Motion Constraint Equation Research Articles

Range Flow Estimation

We discuss the computation of the instantaneous 3D displacement vector fields of deformable surfaces from sequences of range data. We give a novel version of the basic motion constraint equation that can be evaluated directly on the sensor grid. The various forms of the aperture problem encountered are investigated and the derived constraint solutions are solved in a total least squares (TLS) framework. We propose a regularization scheme to compute dense full flow fields from the sparse TLS solutions. The performance of the algorithm is analyzed quantitatively for both synthetic and real data. Finally we apply the method to compute the 3D motion field of living plant leaves.

Mosaic-based positioning and improved motion-estimation methods for automatic navigation of submersible vehicles

Knowledge of the camera trajectory, which may be determined from the motions between consecutive frames of a video clip, can be used to register images for constructing image mosaics. We discuss a mosaic-based positioning framework for building photo-mosaics and concurrently utilizing them for improved positioning. In this approach, the mosaic is directly exploited in bounding the accumulation of position errors as we integrate the incremental motions of the camera. It is also shown that two earlier closed-form solutions for the estimation of motion directly from spatio-temporal image gradients, as for most gradient-based techniques based on the application of linear(ized) image motion constraint equations, are corrupted with systematic biases. These can be reduced significantly by incorporating the higher-order terms. We propose recursive methods to solve the new nonlinear constraint equations, and investigate the performance of the new solutions in a number of experiments with synthetic and real data.

Image Registration, Optical Flow and Local Rigidity

We address the theoretical problems of optical flow estimation and image registration in a multi-scale framework in any dimension. Much work has been done based on the minimization of a distance between a first image and a second image after applying deformation or motion field. Usually no justification is given about convergence of the algorithm used. We start by showing, in the translation case, that convergence to the global minimum is made easier by applying a low pass filter to the images hence making the energy “convex enough”. In order to keep convergence to the global minimum in the general case, we introduce a local rigidity hypothesis on the unknown deformation. We then deduce a new natural motion constraint equation (MCE) at each scale using the Dirichlet low pass operator. This transforms the problem to solving the energy minimization in a finite dimensional subspace of approximation obtained through Fourier Decomposition. This allows us to derive sufficient conditions for convergence of a new multi-scale and iterative motion estimation/registration scheme towards a global minimum of the usual nonlinear energy instead of a local minimum as did all previous methods. Although some of the sufficient conditions cannot always be fulfilled because of the absence of the necessary a priori knowledge on the motion, we use an implicit approach. We illustrate our method by showing results on synthetic and real examples in dimension 1 (signal matching, Stereo) and 2 (Motion, Registration, Morphing), including large deformation experiments.

多体システムのための高速な順動力学解法の開発 １ 動力学の定式化及び他の漸化式解法との比較検討

The aim of this research is to develop an efficient and stable dynamic analysis approach for multibody systems. In this first report, an efficient extension of Rosenthal's order-n algorithm, which is originally limited for only multibody systems with open-loop topology, to multibody systems containing closed-loops is presented. Relative coordinates are employed for modeling systems. Closed topological loops are handled by cut joint technique. Equations of motion and cut joint constraint equations establish a set of differential-algebraic equation (DAE). Then, Rosenthal's order-n algorithm is extended to solve this DAE. Dynamic analysis of a multibody system is carried out to demonstrate efficiency of the proposed approach. Finally, number of arithmetic operations required by the proposed method and other well-known recursive formulations for analyzing a single closed-loop multibody system are counted in order to examine the efficiency of the proposed method compared to other well-known recursive methods

多体システムのための高速な順動力学解法の開発 ２ 幾何学的拘束式違反の対策

Cut joint technique is employed in the first report for extension Rosenthal's order-n algorithm to multibody systems with closed-loop topology. Equations of motion and cut joint constraint equations constitute a set of differential-algebraic equation (DAE) that must be solved when doing dynamic analysis. When numerical method is employed for solving that DAE, cut joint constraint violation occurs. In this second report, methods for eliminating constraint violation are examined. The reliable approach for selecting parameters used in Baumgarte constraint stabilization method for variable step integration methods is proposed. Coordinate projection method is also proposed for constraint violation treatment. Dynamic analysis of a slider crank mechanism is carried out to show the validity of the proposed methods.

A direct violation correction method in numerical simulation of constrained multibody systems

A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method.

Measuring 3‐D plant growth using optical flow

A method is presented for measuring 3-D plant growth using the optical flow computed on an image sequence of a growing corn seedling. Each image in the sequence consists of two views of the same seedling; one view of the corn seedling is front-on while the second view is a orthogonal view (at 90°) of the seedling made by projecting the plant's orthogonal image onto a mirror oriented at 45° with respect to the camera. We compute 3-D velocity (motion) of the corn seedling's tip by using a simple extension of the 2-D motion constraint equation used in optical flow analysis. This method is an extension of the work presented by Barron and Liptay where optical flow was used to measure the 2-D growth (in the vertical plane) of a corn seedling.

Selection of efficient coordinate partitioning methods in flexible multibody systems

In multibody dynamics, differential and algebraic equations which can satisfy both equation of motion and kinematic constraint equation should be solved. To solve these equations, coordinate partitioning method and constraint stabilization method are commonly used. In the coordinate partitioning method, the coordinates are divided into independent and dependent and coordinates. The most typical coordinate partitioning method are LU decomposition, QR decomposition, and SVD (singular value decomposition). The objective of this research is to find an efficient coordinate partitioning method in the dynamic analysis of flexible multibody systems. Comparing two coordinate partitioning methods, i.e. LU and QR decomposition in the flexible multibody systems, a new hybrid coordinate partitioning method is suggested for the flexible multibody analysis.

Studies on mechanics problem of dynamic walking of anthropomorphic biped robots

This paper deals with the mechanics problem of dynamic walking of anthropomorphic biped robots. Through analysing the mechanics system of this kind of robots in detail, the motion constraint equations are established, three mechanics laws describing the relationships among the external forces are obtained, and the approaches to measuring and computing these forces using the pressure sensors are proposed. The results not only reveal the mechanics characteristics of dynamic walking of anthropomorphic biped robots but also are of an important significance in realizing the on-line computation and real-time control of their dunamic walking.

A General Motion Model and Spatio-Temporal Filters forComputing Optical Flow

Traditional optical flow algorithms assume local image translational motion and apply simple image filtering techniques. Recent studies have taken two separate approaches toward improving the accuracy of computed flow: the application of spatio-temporal filtering schemes and the use of advanced motion models such as the affine model. Each has achieved some improvement over traditional algorithms in specialized situations but the computation of accurate optical flow for general motion has been elusive. In this paper, we exploit the interdependency between these two approaches and propose a unified approach. The general motion model we adopt characterizes arbitrary 3-D steady motion. Under perspective projection, we derive an image motion equation that describes the spatio-temporal relation of gray-scale intensity in an image sequence, thus making the utilization of 3-D filtering possible. However, to accommodate this motion model, we need to extend the filter design to derive additional motion constraint equations. Using Hermite polynomials, we design differentiation filters, whose orthogonality and Gaussian derivative properties insure numerical stability; a recursive relation facilitates application of the general nonlinear motion model while separability promotes efficiency. The resulting algorithm produces accurate optical flow and other useful motion parameters. It is evaluated quantitatively using the scheme established by Barron et al. (1994) and qualitatively with real images.

The effect of spatial filtering on accuracy of motion constraint equation

The motion constraint equation has been widely investigated to find optical flow, but little attention has been paid to accuracy and reliability of the constraint equation. In this paper, the bound on accuracy of the constraint equation is derived and the effect of spatial filtering on the bound is examined.

Motion constraint equations based on constant image irradiance

From the basic assumption of constant image irradiance used by Horn and Schunck to derive the classic motion constraint equation, infinite sets of independent motion constraint equations are derived, relating image irradiance with limited order optic flow. Alternatives to the classic motion constraint equation are, furthermore, pointed out, relating optic flow with length of iso-irradiance contours, area and volume measures and their partial derivatives.

Multimodal estimation of discontinuous optical flow using Markov random fields

The estimation of dense velocity fields from image sequences is basically an ill-posed problem, primarily because the data only partially constrain the solution. It is rendered especially difficult by the presence of motion boundaries and occlusion regions which are not taken into account by standard regularization approaches. In this paper, the authors present a multimodal approach to the problem of motion estimation in which the computation of visual motion is based on several complementary constraints. It is shown that multiple constraints can provide more accurate flow estimation in a wide range of circumstances. The theoretical framework relies on Bayesian estimation associated with global statistical models, namely, Markov random fields. The constraints introduced here aim to address the following issues: optical flow estimation while preserving motion boundaries, processing of occlusion regions, fusion between gradient and feature-based motion constraint equations. Deterministic relaxation algorithms are used to merge information and to provide a solution to the maximum a posteriori estimation of the unknown dense motion field. The algorithm is well suited to a multiresolution implementation which brings an appreciable speed-up as well as a significant improvement of estimation when large displacements are present in the scene. Experiments on synthetic and real world image sequences are reported. >

Experimental evaluation of motion constraint equations

The motion constraint equation was originally proposed by Horn and Schunck in 1981 for the computation of optical flow. There have been arguments that this equation, E x u + E y v + E t = 0, is based on simplified assumptions which make it inaccurate for real world applications. Several alternative constraint equations have been proposed and are aimed at modeling motion more accurately. These equations have been used in the computation of optical flow. One basic question we address in this paper is the validity of these constraint equations. We apply the ray tracing technique to synthesize images for which to evaluate the three constraint equations proposed by Horn and Schunck, Schunck, and Nagel. Experimental results indicate that the original constraint equation appears to be more accurate than the proposed alternatives. Further work on the validation of constraint equations is much needed.

On the use of time varying shading and surface rim irradiance

The question of using time varying image irradiance, produced by a static object and a distant moving light source, and the question of using image irradiance of the visual rim, are addressed. Motion constraint equations relating image irradiance with light source position and motion, and with surface orientation and visual surface curvature, are derived. Possible linear determination schemes are pointed out.

Fundamentals of texture flow equations in vision calculus

The question of texture flow equations for a textured surface moving relatively to a perspective camera is considered. Motion constraint equations involving projected image texture density and optic flow are derived and briefly compared with the classic motion constraint equation involving photometric image irradiance and optic flow. It is pointed out that these flow equations are significantly different by nature and by content.

Algorithm-generating constraints and quantum equations of motion from the fermionic Hamiltonian of chiral two-dimensional QCD.

Working with the fermionic Hamiltonian of chiral two-dimensional QCD, we obtain, following a Dirac-type algorithm, the constraint structure and equations of motion on the quantum level. The computation of the anomalous Poisson brackets is based on a method proposed previously. Because of a remarkable cancellation of anomalous contributions, one recovers Maxwell's equations.

Image irradiance equations for a zooming camera

Image irradiance equations for a camera, zooming onto a static surface, are discussed. Surface range may be determined directly from image irradiance. The classic motion constraint equation is not valid during zooming; its exact and approximate extension are discussed.

Thermoelastic Analysis of Flexible Multibody Machine-Tool Mechanisms

This investigation is concerned with the thermoelastic analysis of flexible multibody machine-tool mechanisms using the finite-element method. Thermoelastic response of the machine tool is obtained by discretizing the machine tool into a number of simple elements and calculating the thermal stress and strain of each element under the average temperature rise. The generalized thermoelastic forces associated with the generalized elastic coordinates are then determined using the virtual work. The nonlinear dynamic behavior of the machine-tool mechanism due to the application of a constant cutting force as well as a chattering (dynamic) force, with and without thermal effects, is analyzed. The chattering force is obtained by considering the cutting force variations due to the variation of undeformed chip thickness and the rate of penetration of the tool, as a result of tool deformation. In this investigation, the machine-tool mechanism is considered as a multibody system consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Bending and axial deformation of the elastic bodies in the system are considered. Component mode synthesis techniques are employed in order to reduce the number of elastic coordinates and the system differential equations of motion and nonlinear algebraic constraint equations are written in terms of a coupled set of reference and modal elastic coordinates. The formulation is exemplified using a crank-shaper mechanism wherein the flexibility of the tool as well as the flexibility of the mechanism links are considered.

A motion constraint equation under space-varying or time-varying illumination

This paper presents a new motion constraint equation under space-varying or time-varying illumination and indicates that the motion constraint equation derived by Horn and Schunck is a special case of our equation.