Coplanar Points Research Articles

Influence of Solar Photogravitation on the Motion of Comet Hale-Bopp

On the basis of photogravitational theory, the influence of solar photogravitation on the motion of comet Hale-Bopp is discussed. It is shown that for plasma tail of comet Hale-Bopp there is only one collinear point L3 and for a few of dust tail there can be two coplanar points L6, L7 and three collinear points L21, L22, L3. The regions in which the coplanar points exist and the dust grain of comet Hale-Bopp can move are also obtained.

On the correspondence problem for wide angular separation of non-coplanar points

We describe a solution to the problem of stereo matching for non-coplanar points and wide angular separation of uncalibrated cameras. The image projections of four corresponding coplanar points, with no three collinear, are assumed to be known. The ‘virtual image’ that results by 2D warping of the first image towards the second one according to the homography of a known plane in 3D is used. Then, instead of trying to match points between the two images, we exploit the fact that all corresponding points between the virtual and the second image are in alignment. It is shown how the matching problem can be cast into an optimisation one, and a cost function for evaluating each possible solution is derived. For increased robustness to outliers a Hough-Transform-like algorithm is suggested for obtaining the optimal solution, and a randomised scheme is shown to be efficient for practical applications. Experimental results on both synthetic and real data are also included.

Efficient Invariant Representations

Invariant representations are frequently used in computer vision algorithms to eliminate the effect of an unknown transformation of the data. These representations, however, depend on the order in which the features are considered in the computations. We introduce the class of projective/permutation p^2-invariants which are insensitive to the labeling of the feature set. A general method to compute the p^2-invariant of a point set (or of its dual) in the n-dimensional projective space is given. The one-to-one mapping between n + 3 points and the components of their p^2-invariant representation makes it possible to design correspondence algorithms with superior tolerance to positional errors. An algorithm for coplanar points in projective correspondence is described as an application, and its performance is investigated. The use of p^2-invariants as an indexing tool in object recognition systems may also be of interest.

Distributions of projective invariants and model-based machine vision

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

Distributions of projective invariants and model-based machine vision

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

The Contribution of Provisual Actions to Judgments of form in Depth

An action is ‘provisual’ if it is meant to improve the information that can be obtained through vision. Eye and head movements are the most studied examples, but it is more a general phenomenon, including for example manoeuvring for position. The effect of allowing provisual manipulation was compared with more traditional ways of facilitating perception of form in depth. Stimuli were parallel projections of a square and a rectangle (whose proportions ranged from 1 : 1.1 to 1 : 1.4), both slanted in depth, presented side by side on a computer screen. Nine subjects judged which was which. Information varied along three dimensions: (1) surface type—outline, shaded, and textured; (2) dimensionality—structures were either 2-D, or had a ‘leg’ at each corner (normal to the surface); (3) movement—static, passive (rotary) motion, and provisual (where subjects manoeuvred the objects using a mouse). Surface effects were weak ( p=0.047). Dimensionality had a stronger effect ( p=0.0003). The effect of passive movement was comparable, but provisual manipulation gave much better discrimination, irrespective of surface or dimensionality, reaching a ceiling with rectangle proportions around 1 : 1.3. The subjects' usual strategy was to manoeuvre structures into a roughly frontoparallel position. The judgment is almost trivial in that position, but finding it is an interesting theoretical problem—particularly given the demonstration that it can be achieved from parallel projections of a few coplanar points. A model is being developed based on neural networks capable of learning the probabilities associated with viewpoints and trajectories.

Electric field line diagrams don’t work

Electric fields produced by coplanar point charges have often been represented by field line diagrams that depict two-dimensional slices of the three-dimensional field. Serious problems with these ‘‘conventional’’ field line diagrams (CFLDs) have been overlooked. Two of these problems, ‘‘equatorial clumping’’ and ‘‘false monopole moment,’’ occur because a two-dimensional slice lacks information vital to the accurate representation of an inherently three-dimensional field. Equatorial clumping causes most CFLDs to exhibit unphysical behavior such as irregular spacing between field lines terminating on negative charges. CFLDs can also mistakenly indicate that a neutral charge distribution has a significant monopole moment. Such phenomena make the visual estimation of local field strengths impossible and render CFLDs of little utility for representing three-dimensional fields. While these ‘‘projection’’ problems can be avoided by using two-dimensional field line diagrams to represent two-dimensional (1/r) electric fields, or by using three-dimensional field line diagrams to represent three-dimensional fields, other forms of distortion generally remain.

Three‐dimensional measurement using a single image

This study describes a new measurement technique using a single 2-D image of four coplanar points, which are arranged in a square of known size, to measure geometric features on an object, some of which may be hidden from the view of the camera. A new method is also proposed that uses simple equations and an iteration technique to find 3-D coordinates of points on the object relative to a reference point on the object. © 1996 Society of Photo-Optical Instrumentation Engineers. Subject terms: three-dimensional measurement, perspective four-point problem, image analysis.

Iterative Pose Estimation Using Coplanar Feature Points

This paper presents a new method for the computation of the position and orientation of a camera with respect to a known object, using four or morecoplanarfeature points. Starting with the scaled orthographic projection approximation, this method iteratively refines up to two different pose estimates, and provides an associated quality measure for each pose. When the camera distance is large compared with the object depth, or when the accuracy of feature point extraction is low because of image noise, the quality measures for the two poses are similar, and the two pose estimates are plausible interpretations of the available information. In contrast, known methods using a closed form pose solution for four coplanar points are not robust for distant objects in the presence of image noise because they provide only one of the two possible poses and may choose the wrong pose.

New algorithm for calculating an invariant of 3D point sets from a single view

The invariant used as an index has shown many advantages over the pose dependent methods in model-based object recognition. Although perspective, or even weak perspective invariants do not exist for general three-dimensional point sets from a single view, invariants do exist for structured three-dimensional point sets. However, such invariants are not easy to derive. A new interpretation of calculating invariants for a special structure of three-dimensional objects is presented. The 3D invariant structure proposed by Rothwell requires seven points that lie on the vertices of a six-sided polyhedral and is applicable to position free objects. In comparison, the proposed algorithm requires only six points on adjacent (virtual) planes that provide two sets of four coplanar points and does not require the position free condition. Hence it is applicable to a wider class of objects. The algorithm is demonstrated on images from real scenes.

Quantitative planar region detection

This paper presents a means of segmenting planar regions from two views of a scene using point correspondences. The initial selection of groups of coplanar points is performed on the basis of conservation of two five point projective invariants (groups for which this invariant is conserved are assumed to be coplanar). The five point correspondences are used to estimate a projectivity which is used to predict the change in position of other points assuming they lie on the same plane as the original four. The variance in any points new position is then used to define a distance threshold between actual and predicted position which is used as a coplanarity test to find extended planar regions. If two distinct planar regions can be found then a novel motion direction estimator suggests itself. The projection of the line of intersection of two planes in an image may also be recovered. An analytical error model is derived which relates image uncertainty in a corner's position to genuine perpendicular height of a point above a given plane in the world. The model may be used for example to predict the performance of given stereo ground plane prediction system or a monocular drivable region detection system on and AGV. The model may also be used in reverse to determine the camera resolution required if a vehicle in motion is to resolve obstacles of a given height a given distance from it.

Influence of temperature and interelectrode distance on the negative differential resistance in metal-chalcogenide glassy semiconductors

An investigation has been carried out, to elucidate some aspects of the current-controlled negative differential resistance (CCNDR) effect in bulk metal-chalcogenide glassy semiconductors. Because this phenomenon has been shown to be mainly of a thermal nature, a model, from the thermodynamic point of view, was developed, including some aspects related to the thermistors theory. The main conclusion from this model is the appearance of a current filament, which showed up when the material switched from the high electrical resistance to the low electrical resistance state, forming a crystalline filament between both electrodes. The variation of the CCNDR parameters with temperature and interelectrodic distance was studied, using both coplanar point electrodes and coplanar disc electrodes. The experimental results show a good agreement with the expected behaviour from the proposed thermal model (especially when natural convection was considered as the heat-exchanging process between the material and the ambient surrounding). In addition an algorithm was found to simulate the phenomenon computationally, using the experimentally determined physical parameters for the samples under study.

Point configuration invariants under simultaneous projective and permutation transformations

The projective invariants used in computer vision today are permutation-sensitive since their value depends on the order in which the features were considered in the computation. We derive, using tools from representation theory, the projective and permutation ( p 2) invariants of the four collinear and the five coplanar points configurations. The p 2-invariants are insensitive to both projective transformations and changes in the labeling of the points. When used as model database indexing functions in object recognition systems, the p 2-invariants yield a significant speedup. Permutation invariants for affine transformations are also discussed.

A position estimation system for mobile robots using a monocular image of a 3-D landmark

SUMMARYThis paper presents an absolute position estimation system for a mobile robot moving on a flat surface. In this system, a 3-D landmark with four coplanar points and a non-coplanar point is utilized to improve the accuracy of position estimation and to guide the robot during navigation. Applying theoretical analysis, we investigate the image sensitivity of the proposed 3-D landmark compared with the conventional 2-D landmark. In the camera calibration stage of the experiments, we employ a neural network as a computational tool. The neural network is trained from a set of learning data collected at various points around the mark so that the extrinsic and intrinsic parameters of the camera system can be resolved. The overall estimation algorithm from the mark identification to the position determination is implemented in a 32-bit personal computer with an image digitizer and an arithmetic accelerator. To demonstrate the effectiveness of the proposed 3-D landmark and the neural network-based calibration scheme, a series of navigation experiments were performed on a wheeled mobile robot (LCAR) in an indoor environment. The results show the feasibility of the position estimation system applicable to mobile robot's real-time navigation.

Planar region detection and motion recovery

This paper presents a means of segmenting planar regions from two views of a scene using point correspondences. The initial selection of groups of coplanar points is performed on the basis of conservation of two five point projective invariants (groups for which this invariant is conserved are assumed to be coplanar). The correspondences for four of the five points are used to define a projectivity which is used to predict the change in position of other points assuming they lie on the same plane as the original four. A distance threshold between actual and predicted position is used to find extended planar regions. If two distinct planar regions can be found then a novel motion direction estimator suggests itself.

Motion and structure from point correspondences with error estimation: planar surfaces

The determination of motion and structure of a planar scene from monocular image sequences, is studied. A new and simpler linear algorithm, that gives closed-form solutions for motion and structure parameters using point correspondences between two images, assuming that the coplanar points undergo a rigid motion in 3-D space, is presented. A series of analytical results is established. From two perspective views of a planar scene which is undergoing a rigid motion, there are generally two (normalized) interpretations for motion parameters and the positions of the object plane. These two interpretations, one vertical and the other illusive, are both valid in the sense that they render the same pair of images. The authors identify all the special cases in which the number of interpretations is not two, and derive necessary and sufficient geometrical conditions for those special cases to occur. The approach to error estimation is based on the first-order perturbation. The estimated errors provide quantitative assessment for the accuracy of the solutions. They also indicate degenerate or nearly degenerate configurations in the presence of noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The implementation of an algorithm to find the convex hull of a set of three-dimensional points

A detailed description of the implementation of a three-dimensional convex hull algorithm is given. The problems experienced in the production and testing of a correct and robust implementation of a geometric algorithm are discussed. Attention is paid to those issues that are often brushed over in the theoretical descriptions but cause errors in a real computation. These include degeneracies such as coplanar points, floating-point errors, and other special, but not necessarily degenerate, cases.

Bond Stress/Deformation in Pull‐Out Masonry Specimens

An experimental investigation to determine bond stress, free-end slip, and relative deformation between a reference plane in the masonry and an originally coplanar point on the steel (slip) is conducted. Stack pull-out specimens constructed of grouted nominal 6-in concrete or clay masonry units, reinforced with #4 or #7 bars, are subjected to monotonic or cyclic loadings. The method for evaluating slip incorporated slip-wires connected to linear variable differential transformers (LVDTs). Strain gages provide strain in the steel bars from which stresses in the reinforcing bar and bond stresses can be calculated. Free-end slip information is also collected. These data are useful in developing an understanding of the physical mechanism associated with bond, slip, and bond deterioration. The bond stress, which develops between the reinforcement and the surrounding grout, is a complex phenomenon dominated by the interlocking action of the reinforcing bar ribs embedded in the grout mass. To determine bond stresses and slip, six specimens are constructed and tested. Three specimens of 16 in length are reinforced with a #4 bar, and three, of 32 in length are reinforced with a #7 bar. The bond stress and slip distribution curves are evaluated from experimental data. In addition, short embedment specimens, consisting of prisms of 6-in concrete masonry units (nominal size 6×8×8) are tested. The embedment length is varied from 1 to 6 in. Ten specimen configurations was replicated three times. The nominal bond stress and free-end slip are obtained.

A general photogrammetric method for determining object position and orientation

The author presents a general method for determining the three-dimensional position and orientation of an object relative to a camera based on a two-dimensional image of known feature points located on the object. In contrast to the conventional approach, the method does not make use of the collinearity condition. Instead, the algebraic structure of the problem is fully exploited in arriving at a solution which is independent of the configuration of the feature points. Although the method is applicable to any number or distribution of feature points, no more than five points are needed from a numerical standpoint; three or four points will typically suffice. A necessary condition for the existence of the solution is obtained for the first time, and a rigorous proof of uniqueness is presented for the case of four coplanar points. Simulation results have shown that, in the case of four feature points, nonplanar configurations generally outperform coplanar point sets in both accuracy and robustness in the presence of image noise. >

The 3‐vertex single source Weber location problem

An analytic solution is obtained for the location of a source point when the sum of the weighted distances from three fixed coplanar points is a minimum, in terms of the Cartesian coordinates of the fixed points and the weights. The solution is applied when the weights are discontinuous across circles with centres at the fixed points, and also for a class of problems when the weights are continuous, but more than one source and more than three fixed points occur.