Geometric Inconsistencies Research Articles

Optimal simplification of polygonal chains for subpixel-accurate rendering

For a given polygonal chain, we study the min-# problem, which consists of finding an approximate and ordered subchain with a minimum number of vertices under a given approximation criterion. We propose the first near-linear time algorithm for the min-# problem that ensures optimality in the number of vertices and that retains the shape of the input polygonal chain at the same time. Previous algorithms with near-linear time performance produced geometrical inconsistencies and former methods used to preserve the features of the original chain required a quadratic time complexity. When we set the approximation error equal to the pixel pitch, our simplification criterion guarantees that the rendering of the simplification lies at most half a pixel away from the original chain, contrary to other methods that do not offer a direct control over the rendering. Thus, we avoid producing visible topological inconsistencies. Moreover, our method is based on basic data structures, which leads to a simple and efficient implementation.

Active Contours Under Topology Control—Genus Preserving Level Sets

We present a novel framework to exert topology control over a level set evolution. Level set methods offer several advantages over parametric active contours, in particular automated topological changes. In some applications, where some a priori knowledge of the target topology is available, topological changes may not be desirable. This is typically the case in biomedical image segmentation, where the topology of the target shape is prescribed by anatomical knowledge. However, topologically constrained evolutions often generate topological barriers that lead to large geometric inconsistencies. We introduce a topologically controlled level set framework that greatly alleviates this problem. Unlike existing work, our method allows connected components to merge, split or vanish under some specific conditions that ensure that the genus of the initial active contour (i.e. its number of handles) is preserved. We demonstrate the strength of our method on a wide range of numerical experiments and illustrate its performance on the segmentation of cortical surfaces and blood vessels.

Geometry of giant star model atmospheres: a consistency test

We investigate the effect of a geometric inconsistency in the calculation of synthetic spectra of giant stars. Spectra computed with model atmospheres calculated in spherical geometry while using the plane-parallel approximation for line formation calculations (s_p), as well as the fully plane-parallel case (p_p), are compared to the consistently spherical case (s_s). We present abundance differences for solar metallicity models with Teff ranging from 4000 to 6500 K and logg from 0.5 to 3.0 [cgs]. The effects are smaller for s_p calculations (-0.1 dex in the worst case) than for the p_p case (up to +0.35 dex for minority species and at most -0.04 dex for majority species), both with respect to the s_s case. In the s_p case the differences increase slightly with temperature, while in the p_p case they show a more complex behaviour. In both cases the effects decrease with increasing logg and increase with equivalent width. Thus, within the parameter range of F, G and K giants, consistency seems to be less important than using a spherical model atmosphere. The abundance differences due to sphericity effects presented here can be used for error estimation in abundance studies relying on plane-parallel modelling.

DIRAC FIELDS ON SPACELIKE HYPERSURFACES, THEIR REST-FRAME DESCRIPTION AND DIRAC OBSERVABLES

Grassmann-valued Dirac fields together with the electromagnetic field (the pseudo-classical basis of QED) are reformulated on spacelike hypersurfaces in Minkowski space–time and then restricted to Wigner hyperplanes to get their description in the rest-frame Wigner-covariant instant form of dynamics. The canonical reduction to the Wigner-covariant Coulomb gauge is done in the rest frame. It is shown, on the basis of a geometric inconsistency, that the description of fermions is incomplete, because there is no bosonic carrier of the spin structure describing the trajectory of the electric current in Minkowski space–time, as it was already emphasized in connection with the first quantization of spinning particles in a previous paper.

Least squares restoration of Tertiary thrust sheets in map view, Tajik depression, central Asia

The Tajik depression, located west of the Pamirs and south of the Tien Shan, is a compressional intermontane basin, bounded by basement overthrusts and filled with Mesozoic and Cenozoic sediments. The internal structure is typical of a thin‐skinned fold and thrust belt. Kinematic data available in the literature suggest that indentation of the Pamirs into Asia during the Cenozoic collision of India and Asia has been accommodated in various ways within the depression, including westward extrusion, thickening combined with wrenching along N‐S folds and thrusts, and counterclockwise rotations. These various deformation processes can be analyzed and quantified by reconstruction of the predeformed state of the depression. However, the combination of thrusting, wrenching and block rotations implies a nonplane deformation, which cannot be restored properly using balanced cross sections alone. We have therefore developed a numerical method for restoration of stratigraphic surfaces, designed for regions of nonplane compressional tectonics. The deformed region is represented in map view as a mosaic of fault‐bounded blocks, overlapping each other along the faults. Blocks are separately unfolded and then numerically packed together by least squares minimization of overlaps, yielding fields of finite horizontal translations and rotations about vertical axes. To analyze the deformation postdating the collision of India and Asia, we have restored a stratigraphic surface at the base of the Cenozoic. First, in order to test the numerical method, we restored a map that had previously been restored by a purely manual method. Restoration of a second map, drawn from newly available subsurface data, leads to geometrical inconsistencies: overlaps and gaps, which cannot be reduced, remain in the restored state. After correction of these inconsistencies, restoration yields a complex mode of deformation for the depression. Individual thrust slices have undergone counterclockwise rotations about vertical axes, the magnitudes increasing from west to east, up to a maximum of 40° near the Pamirs. Horizontal shortening is 150 km (35%) in the center of the depression and 240 km (85%) in the northeastern part, between the Pamirs and the Tien Shan. Strike slip motions, commonly not revealed by balancing cross sections, are associated with thrusting on faults striking N‐S. Independent paleomagnetic data and slip directions on small‐scale faults provide positive checks on the results of our restoration.

Algebraic geometry and group theory in geometric constraint satisfaction for computer-aided design and assembly planning

Mechanical design and assembly planning inherently involve geometric constraint satisfaction or scene feasibility (GCS/SF) problems. Such problems imply the satisfaction of proposed relations placed between undefined geometric entities in a given scenario. If the degrees of freedom remaining in the scene are compatible with the proposed relations or constraints, a set of entities is produced that populate the scenario satisfying the relations. Otherwise, a diagnostic of inconsistency of the problem is emitted. This problem appears in various forms in assembly planning (assembly model generation), process planning, constraint driven design, computer vision, etc. Previous attempts at solution using separate numerical, symbolic or procedural approaches suffer serious shortcomings in characterizing the solution space, in dealing simultaneously with geometric (dimensional) and topological (relational) inconsistencies, and in completely covering the possible physical variations of the problem. This investigatio...

Geometric consistency for regression model estimation and testing in climatology and meteorology

Abstract The effects of outliers on linear regression are examined. The sensitivity of classical least‐squares (LS) procedures to outliers is shown to be associated with the geometric inconsistency between the data space and the analysis space. This is illustrated for both estimation and inference. A geometrically consistent procedure based on the Euclidean distance is proposed. This procedure involves the least absolute deviation (LAD) regression and a new permutation test for matched pairs (PTMP). Comparisons made with LS techniques demonstrate that the proposed procedure is more resistant to the existence of outliers in the data set and leads to more intuitive results. Applications and illustrations using meteorological and climatological data are also discussed.

The Use of Objects' Faces in Interpreting Line Drawings

The two parts, which this paper is composed of, deal each with scene interpretation via gaining understanding of the faces of the objects in the scene. The first part extends the set of rules defined in a previous work regarding the assembling of all lines belonging to the same face. The set of rules, originally defined for curved object, can be extended if we confine ourselves to polyhedra. In the second part, a new concept is defined and developed, which leads to a new way of looking at polyhedral line drawings. It puts under the same roof almost all consistency checks known for polyhedra, in a natural and simple way. Geometric inconsistencies as well as interpretations inconsistencies are treated uniformly and in a straightforward manner. Through this concept a way is suggested for acquiring some understanding of back faces, and for suggesting plausible interpretation for them. The generality of this concept is demonstrated through the fact that previously known catalogs of labeled junctions can be derived directly from this concept.

Folded Faults and Sequence of Thrusting in Alberta Foothills

Thrust faults in the Alberta Foothills were folded by development of step-thrusts and other thrust-generated structures beneath them. Because step-thrusting is a pattern of faulting found in layered rocks of varying competence, folded faults may be expected in any thrust belt where the stratigraphic sequence is in-homogeneous and higher thrusts are emplaced before underlying ones. By invoking folded faults, several structures in the Alberta Foothills can be reinterpreted without the geologic and geometric inconsistencies inherent in existing interpretations. Although a west-to-east sequence of thrust emplacement in the Alberta Foothills is implicit in the mechanics of folding of thrusts, some structural relations indicate an east-to-west sequence. The apparent contradiction may be resolved by the proposition that the time taken for a thrust to form, spreading both across and along strike, is greater than the time interval between initiation of successive thrusts.

Operating Speed Based Alignment Consistency Evaluation Using Driving Simulator: Case Studies from Taigan Freeway in Jiangxi, China

Geometric design consistency refers to the conformance of a proposed or existing freeway’s geometry with the driver’s expectancy. Considerable efforts have been undertaken to develop potential quantitative consistency measures and models to estimate them. However, little work has been focused on the facts of freeways in China, especially in mountainous areas. The primary objective of this study is to examine and quantify the design consistency of a freeway, using an interactive driving simulator, so as to identify any inconsistency on the roadway geometry and improve its overall safety performance. More specifically, the techniques to evaluate the consistency of a geometric design depend on three criteria: design consistency, operating speed consistency, and consistency in driving dynamics, as well as three levels of consistency: good, acceptable and poor. In the case of Taigan Freeway, a part of G45 Daguang Freeway in Jiangxi, China, a 21km long segment has been divided into 38 sections including 22 curves and 16 tangents, and transferred into the graphics models in the simulator system. There were 42 drivers required to take part in the simulation experiment and the speed, location of vehicle, and other real time data were recorded as well. The findings have shown that these proposed measures and standards can identify a geometric inconsistency more effectively when there is a large rate of change in the alignment elements for a successive roadway segment.