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OpenCV
4.9.0
Open Source Computer Vision
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OpenCV
4.9.0
Open Source Computer Vision
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#include <opencv2/core/dualquaternion.hpp>
Public Member Functions | |
| 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. More... | |
| DualQuat (const Vec< _Tp, 8 > &q) | |
| create from a double or float vector. More... | |
| DualQuat< _Tp > | conjugate () const |
| return the conjugate of a dual quaternion. \[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \] More... | |
| _Tp | dot (DualQuat< _Tp > p) const |
| return the dot product of two dual quaternion. More... | |
| DualQuat< _Tp > | exp () const |
| return the value of exponential function value More... | |
| Quat< _Tp > | getDualPart () const |
| return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat(). More... | |
| Quat< _Tp > | getRealPart () const |
| return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat(). More... | |
| Quat< _Tp > | getRotation (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| return the rotation in quaternion form. More... | |
| Vec< _Tp, 3 > | getTranslation (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| return the translation vector. The rotation \(r\) in this dual quaternion \(\sigma\) is applied before translation \(t\). The dual quaternion \(\sigma\) is defined as \[\begin{equation} \begin{split} \sigma &= p + \epsilon q \\ &= r + \frac{\epsilon}{2}{t}r. \end{split} \end{equation}\] Thus, the translation can be obtained as follows \[t = 2qp^*.\] More... | |
| DualQuat< _Tp > | inv (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| 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, \[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\] More... | |
| DualQuat< _Tp > | log (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| return the value of logarithm function value More... | |
| DualQuat< _Tp > | norm () const |
| return the norm \(||\sigma||\) of dual quaternion \(\sigma = p + \epsilon q\). \[ \begin{equation} \begin{split} ||\sigma|| &= \sqrt{\sigma * \sigma^*} \\ &= ||p|| + \epsilon \frac{p \cdot q}{||p||}. \end{split} \end{equation} \] 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. \[ ||\sigma|| = [||p||, 0, 0, 0, \frac{p \cdot q}{||p||}, 0, 0, 0].\] More... | |
| DualQuat< _Tp > | normalize () const |
| return a normalized dual quaternion. A dual quaternion can be expressed as \[ \begin{equation} \begin{split} \sigma &= p + \epsilon q\\ &=||\sigma||\left(r+\frac{1}{2}tr\right) \end{split} \end{equation} \] 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: \[ \renewcommand{\Im}{\operatorname{Im}} \begin{equation} \begin{split} \sigma_{u}=\frac{\sigma}{||\sigma||}&=\frac{p + \epsilon q}{||p||+\epsilon\frac{p\cdot q}{||p||}}\\ &=\frac{p}{||p||}+\epsilon\left(\frac{q}{||p||}-p\frac{p\cdot q}{||p||^3}\right)\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\left(qp^{*}-p\cdot q\right)\frac{p}{||p||}\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\Im(qp^*)\frac{p}{||p||}.\\ \end{split} \end{equation} \] As expected, the real part is a rotation and dual part is a pure quaternion. More... | |
| DualQuat< _Tp > | operator* (const DualQuat< _Tp > &) const |
| Multiplication operator of two dual quaternions q and p. Multiplies values on either side of the operator. More... | |
| DualQuat< _Tp > & | 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. More... | |
| DualQuat< _Tp > | 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. More... | |
| DualQuat< _Tp > | operator+ (const DualQuat< _Tp > &) const |
| 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 \(q_i\). More... | |
| DualQuat< _Tp > & | 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. More... | |
| DualQuat< _Tp > | operator- () const |
| Return opposite dual quaternion \(-p\) which satisfies \(p + (-p) = 0.\). More... | |
| DualQuat< _Tp > | operator- (const DualQuat< _Tp > &) const |
| 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 \(-q_i\). More... | |
| DualQuat< _Tp > & | 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. More... | |
| DualQuat< _Tp > | operator/ (const _Tp s) const |
| Division operator of a dual quaternions and a scalar. It divides left operand with the right operand and assign the result to left operand. More... | |
| DualQuat< _Tp > | operator/ (const DualQuat< _Tp > &) const |
| Division operator of two dual quaternions p and q. Divides left hand operand by right hand operand. More... | |
| DualQuat< _Tp > & | 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. More... | |
| Quat< _Tp > & | 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. More... | |
| bool | operator== (const DualQuat< _Tp > &) const |
| 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. More... | |
| DualQuat< _Tp > | power (const _Tp t, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| return the value of \(p^t\) where p is a dual quaternion. This could be calculated as: \[ p^t = \exp(t\ln p) \] More... | |
| DualQuat< _Tp > | power (const DualQuat< _Tp > &q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as: \[ p^q = \exp(q\ln p) \] More... | |
| Affine3< _Tp > | toAffine3 (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| Transform this dual quaternion to a instance of Affine3. More... | |
| Matx< _Tp, 4, 4 > | toMat (QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT) const |
| Transform this dual quaternion to a affine transformation matrix the form of matrix, see createFromMat(). More... | |
| Vec< _Tp, 8 > | toVec () const |
| Transform this dual quaternion to a vector. More... | |
Static Public Member Functions | |
| static DualQuat< _Tp > | createFromAffine3 (const Affine3< _Tp > &R) |
| create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat() More... | |
| static DualQuat< _Tp > | 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 \[\begin{equation} \begin{split} \sigma &= r + \frac{\epsilon}{2}\boldsymbol{t}r \\ &= [\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})] + \frac{\epsilon}{2}[0, \boldsymbol{t}][[\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})]]\\ &= \cos(\frac{\theta}{2}) + \boldsymbol{u}\sin(\frac{\theta}{2}) + \frac{\epsilon}{2}(-(\boldsymbol{t} \cdot \boldsymbol{u})\sin(\frac{\theta}{2}) + \boldsymbol{t}\cos(\frac{\theta}{2}) + \boldsymbol{u} \times \boldsymbol{t} \sin(\frac{\theta}{2})). \end{split} \end{equation}\] More... | |
| static DualQuat< _Tp > | 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 \[ \begin{bmatrix} 1-2(e_2^2 +e_3^2) &2(e_1e_2-e_0e_3) &2(e_0e_2+e_1e_3) &\Delta x\\ 2(e_0e_3+e_1e_2) &1-2(e_1^2+e_3^2) &2(e_2e_3-e_0e_1) &\Delta y\\ 2(e_1e_3-e_0e_2) &2(e_0e_1+e_2e_3) &1-2(e_1^2-e_2^2) &\Delta z\\ 0&0&0&1 \end{bmatrix} \] 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 \[ \begin{bmatrix} x_0& x_1& x_2&...&x_n\\ y_0& y_1& y_2&...&y_n\\ z_0& z_1& z_2&...&z_n\\ 1&1&1&...&1 \end{bmatrix} \] where the same subscript represent the same point. The size of A should be \([4,n]\). and the same size for matrix new_A. More... | |
| static DualQuat< _Tp > | 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 \[ \begin{equation} \begin{split} \sigma &=\boldsymbol{p} + \epsilon \boldsymbol{q}\\ &= \cos\hat{\frac{\theta}{2}}+\overline{\hat{l}}\sin\frac{\hat{\theta}}{2} \end{split} \end{equation} \] where \(\hat{\theta}\) is dual angle and \(\overline{\hat{l}}\) is dual axis: \[ \hat{\theta}=\theta + \epsilon d,\\ \overline{\hat{l}}= \hat{l} +\epsilon m. \] In this representation, \(\theta\) is rotation angle and \((\hat{l},m)\) is the screw axis, d is the translation distance along the axis. More... | |
| static DualQuat< _Tp > | 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: \[\begin{equation} \begin{split} p &= w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}\\ q &= w\_ + x\_\boldsymbol{i} + y\_\boldsymbol{j} + z\_\boldsymbol{k}. \end{split} \end{equation} \] The p and q are the real part and dual part respectively. More... | |
| static DualQuat< _Tp > | 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. More... | |
| template<int cn> | |
| static DualQuat< _Tp > | 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 \[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \] . More... | |
| static DualQuat< _Tp > | 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 \[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \] . More... | |
| static DualQuat< _Tp > | 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: \[ ScLERP(t;\sigma_1,\sigma_2) = \sigma_1 * (\sigma_1^{-1} * \sigma_2)^t, t\in[0,1] \] . More... | |
Public Attributes | |
| _Tp | w |
| _Tp | w_ |
| _Tp | x |
| _Tp | x_ |
| _Tp | y |
| _Tp | y_ |
| _Tp | z |
| _Tp | z_ |
Static Public Attributes | |
| static constexpr _Tp | CV_DUAL_QUAT_EPS = (_Tp)1.e-6 |
Friends | |
| template<typename T > | |
| DualQuat< T > | conjugate (const DualQuat< T > &dq) |
| return the conjugate of a dual quaternion. \[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \] More... | |
| template<typename T > | |
| 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. More... | |
| template<typename T > | |
| 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. More... | |
| template<typename T > | |
| 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. More... | |
| template<typename T > | |
| 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. More... | |
| template<typename T > | |
| 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. More... | |
| template<typename T > | |
| 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. More... | |
| template<typename S > | |
| std::ostream & | cv::operator<< (std::ostream &, const DualQuat< S > &) |
| template<typename T > | |
| DualQuat< T > | exp (const DualQuat< T > &dq) |
| return the value of exponential function value More... | |
| template<typename T > | |
| 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 \[\sigma^{-1} = \frac{\sigma^*}{||\sigma||^2}, \] or equivalentlly, \[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\] More... | |
| template<typename T > | |
| DualQuat< T > | log (const DualQuat< T > &dq, QuatAssumeType assumeUnit) |
| return the value of logarithm function value More... | |
| template<typename T > | |
| 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: \[ p^t = \exp(t\ln p) \] More... | |
| template<typename T > | |
| 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: \[ p^q = \exp(q\ln p) \] More... | |
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:
\[ \begin{equation} \begin{split} \sigma &= \left(r+\frac{\epsilon}{2}tr\right)\\ &= [w, x, y, z, w\_, x\_, y\_, z\_] \end{split} \end{equation} \]
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:
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()
or dqblend().
With more than two dual quaternions to be blended, you can use generalize linear dual quaternion blending with the corresponding weights, i.e. gdqblend().
| cv::DualQuat< _Tp >::DualQuat | ( | ) |
| cv::DualQuat< _Tp >::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.
| cv::DualQuat< _Tp >::DualQuat | ( | const Vec< _Tp, 8 > & | q | ) |
create from a double or float vector.
| DualQuat<_Tp> cv::DualQuat< _Tp >::conjugate | ( | ) | const |
return the conjugate of a dual quaternion.
\[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \]
|
static |
create dual quaternion from an affine matrix. The definition of affine matrix can refer to createFromMat()
|
static |
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
\[\begin{equation} \begin{split} \sigma &= r + \frac{\epsilon}{2}\boldsymbol{t}r \\ &= [\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})] + \frac{\epsilon}{2}[0, \boldsymbol{t}][[\cos(\frac{\theta}{2}), \boldsymbol{u}\sin(\frac{\theta}{2})]]\\ &= \cos(\frac{\theta}{2}) + \boldsymbol{u}\sin(\frac{\theta}{2}) + \frac{\epsilon}{2}(-(\boldsymbol{t} \cdot \boldsymbol{u})\sin(\frac{\theta}{2}) + \boldsymbol{t}\cos(\frac{\theta}{2}) + \boldsymbol{u} \times \boldsymbol{t} \sin(\frac{\theta}{2})). \end{split} \end{equation}\]
| angle | rotation angle. |
| axis | rotation axis. |
| translation | a vector of length 3. |
|
static |
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
\[ \begin{bmatrix} 1-2(e_2^2 +e_3^2) &2(e_1e_2-e_0e_3) &2(e_0e_2+e_1e_3) &\Delta x\\ 2(e_0e_3+e_1e_2) &1-2(e_1^2+e_3^2) &2(e_2e_3-e_0e_1) &\Delta y\\ 2(e_1e_3-e_0e_2) &2(e_0e_1+e_2e_3) &1-2(e_1^2-e_2^2) &\Delta z\\ 0&0&0&1 \end{bmatrix} \]
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
\[ \begin{bmatrix} x_0& x_1& x_2&...&x_n\\ y_0& y_1& y_2&...&y_n\\ z_0& z_1& z_2&...&z_n\\ 1&1&1&...&1 \end{bmatrix} \]
where the same subscript represent the same point. The size of A should be \([4,n]\). and the same size for matrix new_A.
| _R | 4x4 matrix that represents rotations and translation. |
|
static |
A dual quaternion is a vector in form of
\[ \begin{equation} \begin{split} \sigma &=\boldsymbol{p} + \epsilon \boldsymbol{q}\\ &= \cos\hat{\frac{\theta}{2}}+\overline{\hat{l}}\sin\frac{\hat{\theta}}{2} \end{split} \end{equation} \]
where \(\hat{\theta}\) is dual angle and \(\overline{\hat{l}}\) is dual axis:
\[ \hat{\theta}=\theta + \epsilon d,\\ \overline{\hat{l}}= \hat{l} +\epsilon m. \]
In this representation, \(\theta\) is rotation angle and \((\hat{l},m)\) is the screw axis, d is the translation distance along the axis.
| 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. |
|
static |
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:
\[\begin{equation} \begin{split} p &= w + x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k}\\ q &= w\_ + x\_\boldsymbol{i} + y\_\boldsymbol{j} + z\_\boldsymbol{k}. \end{split} \end{equation} \]
The p and q are the real part and dual part respectively.
| realPart | a quaternion, real part of dual quaternion. |
| dualPart | a quaternion, dual part of dual quaternion. |
| _Tp cv::DualQuat< _Tp >::dot | ( | DualQuat< _Tp > | p | ) | const |
return the dot product of two dual quaternion.
| p | other dual quaternion. |
|
static |
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.
| 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. |
| DualQuat<_Tp> cv::DualQuat< _Tp >::exp | ( | ) | const |
return the value of exponential function value
|
static |
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
\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]
.
| 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. |
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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
\[ gDQB(\boldsymbol{w};{\boldsymbol{q}}_1,...,{\boldsymbol{q}}_n)=\frac{w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n} {||w_1{\boldsymbol{q}}_1+...+w_n{\boldsymbol{q}}_n||}. \]
.
| 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. |
| Quat<_Tp> cv::DualQuat< _Tp >::getDualPart | ( | ) | const |
return a quaternion which represent the dual part of dual quaternion. The definition of dual part is in createFromQuat().
| Quat<_Tp> cv::DualQuat< _Tp >::getRealPart | ( | ) | const |
return a quaternion which represent the real part of dual quaternion. The definition of real part is in createFromQuat().
| Quat<_Tp> cv::DualQuat< _Tp >::getRotation | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
return the rotation in quaternion form.
| Vec<_Tp, 3> cv::DualQuat< _Tp >::getTranslation | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
return the translation vector. The rotation \(r\) in this dual quaternion \(\sigma\) is applied before translation \(t\). The dual quaternion \(\sigma\) is defined as
\[\begin{equation} \begin{split} \sigma &= p + \epsilon q \\ &= r + \frac{\epsilon}{2}{t}r. \end{split} \end{equation}\]
Thus, the translation can be obtained as follows
\[t = 2qp^*.\]
| assumeUnit | if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations. |
| DualQuat<_Tp> cv::DualQuat< _Tp >::inv | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
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,
\[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\]
| assumeUnit | if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations. |
| DualQuat<_Tp> cv::DualQuat< _Tp >::log | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
return the value of logarithm function value
| assumeUnit | if QUAT_ASSUME_UNIT, this dual quaternion assume to be a unit dual quaternion and this function will save some computations. |
| DualQuat<_Tp> cv::DualQuat< _Tp >::norm | ( | ) | const |
return the norm \(||\sigma||\) of dual quaternion \(\sigma = p + \epsilon q\).
\[ \begin{equation} \begin{split} ||\sigma|| &= \sqrt{\sigma * \sigma^*} \\ &= ||p|| + \epsilon \frac{p \cdot q}{||p||}. \end{split} \end{equation} \]
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.
\[ ||\sigma|| = [||p||, 0, 0, 0, \frac{p \cdot q}{||p||}, 0, 0, 0].\]
| DualQuat<_Tp> cv::DualQuat< _Tp >::normalize | ( | ) | const |
return a normalized dual quaternion. A dual quaternion can be expressed as
\[ \begin{equation} \begin{split} \sigma &= p + \epsilon q\\ &=||\sigma||\left(r+\frac{1}{2}tr\right) \end{split} \end{equation} \]
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:
\[ \renewcommand{\Im}{\operatorname{Im}} \begin{equation} \begin{split} \sigma_{u}=\frac{\sigma}{||\sigma||}&=\frac{p + \epsilon q}{||p||+\epsilon\frac{p\cdot q}{||p||}}\\ &=\frac{p}{||p||}+\epsilon\left(\frac{q}{||p||}-p\frac{p\cdot q}{||p||^3}\right)\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\left(qp^{*}-p\cdot q\right)\frac{p}{||p||}\\ &=\frac{p}{||p||}+\epsilon\frac{1}{||p||^2}\Im(qp^*)\frac{p}{||p||}.\\ \end{split} \end{equation} \]
As expected, the real part is a rotation and dual part is a pure quaternion.
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator* | ( | const DualQuat< _Tp > & | ) | const |
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
\[ \begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation} \]
For example
| DualQuat<_Tp>& cv::DualQuat< _Tp >::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
\[ \begin{equation} \begin{split} p * q &= [A, B][C, D]\\ &=[AC, AD + BC] \end{split} \end{equation} \]
For example
| DualQuat<_Tp> cv::DualQuat< _Tp >::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:
\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]
For example
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator+ | ( | const DualQuat< _Tp > & | ) | const |
| DualQuat<_Tp>& cv::DualQuat< _Tp >::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
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator- | ( | ) | const |
Return opposite dual quaternion \(-p\) which satisfies \(p + (-p) = 0.\).
For example
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator- | ( | const DualQuat< _Tp > & | ) | const |
| DualQuat<_Tp>& cv::DualQuat< _Tp >::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
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator/ | ( | const _Tp | s | ) | const |
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:
\[ \begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_, z\_] / s\\ &=[w/s, x/s, y/s, z/s, w\_/s, x\_/s, y\_/s, z\_/s]. \end{split} \end{equation} \]
For example
| DualQuat<_Tp> cv::DualQuat< _Tp >::operator/ | ( | const DualQuat< _Tp > & | ) | const |
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:
\[ \begin{equation} \begin{split} p / q &= p * q.inv()\\ \end{split} \end{equation} \]
For example
| DualQuat<_Tp>& cv::DualQuat< _Tp >::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:
\[ \begin{equation} \begin{split} p / q&= p * q.inv()\\ \end{split} \end{equation} \]
For example
| Quat<_Tp>& cv::DualQuat< _Tp >::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:
\[ \begin{equation} \begin{split} p / s &= [w, x, y, z, w\_, x\_, y\_ ,z\_] / s\\ &=[w / s, x / s, y / s, z / s, w\_ / \space s, x\_ / \space s, y\_ / \space s, z\_ / \space s]. \end{split} \end{equation} \]
For example
| bool cv::DualQuat< _Tp >::operator== | ( | const DualQuat< _Tp > & | ) | const |
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.
| DualQuat<_Tp> cv::DualQuat< _Tp >::power | ( | const _Tp | t, |
| QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT |
||
| ) | const |
return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:
\[ p^t = \exp(t\ln p) \]
| 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. |
| DualQuat<_Tp> cv::DualQuat< _Tp >::power | ( | const DualQuat< _Tp > & | q, |
| QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT |
||
| ) | const |
return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:
\[ p^q = \exp(q\ln p) \]
| 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. |
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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:
\[ ScLERP(t;\sigma_1,\sigma_2) = \sigma_1 * (\sigma_1^{-1} * \sigma_2)^t, t\in[0,1] \]
.
| 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. |
For example
| Affine3<_Tp> cv::DualQuat< _Tp >::toAffine3 | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
Transform this dual quaternion to a instance of Affine3.
| Matx<_Tp, 4, 4> cv::DualQuat< _Tp >::toMat | ( | QuatAssumeType | assumeUnit = QUAT_ASSUME_NOT_UNIT | ) | const |
Transform this dual quaternion to a affine transformation matrix the form of matrix, see createFromMat().
| Vec<_Tp, 8> cv::DualQuat< _Tp >::toVec | ( | ) | const |
Transform this dual quaternion to a vector.
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return the conjugate of a dual quaternion.
\[ \begin{equation} \begin{split} \sigma^* &= (p + \epsilon q)^* &= (p^* + \epsilon q^*) \end{split} \end{equation} \]
| dq | a dual quaternion. |
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Multiplication operator of a scalar and a dual quaternions. It multiplies right operand with the left operand and assign the result to left operand.
Rule of dual quaternion multiplication with a scalar:
\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]
For example
|
friend |
Multiplication operator of a dual 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:
\[ \begin{equation} \begin{split} p * s &= [w, x, y, z, w\_, x\_, y\_, z\_] * s\\ &=[w s, x s, y s, z s, w\_ \space s, x\_ \space s, y\_ \space s, z\_ \space s]. \end{split} \end{equation} \]
For example
|
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Addition operator of a scalar and a dual quaternions. Adds right hand operand from left hand operand.
For example
|
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Addition operator of a dual quaternions and a scalar. Adds right hand operand from left hand operand.
For example
|
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Subtraction operator of a dual quaternion and a scalar. Subtracts right hand operand from left hand operand.
For example
|
friend |
Subtraction operator of a scalar and a dual quaternions. Subtracts right hand operand from left hand operand.
For example
|
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return the value of exponential function value
| dq | a dual quaternion. |
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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,
\[\sigma^{-1} = p^{-1} - \epsilon p^{-1}qp^{-1}.\]
| dq | a dual quaternion. |
| assumeUnit | if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations. |
|
friend |
return the value of logarithm function value
| dq | a dual quaternion. |
| assumeUnit | if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations. |
|
friend |
return the value of \(p^t\) where p is a dual quaternion. This could be calculated as:
\[ p^t = \exp(t\ln p) \]
| dq | a dual quaternion. |
| t | index of power function. |
| assumeUnit | if QUAT_ASSUME_UNIT, dual quaternion dq assume to be a unit dual quaternion and this function will save some computations. |
|
friend |
return the value of \(p^q\) where p and q are dual quaternions. This could be calculated as:
\[ p^q = \exp(q\ln p) \]
| p | a dual quaternion. |
| q | a dual quaternion. |
| assumeUnit | if QUAT_ASSUME_UNIT, dual quaternion p assume to be a dual unit quaternion and this function will save some computations. |
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| _Tp cv::DualQuat< _Tp >::w |
| _Tp cv::DualQuat< _Tp >::w_ |
| _Tp cv::DualQuat< _Tp >::x |
| _Tp cv::DualQuat< _Tp >::x_ |
| _Tp cv::DualQuat< _Tp >::y |
| _Tp cv::DualQuat< _Tp >::y_ |
| _Tp cv::DualQuat< _Tp >::z |
| _Tp cv::DualQuat< _Tp >::z_ |
1.8.13