scholarly journals An Efficient Closed Form Solution to the Absolute Orientation Problem for Camera with Unknown Focal Length

Sensors ◽  
2021 ◽  
Vol 21 (19) ◽  
pp. 6480
Author(s):  
Kai Guo ◽  
Hu Ye ◽  
Zinian Zhao ◽  
Junhao Gu
Keyword(s):  
Closed Form ◽  
Focal Length ◽  
Closed Form Solution ◽  
Geometric Approach ◽  
Polynomial Equation ◽  
Form Solution ◽  
Camera Model ◽  
Absolute Orientation ◽  
Camera Position ◽  
The Absolute

In this paper we propose an efficient closed form solution to the absolute orientation problem for cameras with an unknown focal length, from two 2D–3D point correspondences and the camera position. The problem can be decomposed into two simple sub-problems and can be solved with angle constraints. A polynomial equation of one variable is solved to determine the focal length, and then a geometric approach is used to determine the absolute orientation. The geometric derivations are easy to understand and significantly improve performance. Rewriting the camera model with the known camera position leads to a simpler and more efficient closed form solution, and this gives a single solution, without the multi-solution phenomena of perspective-three-point (P3P) solvers. Experimental results demonstrated that our proposed method has a better performance in terms of numerical stability, noise sensitivity, and computational speed, with synthetic data and real images.

scholarly journals Decoupled Closed-Form Solution for Humanoid Lower Limb Kinematics

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Alejandro Said ◽  
Ernesto Rodriguez-Leal ◽  
Rogelio Soto ◽  
J. L. Gordillo ◽  
Leonardo Garrido
Keyword(s):  
Closed Form ◽  
Lower Limb ◽  
Closed Form Solution ◽  
Numerical Data ◽  
Geometric Approach ◽  
Form Solution ◽  
Experimental Implementation ◽  
Limb Kinematics ◽  
Position And Orientation ◽  
Lower Limb Kinematics

This paper presents an explicit, omnidirectional, analytical, and decoupled closed-form solution for the lower limb kinematics of the humanoid robot NAO. The paper starts by decoupling the position and orientation analysis from the overall Denavit-Hartenberg (DH) transformation matrices. Here, the joint activation sequence for the DH matrices is based on the geometry of a triangle. Furthermore, the implementation of a forward and a reversed kinematic analysis for the support and swing phase equations is developed to avoid matrix inversion. The allocation of constant transformations allows the position and orientation end-coordinate systems to be aligned with each other. Also, the redefinition of the DH transformations and the use of constraints allow decoupling the shared DOF between the legs and the torso. Finally, a geometric approach to avoid the singularities during the walking process is indicated. Numerical data is presented along with an experimental implementation to prove the validity of the analytical results.

Closed-Form Solution to the Noncoplanar P4P Problem for Uncalibrated Camera

2011 ◽  
Vol 145 ◽  
pp. 6-10
Author(s):  
Yang Guo
Keyword(s):  
Closed Form ◽  
Affine Transformation ◽  
Closed Form Solution ◽  
Polynomial Equation ◽  
Form Solution ◽  
Potential Candidate ◽  
Degree Polynomial ◽  
Fast Determination ◽  
Hand Eye Coordination

This paper presents a closed-form solution to determination of the position and orientation of a perspective camera with two unknown effective focal lengths for the noncoplanar perspective four point (P4P) problem. Given four noncoplanar 3D points and their correspondences in image coordinate, we convert perspective transformation to affine transformation, and formulate the problem using invariance to 3D affine transformation and arrive to a closed-form solution. We show how the noncoplanar P4P problem is cast into the problem of solving an eighth degree polynomial equation in one unknown. This result shows the noncoplanar P4P problem with two unknown effective focal lengths has at most 8 solutions. Last, we confirm the conclusion by an example. Although developed as part of landmark-guided navigation, the solution might well be used for landmark-based tracking problem, hand-eye coordination, and for fast determination of interior and exterior camera parameters. Because our method is based on closed-form solution, its speed makes it a potential candidate for solving above problems.

A Complex Number Approach for Absolute Precision Position Synthesis for Three Precision Positions

1992 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Closed Form ◽  
Complex Number ◽  
Closed Form Solution ◽  
Form Solution ◽  
Path Generation ◽  
Motion Generation ◽  
The Absolute ◽  
Position Problem ◽  
Position Synthesis

Abstract Dyads can be synthesized by prescribing the precision point coordinates and the absolute planar orientations of one dyad vector at each of three precision positions. This differs from traditional complex number methods wherein the vector orientations are described relative to one another. Absolute precision position synthesis can be performed for both motion generation, and path generation with prescribed timing. The method presented uses vector loop equations and complex number notation to produce a closed form solution for the three absolute precision position problem. Absolute precision position synthesis is applicable to cases that require specific coupler geometries. The synthesis of flat-folding mechanisms is an example of one such application.

scholarly journals Closed-form solution of absolute orientation using orthonormal matrices

1988 ◽  
Vol 5 (7) ◽  
pp. 1127 ◽  
Author(s):  
Berthold K. P. Horn ◽  
Hugh M. Hilden ◽  
Shahriar Negahdaripour
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Absolute Orientation

A new closed-form solution to inverse static force analysis of a spatial three-spring system

2014 ◽  
Vol 229 (14) ◽  
pp. 2599-2610 ◽  
Author(s):  
Ying Zhang ◽  
Qizheng Liao ◽  
Hai-Jun Su ◽  
Shimin Wei
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Polynomial Equation ◽  
Form Solution ◽  
Common Point ◽  
Static Force ◽  
Force Analysis ◽  
Intrinsic Geometry ◽  
Equilibrium Configurations ◽  
Spring System

In this paper, a new closed-form solution to the inverse static force analysis of a spatial three-spring system is presented. The system is formed by three springs each of which connects the ground at one end and joins a common point at the other. When a known force is applied to the common point of the system, the goal of inverse static analysis is to determine all the equilibrium configurations. A system of three polynomial equations in three variables is derived based on the geometric constraint and static force balancing. A 20 by 20 Dixon resultant matrices firstly derived from these three polynomials and then reduced to an 18 by 18 matrix. A 46th-degree univariate polynomial equation is yielded from the above 18 by 18 matrix. By further analysis, we found that 24 roots were degenerated and only the remaining 22 roots are the ones for the three-spring system. The result agrees with previous results. At last, two numerical examples are given to verify the elimination procedure. The presented algebraic elimination solution reveals some intrinsic geometry nature of this challenging problem.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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