Solution of the weighted symmetric similarity transformations based on quaternions

Mercan, H.; Akyılmaz, O.; Aydin, Cüneyt

doi:10.1007/s00190-017-1104-0

Cited by 38 publications

(10 citation statements)

References 37 publications

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“…Then, equation () can be reinterpreted as:

{boldv}_{boldx} = {boldA}_{boldx} t + {boldB}_{boldx} x - {boldl}_{boldx}; {boldP}_{boldx}

where

t = {(Δ X_{S}, Δ Y_{S}, Δ Z_{S}, Δ q_{0}, Δ q_{x}, Δ q_{y}, Δ q_{z})}^{normalT}

is the augmented vector; and

A_{x}

represents the coefficient matrix. The linearity of the constraint condition is given (Mercan et al, ) as:

q_{0} Δ q_{0} + q_{x} Δ q_{x} + q_{y} Δ q_{y} + q_{z} Δ q_{z} + w = 0

where

w = (q_{0}^{2} + q_{x}^{2} + q_{y}^{2} + q_{z}^{2} - 1) / 2

. Then, equation () can be expressed as:

Ct + w = 0

where

C = [0 0 0 q_{0} q_{x} q_{y} q_{z}]

represents the coefficient matrix of the condition equation; and

w = [w]

denotes the constant vector.…”

Section: Methodsmentioning

confidence: 99%

A precise visual localisation method for the Chinese Chang’e‐4 Yutu‐2 rover

Liu

Sima

et al. 2020

The Photogrammetric Record

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Precise localisation of the Yutu-2 moon rover plays an important role in path planning, obstacle avoidance and navigating to target features. To provide highprecision localisation information, a stereo bundle adjustment method using the theory of the unit quaternions is presented for the first time. To improve the precision and robustness of the proposed method, the rover's pose, from a visual odometry technique assisted by an inertial measurement unit and the rotation angles of the mast mechanism, is viewed as a pseudo-observation. A reasonable weighting strategy and a rational geometric constraint condition of the stereo cameras is also invoked. Experimental results demonstrate that the proposed method provides more accurate localisation results than either a bundle adjustment alone or a weighted total least-squares method. The proposed method has been successfully used in Chang'e-4 mission operations. (a) Chang'e-4 lander (b) Yutu-2 rover FIG. 1. The Chang'e-4 probe. (a) An image taken by the Yutu-2 rover's Pancam camera of the lander. (b) An image taken by the lander's topographic camera of the rover.The Photogrammetric Record

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“…Then, equation () can be reinterpreted as:

{boldv}_{boldx} = {boldA}_{boldx} t + {boldB}_{boldx} x - {boldl}_{boldx}; {boldP}_{boldx}

where

t = {(Δ X_{S}, Δ Y_{S}, Δ Z_{S}, Δ q_{0}, Δ q_{x}, Δ q_{y}, Δ q_{z})}^{normalT}

is the augmented vector; and

A_{x}

represents the coefficient matrix. The linearity of the constraint condition is given (Mercan et al, ) as:

q_{0} Δ q_{0} + q_{x} Δ q_{x} + q_{y} Δ q_{y} + q_{z} Δ q_{z} + w = 0

where

w = (q_{0}^{2} + q_{x}^{2} + q_{y}^{2} + q_{z}^{2} - 1) / 2

. Then, equation () can be expressed as:

Ct + w = 0

where

C = [0 0 0 q_{0} q_{x} q_{y} q_{z}]

represents the coefficient matrix of the condition equation; and

w = [w]

denotes the constant vector.…”

Section: Methodsmentioning

confidence: 99%

A precise visual localisation method for the Chinese Chang’e‐4 Yutu‐2 rover

Liu

Sima

et al. 2020

The Photogrammetric Record

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show abstract

“…Mazaheri and Habib [27] compared unit quaternion in single photo space resection with existing algorithms and reported the detailed evaluation of the algorithm. Mercan et al [28] presented an iterative algorithm formulated as a GH model of adjustment for the solution of weighted symmetric similarity transformation problems, which takes advantage of quaternion's unique representation of 3D orthogonal rotation matrix. Later, Uygur et al [29] showed how to evaluate the rotation angles and the full covariance matrix of the transformation parameters from the estimation results in asymmetric and symmetric 3D similarity transformations based on quaternions.…”

Section: Quaternion's Application In Point Cloud Registrationmentioning

confidence: 99%

A Closed-Form Solution to Linear Feature-Based Registration of LiDAR Point Clouds

et al. 2021

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Due to the high complexity of geo-spatial entities and the limited field of view of LiDAR equipment, pairwise registration is a necessary step for integrating point clouds from neighbouring LiDAR stations. Considering that accurate extraction of point features is often difficult without the use of man-made reflectors, and the initial approximate values for the unknown transformation parameters must be estimated in advance to ensure the correct operation of those iterative methods, a closed-form solution to linear feature-based registration of point clouds is proposed in this study. Plücker coordinates are used to represent the linear features in three-dimensional space, whereas dual quaternions are employed to represent the spatial transformation. Based on the theory of least squares, an error norm (objective function) is first constructed by assuming that each pair of corresponding linear features is equivalent after registration. Then, by applying the extreme value analysis to the objective function, detailed derivations of the closed-form solution to the proposed linear feature-based registration method are given step by step. Finally, experimental tests are conducted on a real dataset. The derived experimental result demonstrates the feasibility of the proposed solution: By using eigenvalue decomposition to replace the linearization of the objective function, the proposed solution does not require any initial estimates of the unknown transformation parameters, which assures the stability of the registration method.

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“…Mahboub (2016) proposed a WTLS solution to 3D symmetrical similarity transformation without linearization; however, more iterations may be needed than the linearized model method. Mercan et al (2018) proposed a weighted similarity transformation based on quaternions. The algorithm has seven unknowns including the translation parameters and scaled quaternion, and the iterations may lead to divergence.…”

Section: Introductionmentioning

confidence: 99%

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

Zeng

Chang

et al. 2020

Earth Planets Space

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The 3D similarity coordinate transformation is fundamental and frequently encountered in many areas of work such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. The algorithms of 3D similarity transformation are divided into two categories. One is a closed-form algorithm that is straightforward and fast. However, it cannot provide the accuracy information for the transformation parameters. The other category of algorithm is iterative, and this can offer the accuracy information for the transformation parameters. However, the latter usually needs a good initial value of the unknown. Considering the accuracy information for transformation parameters is essential or indispensable from the viewpoint of uncertainty, this contribution proposes a weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors. It is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters. Two cases including the registration of LIDAR points with big rotation angles and a geodetic datum transformation with small rotation angles are demonstrated to validate the new algorithm.

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Solution of the weighted symmetric similarity transformations based on quaternions

Cited by 38 publications

References 37 publications

A precise visual localisation method for the Chinese Chang’e‐4 Yutu‐2 rover

A precise visual localisation method for the Chinese Chang’e‐4 Yutu‐2 rover

A Closed-Form Solution to Linear Feature-Based Registration of LiDAR Point Clouds

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

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