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.

Determination of a closed-form solution for the multidimensional transport equation using a fractional derivative

2002 ◽  
Vol 29 (10) ◽  
pp. 1141-1150 ◽  
Author(s):  
Jorge Zabadal ◽  
Marco Túllio Vilhena ◽  
Cynthia Feijó Segatto ◽  
Rúben Panta Pazos
Keyword(s):  
Fractional Derivative ◽  
Transport Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

Determination of Closed Form Solution for Acceptance Sampling Using ANN

2005 ◽  
Vol 11 (1) ◽  
pp. 43-61 ◽  
Author(s):  
D. Vasudevan ◽  
V. Selladurai ◽  
P. Nagaraj
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Acceptance Sampling

On the Analytical Solution of Externally Pressurized Porous Gas Journal Bearings

1978 ◽  
Vol 100 (3) ◽  
pp. 442-444 ◽  
Author(s):  
B. C. Majumdar
Keyword(s):  
Analytical Solution ◽  
Pressure Distribution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Journal Bearings ◽  
Form Solution ◽  
Performance Characteristics ◽  
Gas Bearing ◽  
Good Agreement

A closed form solution of pressure distribution which leads to the determination of bearing performance characteristics of an externally pressurized porous gas bearing without journal rotation is obtained. A good agreement with a similar available solution confirms the validity of the method.

Determination of Eigenstresses from Curvature Data

1992 ◽  
Vol 276 ◽  
Author(s):  
Mauro Ferrari ◽  
Marie Weber
Keyword(s):  
Structural Analysis ◽  
Closed Form ◽  
Mechanical Systems ◽  
Closed Form Solution ◽  
Form Solution ◽  
Curvature Measurements

ABSTRACTCurvature measurements are generally employed in conjunction with elementary structural analysis to estimate deposition stresses in miniaturized electro-mechanical systems. In this paper the validity of this procedure is discussed by presenting a closed form solution for a bilayer subject to nonuniform intrinsic straining, and comparing the exact stress-curvature relations with the oft-used formulae of Stoney and Brenner-Senderoff.

A Closed-Form Solution to the Crank Position Corresponding to the Maximum Velocity of the Slider in a Centric Slider-Crank Mechanism

2005 ◽  
Vol 128 (3) ◽  
pp. 654-656 ◽  
Author(s):  
W. J. Zhang ◽  
Q. Li
Keyword(s):  
Closed Form ◽  
Classical Problem ◽  
Closed Form Solution ◽  
Maximum Velocity ◽  
Form Solution ◽  
Machine Design ◽  
Rotary Engine ◽  
Engine Systems ◽  
Crank Mechanism

This paper revisits a classical problem in kinematics, specifically determination of the crank position corresponding to the maximum velocity of the slider in the centric slider-crank mechanism. This position is often critical in designing products constructed using the slider-crank mechanism, e.g., industrial sewing machinery, rotary engine systems, etc. In current literature, the numerical, graphical, or approximate closed-form solution to this problem is available. In this paper, an exact closed-form solution is derived. With this new closed-form solution, it is found that there exist significant errors in an approximate closed-form solution which can be found from many machine design text books for a practica1 use.

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.

Direct Position Analysis of the 4–6 Stewart Platforms

1994 ◽  
Vol 116 (1) ◽  
pp. 61-66 ◽  
Author(s):  
Ning-Xin Chen ◽  
Shin-Min Song
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Stewart Platform ◽  
Form Solution ◽  
Parallel Robots ◽  
Geometric Parameters ◽  
Degree Polynomial ◽  
Position Analysis ◽  
Half Angle ◽  
Stewart Platforms

Although Stewart platforms have been applied in the design of aircraft and vehicle simulators and parallel robots for many years, the closed-form solution of direct (forward) position analysis of Stewart platforms has not been completely solved. Up to the present time, only the relatively simple Stewart platforms have been analyzed. Examples are the octahedral, the 3–6 and the 4–4 Stewart platforms, of which the forward position solutions were derived as an eighth or a twelfth degree polynomials with one variable in the form of square of a tan-half-angle. This paper further extends the direct position analysis to a more general case of the Stewart platform, the 4–6 Stewart platforms, in which two pairs of the upper joint centers of adjacent limbs are coincident. The result is a sixteenth degree polynomial in the square of a tan-half-angle, which indicates that a maximum of 32 configurations may be obtained. It is also shown that the previously derived solutions of the 3–6 and 4–4 Stewart platforms can be easily deduced from the sixteenth degree polynomial by setting some geometric parameters be equal to 1 or 0.

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