Class cv::DualQuat#

#include <opencv2/core/dualquaternion.hpp>

Collaboration diagram for cv::DualQuat:

cv::DualQuat< _Tp > Node1 cv::DualQuat< _Tp >   + DualQuat() + DualQuat() + DualQuat() + conjugate() + dot() + exp() + getDualPart() + getRealPart() + getRotation() + getTranslation() and 22 more... + createFromAffine3() + createFromAngleAxisTrans() + createFromMat() + createFromPitch() + createFromQuat() + dqblend() + gdqblend() + gdqblend() + sclerp() Node2 _Tp     Node2->Node1 +CV_DUAL_QUAT_EPS +w +w_ +x +x_ +y +y_ +z +z_

cv::DualQuat< _Tp > Node1 cv::DualQuat< _Tp >   + DualQuat() + DualQuat() + DualQuat() + conjugate() + dot() + exp() + getDualPart() + getRealPart() + getRotation() + getTranslation() and 22 more... + createFromAffine3() + createFromAngleAxisTrans() + createFromMat() + createFromPitch() + createFromQuat() + dqblend() + gdqblend() + gdqblend() + sclerp() Node2 _Tp     Node2->Node1 +CV_DUAL_QUAT_EPS +w +w_ +x +x_ +y +y_ +z +z_

Friends#

Return

Name

Description

DualQuat< T >

conjugate(const DualQuat< T > &dq)

return the conjugate of a dual quaternion.

DualQuat< T >

cv::operator*(const DualQuat< T > &, const T s)

Multiplication operator of a dual quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.

DualQuat< T >

cv::operator*(const T s, const DualQuat< T > &)

Multiplication operator of a scalar and a dual quaternions. It multiplies right operand with the left operand and assign the result to left operand.

DualQuat< T >

cv::operator+(const DualQuat< T > &, const T s)

Addition operator of a dual quaternions and a scalar. Adds right hand operand from left hand operand.

DualQuat< T >

cv::operator+(const T s, const DualQuat< T > &)

Addition operator of a scalar and a dual quaternions. Adds right hand operand from left hand operand.

DualQuat< T >

cv::operator-(const DualQuat< T > &, const T s)

Subtraction operator of a dual quaternion and a scalar. Subtracts right hand operand from left hand operand.

DualQuat< T >

cv::operator-(const T s, const DualQuat< T > &)

Subtraction operator of a scalar and a dual quaternions. Subtracts right hand operand from left hand operand.

std::ostream &

cv::operator<<(std::ostream &, const DualQuat< S > &)

DualQuat< T >

exp(const DualQuat< T > &dq)

return the value of exponential function value

DualQuat< T >

inv(const DualQuat< T > &dq, QuatAssumeType assumeUnit)

if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is

DualQuat< T >

log(const DualQuat< T > &dq, QuatAssumeType assumeUnit)

return the value of logarithm function value

DualQuat< T >

power(const DualQuat< T > &dq, const T t, QuatAssumeType assumeUnit)

return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:

DualQuat< T >

power(const DualQuat< T > &p, const DualQuat< T > &q, QuatAssumeType assumeUnit)

return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:

Detailed Description#

Dual quaternions were introduced to describe rotation together with translation while ordinary quaternions can only describe rotation. It can be used for shortest path pose interpolation, local pose optimization or volumetric deformation. More details can be found

A unit dual quaternion can be classically represented as:

\[\begin{split} \begin{equation} \begin{split} \sigma &= \left(r+\frac{\epsilon}{2}tr\right)\\ &= [w, x, y, z, w\_, x\_, y\_, z\_] \end{split} \end{equation} \end{split}\]
where \(r, t\) represents the rotation (ordinary unit quaternion) and translation (pure ordinary quaternion) respectively.

A general dual quaternions which consist of two quaternions is usually represented in form of:

\[ \sigma = p + \epsilon q \]
where the introduced dual unit \(\epsilon\) satisfies \(\epsilon^2 = \epsilon^3 =...=0\), and \(p, q\) are quaternions.

Alternatively, dual quaternions can also be interpreted as four components which are all dual numbers:

\[ \sigma = \hat{q}_w + \hat{q}_xi + \hat{q}_yj + \hat{q}_zk \]
If we set \(\hat{q}_x, \hat{q}_y\) and \(\hat{q}_z\) equal to 0, a dual quaternion is transformed to a dual number. see normalize().

If you want to create a dual quaternion, you can use:

using namespace cv;
double angle = CV_PI;

// create from eight number
DualQuatd dq1(1, 2, 3, 4, 5, 6, 7, 8); //p = [1,2,3,4]. q=[5,6,7,8]

// create from Vec
Vec<double, 8> v{1,2,3,4,5,6,7,8};
DualQuatd dq_v{v};

// create from two quaternion
Quatd p(1, 2, 3, 4);
Quatd q(5, 6, 7, 8);
DualQuatd dq2 = DualQuatd::createFromQuat(p, q);

// create from an angle, an axis and a translation
Vec3d axis{0, 0, 1};
Vec3d trans{3, 4, 5};
DualQuatd dq3 = DualQuatd::createFromAngleAxisTrans(angle, axis, trans);

// If you already have an instance of class Affine3, then you can use
Affine3d R = dq3.toAffine3();
DualQuatd dq4 = DualQuatd::createFromAffine3(R);

// or create directly by affine transformation matrix Rt
// see createFromMat() in detail for the form of Rt
Matx44d Rt = dq3.toMat();
DualQuatd dq5 = DualQuatd::createFromMat(Rt);

// Any rotation + translation movement can
// be expressed as a rotation + translation around the same line in space (expressed by Plucker
// coords), and here's a way to represent it this way.
Vec3d axis{1, 1, 1}; // axis will be normalized in createFromPitch
Vec3d trans{3, 4 ,5};
axis = axis / std::sqrt(axis.dot(axis));// The formula for computing moment that I use below requires a normalized axis
Vec3d moment = 1.0 / 2 * (trans.cross(axis) + axis.cross(trans.cross(axis)) *
                           std::cos(rotation_angle / 2) / std::sin(rotation_angle / 2));
double d = trans.dot(qaxis);
DualQuatd dq6 = DualQuatd::createFromPitch(angle, d, axis, moment);

A point \(v=(x, y, z)\) in form of dual quaternion is \([1+\epsilon v]=[1,0,0,0,0,x,y,z]\). The transformation of a point \(v_1\) to another point \(v_2\) under the dual quaternion \(\sigma\) is

\[ 1 + \epsilon v_2 = \sigma * (1 + \epsilon v_1) * \sigma^{\star} \]
where \(\sigma^{\star}=p^*-\epsilon q^*.\)

A line in the \(Pl\ddot{u}cker\) coordinates \((\hat{l}, m)\) defined by the dual quaternion \(l=\hat{l}+\epsilon m\). To transform a line,

\[l_2 = \sigma * l_1 * \sigma^*,\]
where \(\sigma=r+\frac{\epsilon}{2}rt\) and \(\sigma^*=p^*+\epsilon q^*\).

To extract the Vec<double, 8> or Vec<float, 8>, see toVec();

To extract the affine transformation matrix, see toMat();

To extract the instance of Affine3, see toAffine3();

If two quaternions \(q_0, q_1\) are needed to be interpolated, you can use sclerp() 

DualQuatd::sclerp(q0, q1, t)
or dqblend(). 
DualQuatd::dqblend(q0, q1, t)
With more than two dual quaternions to be blended, you can use generalize linear dual quaternion blending with the corresponding weights, i.e. gdqblend().

Member Typedef Documentation#

value_type#

typedef _Tp cv::DualQuat::value_type

Constructor & Destructor Documentation#

DualQuat()#

cv::DualQuat::DualQuat()

DualQuat()#

cv::DualQuat::DualQuat(
const _Tp w,
const _Tp x,
const _Tp y,
const _Tp z,
const Tp w,
const Tp x,
const Tp y,
const Tp z )

create from eight same type numbers.

DualQuat()#

cv::DualQuat::DualQuat(const Vec< _Tp, 8 > & q)

create from a double or float vector.

Member Function Documentation#

conjugate()#

DualQuat< _Tp > cv::DualQuat::conjugate()

return the conjugate of a dual quaternion.

(1)#\[\begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation}\]

dot()#

_Tp cv::DualQuat::dot(DualQuat< _Tp > p)

return the dot product of two dual quaternion.

Parameters

  • p — other dual quaternion.

exp()#

DualQuat< _Tp > cv::DualQuat::exp()

return the value of exponential function value

getDualPart()#

Quat< _Tp > cv::DualQuat::getDualPart()

return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat().

See also

createFromQuat, getRealPart

getRealPart()#

Quat< _Tp > cv::DualQuat::getRealPart()

return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat().

See also

createFromQuat, getDualPart

getRotation()#

Quat< _Tp > cv::DualQuat::getRotation(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

return the rotation in quaternion form.

getTranslation()#

Vec< _Tp, 3 > cv::DualQuat::getTranslation(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

return the translation vector. The rotation \(r\) in this dual quaternion \(\sigma\) is applied before translation \(t\). The dual quaternion \(\sigma\) is defined as

(2)#\[\begin{equation} \begin{split} \sigma &= p + \epsilon q \\ &= r + \frac{\epsilon}{2}{t}r. \end{split} \end{equation}\]

Thus, the translation can be obtained as follows

\[ t = 2qp^*. \]

Note

This dual quaternion’s translation is applied after the rotation.

Parameters

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

inv()#

DualQuat< _Tp > cv::DualQuat::inv(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

if \(\sigma = p + \epsilon q\) is a dual quaternion, p is not zero, the inverse dual quaternion is

\[ \sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \]

or equivalentlly,

\[ \sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}. \]

Parameters

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

log()#

DualQuat< _Tp > cv::DualQuat::log(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

return the value of logarithm function value

Parameters

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

norm()#

DualQuat< _Tp > cv::DualQuat::norm()

return the norm \(||\sigma||\) of dual quaternion \(\sigma = p + \epsilon q\).

(3)#\[\begin{equation} \begin{split} ||\sigma|| &= \sqrt{\sigma * \sigma^*} \\ &= ||p|| + \epsilon \frac{p \cdot q}{||p||}. \end{split} \end{equation}\]

Generally speaking, the norm of a not unit dual quaternion is a dual number. For convenience, we return it in the form of a dual quaternion , i.e.

\[ ||\sigma|| = [||p||, 0, 0, 0, \frac{p \cdot q}{||p||}, 0, 0, 0]. \]

Note

The data type of dual number is dual quaternion.

normalize()#

DualQuat< _Tp > cv::DualQuat::normalize()

return a normalized dual quaternion. A dual quaternion can be expressed as

(4)#\[\begin{equation} \begin{split} \sigma &= p + \epsilon q\\ &=||\sigma||\left(r+\frac{1}{2}tr\right) \end{split} \end{equation}\]

where \(r, t\) represents the rotation (ordinary quaternion) and translation (pure ordinary quaternion) respectively, and \(||\sigma||\) is the norm of dual quaternion(a dual number). A dual quaternion is unit if and only if

\[ ||p||=1, p \cdot q=0 \]

where \(\cdot\) means dot product. The process of normalization is

\[ \sigma_{u}=\frac{\sigma}{||\sigma||} \]

Next, we simply proof \(\sigma_u\) is a unit dual quaternion:

\[\begin{split} \renewcommand{\Im}{\operatorname{Im}} \begin{equation} \begin{split} \sigma_{u}=\frac{\sigma}{||\sigma||}&=\frac{p + \epsilon q}{||p||+\epsilon\frac{p\cdot q}{||p||}}\\ &=\frac{p}{||p||}+\epsilon\left(\frac{q}{||p||}-p\frac{p\cdot q}{||p||^3}\right)\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\left(qp^{*}-p\cdot q\right)\frac{p}{||p||}\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\Im(qp^*)\frac{p}{||p||}.\\ \end{split} \end{equation} \end{split}\]

As expected, the real part is a rotation and dual part is a pure quaternion.

operator*()#

DualQuat< _Tp > cv::DualQuat::operator*(const DualQuat< _Tp > &)

Multiplication operator of two dual quaternions q and p. Multiplies values on either side of the operator.

Rule of dual quaternion multiplication: The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus

(5)#\[\begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p * q << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]

operator*=()#

DualQuat< _Tp > cv::DualQuat::operator*=(const _Tp s)

Multiplication assignment operator of a quaternions and a scalar. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication with a scalar:

(6)#\[\begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
p *= s;
std::cout << p << std::endl; //[2, 4, 6, 8, 10, 12, 14, 16]

Note

the type of scalar should be equal to the dual quaternion.

operator*=()#

DualQuat< _Tp > & cv::DualQuat::operator*=(const DualQuat< _Tp > &)

Multiplication assignment operator of two quaternions. It multiplies right operand with the left operand and assign the result to left operand.

Rule of dual quaternion multiplication: The dual quaternion can be written as an ordered pair of quaternions [A, B]. Thus

(7)#\[\begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p *= q;
std::cout << p << std::endl; //[-60, 12, 30, 24, -216, 80, 124, 120]

operator+()#

DualQuat< _Tp > cv::DualQuat::operator+(const DualQuat< _Tp > &)

Addition operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(q_i\).

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p + q << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]

operator+=()#

DualQuat< _Tp > & cv::DualQuat::operator+=(const DualQuat< _Tp > &)

Addition assignment operator of two dual quaternions p and q. It adds right operand to the left operand and assign the result to left operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p += q; // equivalent to p = p + q
std::cout << p << std::endl; //[6, 8, 10, 12, 14, 16, 18, 20]

operator-()#

DualQuat< _Tp > cv::DualQuat::operator-()

Return opposite dual quaternion \(-p\) which satisfies \(p + (-p) = 0.\).

For example

DualQuatd q{1, 2, 3, 4, 5, 6, 7, 8};
std::cout << -q << std::endl; // [-1, -2, -3, -4, -5, -6, -7, -8]

operator-()#

DualQuat< _Tp > cv::DualQuat::operator-(const DualQuat< _Tp > &)

Subtraction operator of two dual quaternions p and q. It returns a new dual quaternion that each value is the sum of \(p_i\) and \(-q_i\).

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p - q << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]

operator-=()#

DualQuat< _Tp > & cv::DualQuat::operator-=(const DualQuat< _Tp > &)

Subtraction assignment operator of two dual quaternions p and q. It subtracts right operand from the left operand and assign the result to left operand.

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p -= q; // equivalent to p = p - q
std::cout << p << std::endl; //[-4, -4, -4, -4, 4, -4, -4, -4]

operator/()#

DualQuat< _Tp > cv::DualQuat::operator/(const _Tp s)

Division operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a scalar:

(8)#\[\begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_, z\_] / s\\ &=[w/s, x/s, y/s, z/s, w\_/s, x\_/s, y\_/s, z\_/s]. \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4]

Note

the type of scalar should be equal to this dual quaternion.

operator/()#

DualQuat< _Tp > cv::DualQuat::operator/(const DualQuat< _Tp > &)

Division operator of two dual quaternions p and q. Divides left hand operand by right hand operand.

Rule of dual quaternion division with a dual quaternion:

(9)#\[\begin{equation} \begin{split} p / q &= p * q.inv()\\ \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
std::cout << p / q << std::endl; // equivalent to p * q.inv()

operator/=()#

Quat< _Tp > & cv::DualQuat::operator/=(const _Tp s)

Division assignment operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a scalar:

(10)#\[\begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_ ,z\_] / s\\ &=[w / s, x / s, y / s, z / s, w\_ / \space s, x\_ / \space s, y\_ / \space s, z\_ / \space s]. \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
double s = 2.0;;
p /= s; // equivalent to p = p / s
std::cout << p << std::endl; //[0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0]

Note

the type of scalar should be equal to the dual quaternion.

operator/=()#

DualQuat< _Tp > & cv::DualQuat::operator/=(const DualQuat< _Tp > &)

Division assignment operator of two dual quaternions p and q; It divides left operand with the right operand and assign the result to left operand.

Rule of dual quaternion division with a quaternion:

(11)#\[\begin{equation} \begin{split} p / q&= p * q.inv()\\ \end{split} \end{equation}\]

For example

DualQuatd p{1, 2, 3, 4, 5, 6, 7, 8};
DualQuatd q{5, 6, 7, 8, 9, 10, 11, 12};
p /= q; // equivalent to p = p * q.inv()
std::cout << p << std::endl;

operator==()#

bool cv::DualQuat::operator==(const DualQuat< _Tp > &)

return true if two dual quaternions p and q are nearly equal, i.e. when the absolute value of each \(p_i\) and \(q_i\) is less than CV_DUAL_QUAT_EPS.

power()#

DualQuat< _Tp > cv::DualQuat::power(
const _Tp t,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:

\[ p^t = \exp(t\ln p) \]

Parameters

  • t — index of power function.

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

power()#

DualQuat< _Tp > cv::DualQuat::power(
const DualQuat< _Tp > & q,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:

\[ p^q = \exp(q\ln p) \]

Parameters

  • q — a dual quaternion

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a dual unit quaternion and this function will save some computations.

toAffine3()#

Affine3< _Tp > cv::DualQuat::toAffine3(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

Transform this dual quaternion to a instance of Affine3.

toMat()#

Matx< _Tp, 4, 4 > cv::DualQuat::toMat(QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT)

Transform this dual quaternion to a affine transformation matrix the form of matrix, see createFromMat().

toVec()#

Vec< _Tp, 8 > cv::DualQuat::toVec()

Transform this dual quaternion to a vector.

createFromAffine3()#

static DualQuat< _Tp > cv::DualQuat::createFromAffine3(const Affine3< _Tp > & R)

create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat()

createFromAngleAxisTrans()#

static DualQuat< _Tp > cv::DualQuat::createFromAngleAxisTrans(
const _Tp angle,
const Vec< _Tp, 3 > & axis,
const Vec< _Tp, 3 > & translation )

create a dual quaternion from a rotation angle \(\theta\), a rotation axis \(\boldsymbol{u}\) and a translation \(\boldsymbol{t}\). It generates a dual quaternion \(\sigma\) in the form of

(12)#\[\begin{equation} \begin{split} \sigma &= r + \frac{\epsilon}{2}\boldsymbol{t}r \\ &= [\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})] + \frac{\epsilon}{2}[0, \boldsymbol{t}][[\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})]]\\ &= \cos(\frac{\theta}{2}) + \boldsymbol{u}\sin(\frac{\theta}{2}) + \frac{\epsilon}{2}(-(\boldsymbol{t} \cdot \boldsymbol{u})\sin(\frac{\theta}{2}) + \boldsymbol{t}\cos(\frac{\theta}{2}) + \boldsymbol{u} \times \boldsymbol{t} \sin(\frac{\theta}{2})). \end{split} \end{equation}\]

Note

Axis will be normalized in this function. And translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

See also

Quat

Parameters

  • angle — rotation angle.

  • axis — rotation axis.

  • translation — a vector of length 3.

createFromMat()#

static DualQuat< _Tp > cv::DualQuat::createFromMat(InputArray _R)

Transform this dual quaternion to an affine transformation matrix \(M\). Dual quaternion consists of a rotation \(r=[a,b,c,d]\) and a translation \(t=[\Delta x,\Delta y,\Delta z]\). The affine transformation matrix \(M\) has the form.

\[\begin{split} \begin{bmatrix} 1-2(e_2^2 +e_3^2) &2(e_1e_2-e_0e_3) &2(e_0e_2+e_1e_3) &\Delta x\\ 2(e_0e_3+e_1e_2) &1-2(e_1^2+e_3^2) &2(e_2e_3-e_0e_1) &\Delta y\\ 2(e_1e_3-e_0e_2) &2(e_0e_1+e_2e_3) &1-2(e_1^2-e_2^2) &\Delta z\\ 0&0&0&1 \end{bmatrix} \end{split}\]

if A is a matrix consisting of n points to be transformed, this could be achieved by

\[ new\_A = M * A \]

where A has the form

\[\begin{split} \begin{bmatrix} x_0& x_1& x_2&...&x_n\\ y_0& y_1& y_2&...&y_n\\ z_0& z_1& z_2&...&z_n\\ 1&1&1&...&1 \end{bmatrix} \end{split}\]

where the same subscript represent the same point. The size of A should be \([4,n]\). and the same size for matrix new_A.

Note

Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

Parameters

  • _R — 4x4 matrix that represents rotations and translation.

createFromPitch()#

static DualQuat< _Tp > cv::DualQuat::createFromPitch(
const _Tp angle,
const _Tp d,
const Vec< _Tp, 3 > & axis,
const Vec< _Tp, 3 > & moment )

A dual quaternion is a vector in form of.

(13)#\[\begin{equation} \begin{split} \sigma &=\boldsymbol{p} + \epsilon \boldsymbol{q}\\ &= \cos\hat{\frac{\theta}{2}}+\overline{\hat{l}}\sin\frac{\hat{\theta}}{2} \end{split} \end{equation}\]

where \(\hat{\theta}\) is dual angle and \(\overline{\hat{l}}\) is dual axis:

\[\begin{split} \hat{\theta}=\theta + \epsilon d,\\ \overline{\hat{l}}= \hat{l} +\epsilon m. \end{split}\]

In this representation, \(\theta\) is rotation angle and \((\hat{l},m)\) is the screw axis, d is the translation distance along the axis.

Note

Translation is applied after the rotation. Use createFromQuat(r, r * t / 2) to create a dual quaternion which translation is applied before rotation.

Parameters

  • angle — rotation angle.

  • d — translation along the rotation axis.

  • axis — rotation axis represented by quaternion with w = 0.

  • moment — the moment of line, and it should be orthogonal to axis.

createFromQuat()#

static DualQuat< _Tp > cv::DualQuat::createFromQuat(
const Quat< _Tp > & realPart,
const Quat< _Tp > & dualPart )

create Dual Quaternion from two same type quaternions p and q. A Dual Quaternion \(\sigma\) has the form:

\[ \sigma = p + \epsilon q \]

where p and q are defined as follows:

(14)#\[\begin{equation} \begin{split} p &= w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}\\ q &= w\_ + x\_\boldsymbol{i} + y\_\boldsymbol{j} + z\_\boldsymbol{k}. \end{split} \end{equation}\]

The p and q are the real part and dual part respectively.

See also

Quat

Parameters

  • realPart — a quaternion, real part of dual quaternion.

  • dualPart — a quaternion, dual part of dual quaternion.

dqblend()#

static DualQuat< _Tp > cv::DualQuat::dqblend(
const DualQuat< _Tp > & q1,
const DualQuat< _Tp > & q2,
const _Tp t,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

The method of Dual Quaternion linear Blending(DQB) is to compute a transformation between dual quaternion \(q_1\) and \(q_2\) and can be defined as:

\[ DQB(t;{\boldsymbol{q}}_1,{\boldsymbol{q}}_2)= \frac{(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2}{||(1-t){\boldsymbol{q}}_1+t{\boldsymbol{q}}_2||}. \]

where \(q_1\) and \(q_2\) are unit dual quaternions representing the input transformations. If you want to use DQB that works for more than two rigid transformations, see gdqblend

See also

gdqblend

Parameters

  • q1 — a unit dual quaternion representing the input transformations.

  • q2 — a unit dual quaternion representing the input transformations.

  • t — parameter \(t\in[0,1]\).

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

gdqblend()#

template<int cn>
static DualQuat< _Tp > cv::DualQuat::gdqblend(
const Vec< DualQuat< _Tp >, cn > & dualquat,
InputArray weights,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.

\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]

Note

the type of weights’ element should be the same as the date type of dual quaternion inside the dualquat.

Parameters

  • dualquat — vector of dual quaternions

  • weights — vector of weights, the size of weights should be the same as dualquat, and the weights should satisfy \(\sum_0^n w_{i} = 1\) and \(w_i>0\).

  • assumeUnit — if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

gdqblend()#

static DualQuat< _Tp > cv::DualQuat::gdqblend(
InputArray dualquat,
InputArray weights,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

The generalized Dual Quaternion linear Blending works for more than two rigid transformations. If these transformations are expressed as unit dual quaternions \(q_1,...,q_n\) with convex weights \(w = (w_1,...,w_n)\), the generalized DQB is simply.

\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]

Note

the type of weights’ element should be the same as the date type of dual quaternion inside the dualquat.

Parameters

  • dualquat — The dual quaternions which have 8 channels and 1 row or 1 col.

  • weights — vector of weights, the size of weights should be the same as dualquat, and the weights should satisfy \(\sum_0^n w_{i} = 1\) and \(w_i>0\).

  • assumeUnit — if QUAT_ASSUME_UNIT, these dual quaternions assume to be unit quaternions and this function will save some computations.

sclerp()#

static DualQuat< _Tp > cv::DualQuat::sclerp(
const DualQuat< _Tp > & q1,
const DualQuat< _Tp > & q2,
const _Tp t,
bool directChange = true,
QuatAssumeType assumeUnit = QUAT_ASSUME_NOT_UNIT )

The screw linear interpolation(ScLERP) is an extension of spherical linear interpolation of dual quaternion. If \(\sigma_1\) and \(\sigma_2\) are two dual quaternions representing the initial and final pose. The interpolation of ScLERP function can be defined as:

\[ ScLERP(t;\sigma_1,\sigma_2) = \sigma_1 * (\sigma_1^{-1} * \sigma_2)^t, t\in[0,1] \]

For example

double angle1 = CV_PI / 2;
Vec3d axis{0, 0, 1};
Vec3d t(0, 0, 3);
DualQuatd initial = DualQuatd::createFromAngleAxisTrans(angle1, axis, t);
double angle2 = CV_PI;
DualQuatd final = DualQuatd::createFromAngleAxisTrans(angle2, axis, t);
DualQuatd inter = DualQuatd::sclerp(initial, final, 0.5);

Parameters

  • q1 — a dual quaternion represents a initial pose.

  • q2 — a dual quaternion represents a final pose.

  • t — interpolation parameter

  • directChange — if true, it always return the shortest path.

  • assumeUnit — if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations.

Member Data Documentation#

CV_DUAL_QUAT_EPS#

static _Tp cv::DualQuat::CV_DUAL_QUAT_EPS = (_Tp)1.e-6

w#

_Tp cv::DualQuat::w

w_#

Tp cv::DualQuat::w

x#

_Tp cv::DualQuat::x

x_#

Tp cv::DualQuat::x

y#

_Tp cv::DualQuat::y

y_#

Tp cv::DualQuat::y

z#

_Tp cv::DualQuat::z

z_#

Tp cv::DualQuat::z

Source file#

The documentation for this class was generated from the following file: