camera calibration with one dimensional objects camera calibration has been studied extensively in computer vision and photogrammetry and the proposed techniques in the literature include those using 3d apparatus two or three planes orthogonal to each other or a plane undergoing a pure translation etc 2d objects planar patterns undergoing unknown motions and 0d features self calibration using unknown scene points yet the paper proposes a new calibration technique using 1d objects points aligned on a line thus filling the missing dimension in calibration in particular we show that camera calibration is not possible with free moving 1d objects but can be solved if one point is fixed a closed form solution is developed if six or more observations of such a 1d object are made for higher accuracy a nonlinear technique based on the maximum likelihood criterion is then used to refine the estimate singularities have also been studied besides the theoretical aspect the proposed technique is also important in practice especially when calibrating multiple cameras mounted apart from each other where the calibration objects are required to be visible simultaneously

rapid and brief communication camera calibration with one dimensional objects moving under gravity camera calibration using one dimensional 1d rigid objects is arresting the attentions of researchers since the easy to construct geometrical structure of the apparatuses in this paper we extend the motion patterns applicable for calibration with the motion of 1d objects we show that a 1d object with three or more markers rotating around one marker which is moving in a plane provides constraint equations on camera intrinsic parameters a stick moving under gravity without other forces acting on performs such a motion simulated tests show the feasibility and numerical robustness of this method

about the critical condition of the q state potts model on the anisotropic hypercubic lattice a critical condition is obtained for the ferromagnetic model on the hypercubic lattice in d dimensions with different potts interactions along the different lattice axes it is done by extending a simple procedure which is shown to be exact for the square lattice in d 2 dimensions

camera calibration using one dimensional information and its applications in both controlled and uncontrolled environments camera model and its calibration are required in many applications for coordinate conversions between the two dimensional image and the real three dimensional world self calibration method is usually chosen for camera calibration in uncontrolled environments because the scene geometry could be unknown however when no reliable feature correspondences can be established or when the camera is static in relation to the majority of the scene self calibration method fails to work on the other hand object based calibration methods are more reliable than self calibration methods due to the existence of the object with known geometry however most object based calibration methods are unable to work in uncontrolled environments because they require the geometric knowledge on calibration objects though in the past few years the simplest geometry required for a calibration object has been reduced to a 1d object with at least three points it is still not easy to find such an object in an uncontrolled environment not to mention the additional metric motion requirement in the existing methods meanwhile it is very easy to find a 1d object with two end points in most scenes thus it would be very worthwhile to investigate an object based method based on such a simple object so that it would still be possible to calibrate a camera when both self calibration and existing object based calibration fail to work we propose a new camera calibration method which requires only an object with two end points the simplest geometry that can be extracted from many real life objects through observations of such a 1d object at different positions orientations on a plane which is fixed in relation to the camera both intrinsic focal length and extrinsic rotation angles and translations camera parameters can be calibrated using the proposed method the proposed method has been tested on simulated data and real data from both controlled and uncontrolled environments including situations where no explicit 1d calibration objects are available e g from a human walking sequence very accurate camera calibration results have been achieved using the proposed method

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

a time integration algorithm for linear transient analysis based on the reproducing kernel method in this paper a new algorithm based on the reproducing kernel method is presented for linear transient analysis using very large time steps in the conventional time integration methods the required accuracy is always achieved by either decreasing the size of time steps or increasing the order of approximations the proposed method however does not require a restriction on the size of the time step the required approximating functions for integration of governing equations are determined using the concept of reproduction of the solution within each time step a recurrence relationship is obtained through solving the governing equation by the point collocation based reproducing kernel method within the time step in collocation points the values of external excitation can be obtained either through the approximation between the values at the beginning and the end of each time step or assigning the exact values of the external force hence one advantage of this new method is the independency of the sampling rates of external excitation from the size of the time step the proposed method is applied for solving first and second order differential equations and advantages of the method are illustrated through a number of numerical examples

self calibration of a moving camera from pointcorrespondences and fundamental matrices we address the problem of estimating three dimensional motion and structure from motion with an uncalibrated moving camera we show that point correspondences between three images and the fundamental matrices computed from these point correspondences are sufficient to recover the internal orientation of the camera its calibration the motion parameters and to compute coherent perspective projection matrices which enable us to reconstruct 3 d structure up to a similarity in contrast with other methods no calibration object with a known 3 d shape is needed and no limitations are put upon the unknown motions to be performed or the parameters to be recovered as long as they define a projective camera the theory of the method which is based on the constraint that the observed points are part of a static scene thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic is first detailed several algorithms are then presented and their performances compared by means of extensive simulations and illustrated by several experiments with real images

the norsett time integration methodology for finite element transient analysis this paper presents a set of methods for time integration of problems arising from finite element semidiscretizations the purpose is to obtain computationally efficient methods which possess higher order accuracy and controllable dissipation in the spurious high modes the methods are developed and analysed by a general collocation methodology which leads to the class of norsett approximants an algorithmic parameter is used to achieve an effective control over numerical dissipation moreover a simple and efficient implementation scheme is presented at each time step algorithms based on p order collocation polynomials require the solution of p sets of linear algebraic equations with the same coefficient matrix in this way a single factorization is needed and no transformations are required to recover the approximate solution at the end or within the time interval to demonstrate the performance of the proposed algorithms a wide experimental evaluation is carried out on typical test problems in finite element transient analysis

weighted similarity invariant linear algorithm for camera calibration with rotating 1 d objects in this paper a weighted similarity invariant linear algorithm for camera calibration with rotating 1 d objects is proposed first we propose a new estimation method for computing the relative depth of the free endpoint on the 1 d object and prove its robustness against noise compared with those used in previous literature the introduced estimator is invariant to image similarity transforms resulting in a similarity invariant linear calibration algorithm which is slightly more accurate than the well known normalized linear algorithm then we use the reciprocals of the standard deviations of the estimated relative depths from different images as the weights on the constraint equations of the similarity invariant linear calibration algorithm and propose a weighted similarity invariant linear calibration algorithm with higher accuracy experimental results on synthetic data as well as on real image data show the effectiveness of our proposed algorithm

a new easy camera calibration technique based on circular points inspired by zhang s work a new easy technique for calibrating a camera based on circular points is proposed the proposed technique only requires the camera to observe a newly designed planar calibration pattern referred to as the model plane hereinafter which includes a circle and a pencil of lines passing through the circle s center at a few at least three different unknown orientations then all the five intrinsic parameters can be determined linearly the main point of our new technique is that it needs to know neither metric measurement on the model plane nor the correspondences between points on the model plane and image ones hence it can be done fully automatically the proposed technique is particularly useful for those people who are not familiar with computer vision experiments with simulated data as well as with real images show that our new technique is robust and accurate

on the percolation threshold for a d dimensional simple hypercubic lattice the authors use 1 sigma expressions sigma 2d 1 to investigate systematically a number of conjectures and approximations for the bond and site percolation thresholds of a d dimensional simple hypercubic lattice the technique provides convincing evidence that none of the conjectures are correct and gives a quantitative understanding of the quality of the approximations

collocation time finite element procedures for integration of strongly nonlinear initial value problems by use of a collocation time finite element formulation a modification of the well known generalized midpoint method is derived the new family possesses an additional free parameter this allows for significant improvement of the numerical solution in strongly nonlinear situations numerical results show the accuracy of the proposed algorithm in comparison to the midpoint method

cayley transformation and numerical stability of calibration equation the application of cayley transformation to enhance the numerical stability of camera calibration is investigated first a new calibration equation called the standard calibration equation is introduced using the cayley transformation and its analytical solution is obtained the standard calibration equation is equivalent to the classical calibration equation but it exhibits remarkable better numerical stability second a one parameter calibration family called the cayley calibration family which is equivalent to the standard calibration equation is obtained using also the cayley transformation and it is found that this family is composed of those infinite homographies whose rotation has the same axis with the rotation between the two given views the condition number of equations in the cayley calibration family varies with the parameter value and an algorithm to determine the best parameter value is provided third the generalized cayley calibration families equivalent to the standard calibration equation are also introduced via generalized cayley transformations an example of the generalized cayley transformations is illustrated called the s cayley calibration family as in the cayley calibration family the numerical stability of equations in a generalized cayley calibration family also depends on the parameter value in addition a more generic calibration family is also proposed and it is proved that the standard calibration equation the cayley calibration family and the s cayley calibration family are all some special cases of this generic calibration family

a time stepping procedure for structural dynamics using gauss point collocation when a cubic function is interpolated between the prescribed initial displacement and velocity and the exact displacement and velocity at the end of a time step for a single degree of freedom system the error or residual in the governing equation is zero at a number of times it is shown that for a general undamped system in the limit as the time step approaches zero these times correspond to gauss points this observation is verified by considering a general collocation procedure in which the displacement in any time step is approximated as a cubic function of time with two coefficients chosen to satisfy the displacement and velocity at the beginning of the time step with the other two coefficients being chosen to satisfy the governing differential equation at any two times it is shown that optimum accuracy is obtained if these points are the gauss points detailed expressions are then presented for this particular case and stability of the algorithm is investigated showing that the procedure is conditionally stable for time steps which are a small proportion of the least period of vibration of the structure the algorithm is considered to be the most accurate possible procedure based on cubic approximation of the displacement

unconditionally stable collocation algorithms for second order initial value problems abstract in this paper unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented the second order equations are manipulated directly if the approximate solution is expressed as a polynomial of degree n 1 there are n unknowns to be determined after taking into account the two given initial conditions it is well known that by suppressing the residuals of the governing equations at n distinct collocation points only the resultant algorithms are only conditionally stable in this paper linear combinations of the residuals at n 1 distinct collocation points are used to solve for the n unknowns the collocation points and the relative weights between the residuals are derived from the weighted residual method the weighting functions are arbitrary polynomials of degree not exceeding n 1 to control the accuracy and stability properties of the resultant algorithms the reduced integration technique is used to evaluate the integrals in the formulation once the reduced integration rules are decided the equivalent collocation form can be derived it is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non linear problems directly numerical examples are given to illustrate the validity of the present formulation

camera calibration with moving one dimensional objects in this paper we show that the rotating 1d calibrating object used in the literature is in essence equivalent to a familiar 2d planar calibration object in addition we also show that when the 1d object undergoes a planar motion rather than rotating around a fixed point such equivalence still holds but the traditional way fails to handle it experiments are carried out to verify the theoretical correctness and numerical robustness of our results

constraints on general motions for camera calibration with one dimensional objects this paper focuses on two problems in camera calibration with one dimensional 1d objects a to find out the general motion patterns well suited for solving the calibration problem and b to improve the robustness and accuracy of the method firstly a sufficient and necessary condition for the solvability of 1d calibration with general motions is proved then the special motion of tossing a 1d object is provided as an example to illustrate the correctness and feasibility of this condition after that some practical issues on obtaining the solution are inspected by avoiding singularities the precision and robustness of the method are improved the relative mean errors are reduced to less than 5 at the noise level of one pixel which surpasses the state of the art methods of the same category

camera calibration with three noncollinear points under special motions plane based 2d camera calibration is becoming a hot research topic in recent years because of its flexibility however at least four image points are needed in every view to denote the coplanar feature in the 2d camera calibration can we do the camera calibration by using the calibration object that only has three points some 1d camera calibration techniques use the setup of three collinear points with known distances but it is a kind of special conditions of calibration object setup how about the general setup three noncollinear points we propose a new camera calibration algorithm based on the calibration objects with three noncollinear points experiments with simulated data and real images are carried out to verify the theoretical correctness and numerical robustness of our results because the objects with three noncollinear points have special properties in camera calibration they are midway between 1d and 2d calibration objects our method is actually a new kind of camera calibration algorithm

unconditionally stable higher order accurate collocation time step integration algorithms for first order equations abstract in this paper unconditionally stable higher order accurate time step integration algorithms for linear first order differential equations based on the collocation method are presented the amplification factor at the end of the spectrum is a controllable algorithmic parameter the collocation parameters for unconditionally stable higher order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor in general when the numerical solution is approximated by a polynomial of degree n this approximation is at least n th order accurate however by using the above collocation parameters the order of accuracy can be improved to 2 n 1 or 2 n the approximate solutions are found to be equivalent to the generalized pade approximations furthermore it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions no modification of the collocation parameters is required

camera calibration with distortion models and accuracy evaluation a camera model that accounts for major sources of camera distortion namely radial decentering and thin prism distortions is presented the proposed calibration procedure consists of two steps 1 the calibration parameters are estimated using a closed form solution based on a distribution free camera model and 2 the parameters estimated in the first step are improved iteratively through a nonlinear optimization taking into account camera distortions according to minimum variance estimation the objective function to be minimized is the mean square discrepancy between the observed image points and their inferred image projections computed with the estimated calibration parameters the authors introduce a type of measure that can be used to directly evaluate the performance of calibration and compare calibrations among different systems the validity and performance of the calibration procedure are tested with both synthetic data and real images taken by tele and wide angle lenses

a theory of self calibration of a moving camera there is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement the epipolar transformation imposes two algebraic constraints on the camera calibration if two epipolar transformations arising from different camera displacements are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four the curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces or by a curve of degree six in the dual of the image plane the curve in the dual plane has one singular point of order three and three singular points of order two if three epipolar transformations are available then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration the two curves have a common singular point of order three experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation the real intersections of the two dual curves are found by locating the zeros of a function defined on the interval 0 2 the intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

the normalised image of the absolute conic and its application for zooming camera calibration we present a novel technique for calibrating a zooming camera based on the invariance properties of the normalised image of the absolute conic niac we show that the camera parameters independent of position orientation and zooming are determined uniquely by the niac and we exploit these invariance properties to develop a stratified calibration method that decouples the calibration parameters the method is organised in three steps i computation of the niac ii computation of the focal length for each image and iii computation of the orientation and the position of the camera the method requires a minimum of three views of a single planar grid experiments with synthetic and real data suggest that the method is competitive with other state of the art plane based zooming calibration methods in the scenarios considered

a transient response analysis in the state space applying the average velocity concept an implicit direct time integration method for obtaining transient responses of general dynamic systems is described the conventional newmark method cannot be directly applied to state space first order differential equations which contain no explicit acceleration terms the method proposed here is the state space newmark method that incorporates the average velocity concept and can be applied to an analysis of general dynamic systems that are expressed by state space first order differential equations it is also readily coded into a program stability and accuracy analyses indicate that the method is numerically unconditionally stable like the conventional newmark method and has a period error of second order accuracy for small damping and fourth order for large damping and an amplitude error of second order regardless of damping in addition its utility and validity are confirmed by two application examples the results suggest that the proposed state space newmark method based on average velocity be generally applied to the analysis of transient responses of general dynamic systems with a high degree of reliability with respect to stability and accuracy

on the location of the critical point of the q state potts model on the hypercubic lattice the critical temperature of the q state ferromagnetic potts model on a hypercubic lattice in d dimensions is expressed as an explicit function of both d and q the author s formula agrees very well with the existing accurate results and numerous numerical studies for particular values of d and q and it seems to be an excellent approximation for all d and q

about the critical condition of the spin s 1 ising model on the anisotropic square and sc lattice the authors extend to the s 1 ising model a simple procedure which was proposed recently for the q state potts model the approximate critical conditions are obtained for the ferromagnetic model on the square and simple cubic lattice