Trihedral Vertices Research Articles

Entanglement susceptibilities and universal geometric entanglement entropy

The entanglement entropy (EE) can measure the entanglement between a spatial subregion and its complement, which provides key information about quantum states. Here, rather than focusing on specific regions, we study how the entanglement entropy changes with small deformations of the entangling surface. This leads to the notion of entanglement susceptibilities. These relate the variation of the EE to the geometric variation of the subregion. We determine the form of the leading entanglement susceptibilities for a large class of scale invariant states, such as groundstates of conformal field theories, and systems with Lifshitz scaling, which includes fixed points governed by disorder. We then use the susceptibilities to derive the universal contributions that arise due to non-smooth features in the entangling surface: corners in 2d, as well as cones and trihedral vertices in 3d. We finally discuss the generalization to Renyi entropies.

Wireframe Projections: Physical Realisability of Curved Objects and Unambiguous Reconstruction of Simple Polyhedra

The reconstruction of an object from a single 2D projection of a 3D wireframe model is a vision problem with applications in CAD/CAM and computer graphics.We propose an algorithm for the interpretation of wireframe projections based on assigning semantic and numerical depth labels to lines. This method allows us to state necessary and sufficient conditions for the physical realisability of a wireframe projection of a curved object. The presence of linear features provides further constraints on the positions of object vertices. For example, each straight line gives rise to a coplanarity constraint between a set of object vertices. We show that extra information, such as vanishing points, parallel lines or user-entered depth-parity information, is sufficient to uniquely determine the face-circuits in wireframe projections of polyhedra with simple trihedral vertices. In fact, a polyhedron with simple trihedral vertices can be unambiguously reconstructed from its 3D wireframe model.

Recognition of polyhedral objects using triplets of projected spatial edges based on a single perspective image

A novel and intuitive mathematical formulation is proposed for the interpretation of the geometric relationships of a triple of spatial edges and their perspective projection forming image lines. No restriction is imposed on the configuration of the triple of spatial edges. An eighth-degree polynomial equation explicitly defined by the space angles between the corresponding three spatial edges in an object frame is derived. The crux of this representation is that the angular attributes of pairs of spatial edges are expressed in an object-independent coordinate system. An effective hypothesis generation scheme is proposed which can take advantage of the representation. In the proposed recognition framework, triples of connected spatial edges and trihedral vertices are employed as key features for model invocation, hypothesis generation and pose estimation. Extensive experiments are performed to verify the plausibility and robustness of the proposed recognition methods. In general, the success rate for identifying and localising polyhedral objects using the proposed recognition method is relatively high.

A comparative study of the robustness of two pose estimation techniques

The paper presents an analysis of the stability of pose estimation. Stability is defined as sensitivity of the pose parameters towards noise in image features used for estimating pose. The specific emphasis of the analysis is on determining {how the stability varies with viewpoint} relative to an object and to understand the relationships between object geometry, viewpoint, and pose stability. Two pose estimation techniques are investigated. One uses a numerical scheme for finding pose parameters; the other is based on closed form solutions. Both are “pose from trihedral vertices” techniques, which provide the rotation part of object pose based on orientations of three edge segments. The analysis is based on generalized sensitivity analysis propagating the uncertainty in edge segment orientations to the resulting effect on the pose parameters. It is shown that there is a precomputable, generic relationship between viewpoint and pose stability, and that there is a drastic difference in stability over the range of viewpoints. This viewpoint variation is shared by the two investigated techniques. Additionally, the paper offers an explicit way to determine the most robust viewpoints directly for any given vertex model. Experiments on real images show that the results of the work can be used to compute the variance in pose parameters for any given pose. For the predicted {instable} viewpoints the variance in pose parameters is on the order of 20 (degrees squared), whereas the variance for robust viewpoints is on the order of 0.05 (degrees squared), i.e., two orders of magnitude difference.

Stereo calibration from correspondences of OTV projections

Stereo images have to be calibrated before stereo vision can recover three-dimensional information of the imaged scene. Position constraints over image point correspondences are traditionally used to solve the calibration problem. A method is described that uses angle constraints over correspondences of a particular type of image features, the projections of orthogonal trihedral vertices (OTV), for calibration. Computations of the rotation matrix and the translation vector are separable and the method has a closed-form solution. It also requires correspondences of only two vertex projections at minimum to recover all the transformation parameters which are recoverable from a stereo image pair. Extensive experimental results, including those on real images, are presented and they show that use of angle constraints is generally more accurate than position constraints alone.

Localization and noise in edge detection

Two aspects of edge detection are analyzed, namely accuracy of localization and sensitivity to noise. The detection of corners and trihedral vertices is analyzed for gradient schemes and zero-crossing schemes. It is shown that neither scheme correctly detects corners of trihedral vertices, but that the gradient schemes are less sensitive to noise. A simple but important conclusion is that the noise present in digital images of typical indoor scenes is small and the signal-to-noise ratio is high. The noise present in digital images is so small as to make the performances of a variety of filters almost indistinguishable. As a consequence small filters can be used and the exact shape of the filter is not critical. >

Configuration Estimate of Trihedral Vertex from a Single View

AbstractIn a previous paper, we introduced “perspective angle transform” (PAT) to analyze the relationship between an angle seen under the condition of “central projection” and an actual angle. The present paper discusses mainly (based on the previous paper) the connotative interpretation of trihedral vertices by using information on an image. This is a problem in reconstructing a three‐dimensional (3‐D) configuration of the trihedral vertex from a central projection image in which all the angles are known. An algorithm to express this problem is introduced so that the problem can be analyzed. An algebraic solution based on the perspective angle transform is given for a case where at least two of the three angles in a trihedral vertex are right angles (i.e., a two‐‐orthogonal vertex or a three‐orthogonal vertex). The number of solutions and the possibility of their interpretations are also analyzed. A one‐dimensional search algorithm (or the examination method for the hypothesis of a normal image) based on the interpretation of the two‐orthogonal vertex is introduced for interpretation of a trihedral vertex. The number of solutions is examined statistically by using a computer simulation. This paper will be useful for analyzing scenes and robot‐vision, whichever images employ central or parallel projection.