheta = cos(theta);
const T sintheta = sin(theta);
R(0, 0) = costheta + wx*wx*(kOne - costheta);
R(1, 0) = wz*sintheta + wx*wy*(kOne - costheta);
R(2, 0) = -wy*sintheta + wx*wz*(kOne - costheta);
R(0, 1) = wx*wy*(kOne - costheta) - wz*sintheta;
R(1, 1) = costheta + wy*wy*(kOne - costheta);
R(2, 1) = wx*sintheta + wy*wz*(kOne - costheta);
R(0, 2) = wy*sintheta + wx*wz*(kOne - costheta);
R(1, 2) = -wx*sintheta + wy*wz*(kOne - costheta);
R(2, 2) = costheta + wz*wz*(kOne - costheta);
} else {
// Near zero, we switch to using the first order Taylor expansion.
R(0, 0) = kOne;
R(1, 0) = angle_axis[2];
R(2, 0) = -angle_axis[1];
R(0, 1) = -angle_axis[2];
R(1, 1) = kOne;
R(2, 1) = angle_axis[0];
R(0, 2) = angle_axis[1];
R(1, 2) = -angle_axis[0];
R(2, 2) = kOne;
}
}
template <typename T>
inline void EulerAnglesToRotationMatrix(const T* euler,
const int row_stride_parameter,
T* R) {
EulerAnglesToRotationMatrix(euler, RowMajorAdapter3x3(R));
}
template <typename T, int row_stride, int col_stride>
void EulerAnglesToRotationMatrix(
const T* euler,
const MatrixAdapter<T, row_stride, col_stride>& R) {
const double kPi = 3.14159265358979323846;
const T degrees_to_radians(kPi / 180.0);
const T pitch(euler[0] * degrees_to_radians);
const T roll(euler[1] * degrees_to_radians);
const T yaw(euler[2] * degrees_to_radians);
const T c1 = cos(yaw);
const T s1 = sin(yaw);
const T c2 = cos(roll);
const T s2 = sin(roll);
const T c3 = cos(pitch);
const T s3 = sin(pitch);
R(0, 0) = c1*c2;
R(0, 1) = -s1*c3 + c1*s2*s3;
R(0, 2) = s1*s3 + c1*s2*c3;
R(1, 0) = s1*c2;
R(1, 1) = c1*c3 + s1*s2*s3;
R(1, 2) = -c1*s3 + s1*s2*c3;
R(2, 0) = -s2;
R(2, 1) = c2*s3;
R(2, 2) = c2*c3;
}
template <typename T> inline
void QuaternionToScaledRotation(const T q[4], T R[3 * 3]) {
QuaternionToScaledRotation(q, RowMajorAdapter3x3(R));
}
template <typename T, int row_stride, int col_stride> inline
void QuaternionToScaledRotation(
const T q[4],
const MatrixAdapter<T, row_stride, col_stride>& R) {
// Make convenient names for elements of q.
T a = q[0];
T b = q[1];
T c = q[2];
T d = q[3];
// This is not to eliminate common sub-expression, but to
// make the lines shorter so that they fit in 80 columns!
T aa = a * a;
T ab = a * b;
T ac = a * c;
T ad = a * d;
T bb = b * b;
T bc = b * c;
T bd = b * d;
T cc = c * c;
T cd = c * d;
T dd = d * d;
R(0, 0) = aa + bb - cc - dd; R(0, 1) = T(2) * (bc - ad); R(0, 2) = T(2) * (ac + bd); // NOLINT
R(1, 0) = T(2) * (ad + bc); R(1, 1) = aa - bb + cc - dd; R(1, 2) = T(2) * (cd - ab); // NOLINT
R(2, 0) = T(2) * (bd - ac); R(2, 1) = T(2) * (ab + cd); R(2, 2) = aa - bb - cc + dd; // NOLINT
}
template <typename T> inline
void QuaternionToRotation(const T q[4], T R[3 * 3]) {
QuaternionToRotation(q, RowMajorAdapter3x3(R));
}
template <typename T, int row_stride, int col_stride> inline
void QuaternionToRotation(const T q[4],
const MatrixAdapter<T, row_stride, col_stride>& R) {
QuaternionToScaledRotation(q, R);
T normalizer = q[0]*q[0] + q[1]*q[1] + q[2]*q[2] + q[3]*q[3];
normalizer = T(1) / normalizer;
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 3; ++j) {
R(i, j) *= normalizer;
}
}
}
template <typename T> inline
void UnitQuaternionRotatePoint(const T q[4], const T pt[3], T result[3]) {
const T t2 = q[0] * q[1];
const T t3 = q[0] * q[2];
const T t4 = q[0] * q[3];
const T t5 = -q[1] * q[1];
const T t6 = q[1] * q[2];
const T t7 = q[1] * q[3];
const T t8 = -q[2] * q[2];
const T t9 = q[2] * q[3];
const T t1 = -q[3] * q[3];
result[0] = T(2) * ((t8 + t1) * pt[0] + (t6 - t4) * pt[1] + (t3 + t7) * pt[2]) + pt[0]; // NOLINT
result[1] = T(2) * ((t4 + t6) * pt[0] + (t5 + t1) * pt[1] + (t9 - t2) * pt[2]) + pt[1]; // NOLINT
result[2] = T(2) * ((t7 - t3) * pt[0] + (t2 + t9) * pt[1] + (t5 + t8) * pt[2]) + pt[2]; // NOLINT
}
template <typename T> inline
void QuaternionRotatePoint(const T q[4], const T pt[3], T result[3]) {
// 'scale' is 1 / norm(q).
const T scale = T(1) / sqrt(q[0] * q[0] +
q[1] * q[1] +
q[2] * q[2] +
q[3] * q[3]);
// Make unit-norm version of q.
const T unit[4] = {
scale * q[0],
scale * q[1],
scale * q[2],
scale * q[3],
};
UnitQuaternionRotatePoint(unit, pt, result);
}
template<typename T> inline
void QuaternionProduct(const T z[4], const T w[4], T zw[4]) {
zw[0] = z[0] * w[0] - z[1] * w[1] - z[2] * w[2] - z[3] * w[3];
zw[1] = z[0] * w[1] + z[1] * w[0] + z[2] * w[3] - z[3] * w[2];
zw[2] = z[0] * w[2] - z[1] * w[3] + z[2] * w[0] + z[3] * w[1];
zw[3] = z[0] * w[3] + z[1] * w[2] - z[2] * w[1] + z[3] * w[0];
}
// xy = x cross y;
template<typename T> inline
void CrossProduct(const T x[3], const T y[3], T x_cross_y[3]) {
x_cross_y[0] = x[1] * y[2] - x[2] * y[1];
x_cross_y[1] = x[2] * y[0] - x[0] * y[2];
x_cross_y[2] = x[0] * y[1] - x[1] * y[0];
}
template<typename T> inline
T DotProduct(const T x[3], const T y[3]) {
return (x[0] * y[0] + x[1] * y[1] + x[2] * y[2]);
}
template<typename T> inline
void AngleAxisRotatePoint(const T angle_axis[3], const T pt[3], T result[3]) {
const T theta2 = DotProduct(angle_axis, angle_axis);
if (theta2 > T(std::numeric_limits<double>::epsilon())) {
// Away from zero, use the rodriguez formula
//
// result = pt costheta +
// (w x pt) * sintheta +
// w (w . pt) (1 - costheta)
//
// We want to be careful to only evaluate the square root if the
// norm of the angle_axis vector is greater than zero. Otherwise
// we get a division by zero.
//
const T theta = sqrt(theta2);
const T costheta = cos(theta);
const T sintheta = sin(theta);
const T theta_inverse = T(1.0) / theta;
const T w[3] = { angle_axis[0] * theta_inverse,
angle_axis[1] * theta_inverse,
angle_axis[2] * theta_inverse };
// Explicitly inlined evaluation of the cross product for
// performance reasons.
const T w_cross_pt[3] = { w[1] * pt[2] - w[2] * pt[1],
w[2] * pt[0] - w[0] * pt[2],
w[0] * pt[1] - w[1] * pt[0] };
const T tmp =
(w[0] * pt[0] + w[1] * pt[1] + w[2] * pt[2]) * (T(1.0) - costheta);
result[0] = pt[0] * costheta + w_cross_pt[0] * sintheta + w[0] * tmp;
result[1] = pt[1] * costheta + w_cross_pt[1] * sintheta + w[1] * tmp;
result[2] = pt[2] * costheta + w_cross_pt[2] * sintheta + w[2] * tmp;
} else {
// Near zero, the first order Taylor approximation of the rotation
// matrix R corresponding to a vector w and angle w is
//
// R = I + hat(w) * sin(theta)
//
// But sintheta ~ theta and theta * w = angle_axis, which gives us
//
// R = I + hat(w)
//
// and actually performing multiplication with the point pt, gives us
// R * pt = pt + w x pt.
//
// Switching to the Taylor expansion near zero provides meaningful
// derivatives when evaluated using Jets.
//
// Explicitly inlined evaluation of the cross product for
// performance reasons.
const T w_cross_pt[3] = { angle_axis[1] * pt[2] - angle_axis[2] * pt[1],
angle_axis[2] * pt[0] - angle_axis[0] * pt[2],
angle_axis[0] * pt[1] - angle_axis[1] * pt[0] };
result[0] = pt[0] + w_cross_pt[0];
result[1] = pt[1] + w_cross_pt[1];
result[2] = pt[2] + w_cross_pt[2];
}
}
} // namespace ceres
#endif // CERES_PUBLIC_ROTATION_H_