Reasoning with uncertain points, straight lines, and straight line segments

Decisions based on basic geometric entities can only be optimal, if their uncertainty is propagated trough the entire reasoning chain. This concerns the construction of new entities from given ones, the testing of geometric relations between geometric entities, and the parameter estimation of geometric entities based on spatial relations which have been found to hold. Basic feature extraction procedures often provide measures of uncertainty. These uncertainties should be incorporated into the representation of geometric entities permitting statistical testing, eliminates the necessity of specifying non-interpretable thresholds and enables statistically optimal parameter estimation. Using the calculus of homogeneous coordinates the power of algebraic projective geometry can be exploited in these steps of image analysis. This review collects, discusses and evaluates the various representations of uncertain Preprint submitted to Elsevier 23 July 2009geometric entities in 2D together with their conversions. The representations are extended to achieve a consistent set of representations allowing geometric reasoning. The statistical testing of geometric relations is presented. Furthermore, a generic estimation procedure is provided for multiple uncertain geometric entities based on possibly correlated observed geometric entities and geometric constraints. Key words: spatial reasoning, uncertainty, homogeneous coordinates, geometric entitie

Markierungseinrichtung und Verfahren zur Markierung

Die Erfindung betrifft eine Markierungseinrichtung (1) mit zumindest einem Reflektor (2) und zumindest einer Haltevorrichtung (3), wobei die Haltevorrichtung (3) zumindest eine Basis (31) und zumindest einen Adapter (32) enthält, wobei die Basis (31) eine zumindest teiweise plane Unterseite (312) aufweist und dazu eingerichtet ist, auf den zu vermessenden Punkt (4) aufgelegt zu werden, der Adapter (32) dazu eingerichtet ist, auf der Basis (31) aufgelegt zu werden und der Reflektor (2) dazu eingerichtet ist, auf den Adapter (32) aufgelegt zu werden. Weiterhin betrifft die Erfindung ein Verfahren zur Markierung eines zu vermessenden Punktes

Utilizing the uncertainty of polyhedra for the reconstruction of buildings

The reconstruction of urban areas suffers from the dilemma of modeling urban structures in a generic or specific way, thus being too unspecific or too restrictive. One approach is to model and to instantiate buildings as arbitrarily shaped polyhedra and to recognize comprised man-made structures in a subsequent stage by geometric reasoning. To do so, we assume the existence of boundary representations for buildings with vertical walls and horizontal ground floors. To stay generic and to avoid the use of templates for pre-defined building primitives, no further assumptions for the buildings’ outlines and the planar roof areas are made. Typically, roof areas are derived interactively or in an automatic process based on given point clouds or digital surface models. Due to the mensuration process and the assumption of planar boundaries, these planar faces are uncertain. Thus, a stochastic geometric reasoning process with statistical testing is appropriate to detected man-made structures followed by an adjustment to enforce the deduced geometric constraints. Unfortunately, city models usually do not feature information about the uncertainty of geometric entities. We present an approach to specify the uncertainty of the planes corresponding to the planar patches, i.e., polygons bounding a building, analytically. This paves the way to conduct the reasoning process with just a few assumptions. We explicate and demonstrate the approach with real data

Design of orientation assessment functions for gestalt-grouping utilizing labeled sample-data

Psychological evidence is given that perceptual grouping is an important help for various visual tasks. Object recognition and land use classification from remotely sensed imagery is an example. In machine vision, such a grouping process can be implemented by coding Gestalt laws such as proximity, symmetry, or good continuation. Since geometric relations are rarely fulfilled exactly, soft membership functions are utilized called Gestalt assessments. Hierarchical grouping is possible on increasing scales. Such an approach to hierarchical Gestalt grouping is modified in this paper. In its original form, the approach uses rather heuristic default assessment functions, which are a possible choice as long as no labeled example data are given. The assessment functions can be parameterized so as to improve the perceptual grouping, guiding it by the Gestalten salient to human perception. To this end, we use orientation statistics from the publicly available data set given for the ICCV symmetry recognition competition 2017. Also, with a particular recognition task at hand, labeled example data can serve as the desired foreground. Here we use the ground-truth layer for buildings of the Vaihingen benchmark of the ISPRS. A mixture distribution containing two von Mises-distributions and the uniform component for the clutter in the background is fitted using expectation maximization

Geometric reasoning with uncertain polygonal faces

The reconstruction of urban areas suffers from the dilemma of modeling urban structures in a generic or specific way, thus being too unspecific or too restrictive. One approach to overcome this dilemma is to model and to instantiate buildings as arbitrarily shaped polyhedra and to recognize man-made structures in a subsequent stage by geometric reasoning. Thus, the existence of unconstrained boundary representations for buildings is assumed. To stay generic and to avoid the use of templates for pre-defined building primitives, no assumptions for the buildings' outlines and the planar roof areas are made. Typically, roof areas are derived interactively or in an automatic process based on given point clouds or digital surface models. Due to the measurement process and the assumption of planar boundaries, these planar faces are uncertain. Thus, a stochastic geometric reasoning process with statistical testing is appropriate to detected man-made structures followed by an adjustment to enforce the deduced geometric constraints. Unfortunately, city models usually do not feature information about the uncertainty of geometric entities. We present an approach to specify the uncertainty of the planes corresponding to the planar patches, i.e., polygons bounding a building, analytically. This paves the way to conduct the reasoning process with just a few assumptions. We describe and demonstrate the approach with real data

CONTINUOUS SELF-CALIBRATION AND EGO-MOTION DETERMINATION OF A MOVING CAMERA BY OBSERVING A PLANE

In many vision applications the depth relief of an observed scene is small compared with the extent of the image. Beside man-made environments these scenes may be approximated by a plane. Due to environmental influences camera parameters can gradually change which motivates the need for a continuous self-calibration. Based on the theory of recursive parameter estimation we present an update scheme for the parameters to handle endless video streams in real time. The geometric parametrization of the frame to frame homographies allows to incorporate information of other sensors. The application is the visual navigation of robots and unmanned aerial vehicles moving above nearly planar environments. The approach will be empirically evaluated with a synthetic data set and demonstrated with a real data set. A typical example is given by a real data set of an indoor robot observing the ground floor with one camera.

Simplification of polygonal chains by enforcing few distinctive edge directions

Simplification of polygonal chains by reducing the number of vertices becomes challenging when additionally dominant edge expansions of the polygonal chains shall be exposed. Such simplifications are often sought in order to generalize polygonal chains representing borders or medial axes of man-made structures such as buildings or road networks, observed in aerial images. In this paper, we present two methods that reduce the number of vertices in polygonal chains meanwhile featuring additional properties: First, the resulting polygonal chains are irrespective of coordinate axes as pixel-based approaches tend to produce. Second, the simplified chains keep the rough shape of the initial ones and emphasize dominating edge expansions. Optionally, detected perpendicularities may be enforced. Third, polygons with holes are supposed to exhibit parallel segments in interior and exterior polygonal chains. Our methods treat the associated polygonal chains simultaneously by emphasizing common distinctive directions