Planar shape manipulation using approximate geometric primitives

We present robust algorithms for set operations and Euclidean transformations of curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is \bigo(n\log n+k) for an input of size $n$ with k=\bigo(n^2) consistency violations. The output is as accurate as the geometric primitives. We validate our algorithms in floating point using sequences of six set operations and Euclidean transforms on shapes bounded by curves of algebraic degree~1 to~6. We test generic and degenerate inputs. Keywords: robust computational geometry, plane subdivisions, set operations

Drawing Planar Graphs with Few Geometric Primitives

We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2017

Determining geometric primitives for a 3D GIS : easy as 1D, 2D, 3D?

Acquisition techniques such as photo modelling, using SfM-MVS algorithms, are being applied increasingly in several fields of research and render highly realistic and accurate 3D models. Nowadays, these 3D models are mainly deployed for documentation purposes. As these data generally encompass spatial data, the development of a 3D GIS would allow researchers to use these 3D models to their full extent. Such a GIS would allow a more elaborate analysis of these 3D models and thus support the comprehension of the objects that the features in the model represent. One of the first issues that has to be tackled in order to make the resulting 3D models compatible for implementation in a 3D GIS is the choice of a certain geometric primitive to spatially represent the input data. The chosen geometric primitive will not only influence the visualisation of the data, but also the way in which the data can be stored, exchanged, manipulated, queried and understood. Geometric primitives can be one-, two- and three-dimensional. By adding an extra dimension, the complexity of the data increases, but the user is allowed to understand the original situation more intuitively. This research paper tries to give an initial analysis of 1D, 2D and 3D primitives in the framework of the integration of SfM-MVS based 3D models in a 3D GIS

Semi-automated stereoradiographic upper limb 3D reconstructions using a combined parametric and statistical model: a preliminary study

PURPOSE: Quantitative assessment of 3D clinical indices may be crucial for elbow surgery planning. 3D parametric modeling from bi-planar radiographs was successfully proposed for spine and lower limb clinical investigation as an alternative for CT-scan. The aim of this study was to adapt this method to the upper limb with a preliminary validation. METHODS: CT-scan 3D models of humerus, radius and ulna were obtained from 20 cadaveric upper limbs and yielded parametric models made of geometric primitives. Primitives were defined by descriptor parameters (diameters, angles...) and correlations between these descriptors were found. Using these correlations, a semi-automated reconstruction method of humerus using bi-planar radiographs was achieved: a 3D personalized parametric model was built, from which clinical parameters were computed [orientation and projections on bone surface of trochlea sulcus to capitulum (CTS) axis, trochlea sulcus anterior offset and width of distal humeral epiphysis]. This method was evaluated by accuracy compared to CT-scan and reproducibility. RESULTS: Points-to-surface mean distance was 0.9 mm (2 RMS = 2.5 mm). For clinical parameters, mean differences were 0.4-1.9 mm and from 1.7° to 2.3°. All parameters except from angle formed by CTS axis and bi-epicondylar axis in transverse plane were reproducible. Reconstruction time was about 5 min. CONCLUSIONS: The presented method provides access to morphological upper limb parameters with very low level of radiation. Preliminary in vitro validation for humerus showed that it is fast and accurate enough to be used in clinical daily practice as an alternative to CT-scan for total elbow arthroplasty pre operative evaluation

3d Modelling with Linear Approaches Using Geometric Primitives

In this paper, we study linear approaches for 3D model acquisition from non-calibrated images. First, the intrinsic andextrinsic camera calibration is taken into consideration. In particular, we study the use of a specific calibrationprimitive: the parallelepiped. Parallelepipeds are frequently present in man-made environments and naturally encode theaffine structure of the scene. Any information about their euclidean structure (angles or ratios of edge lengths), possiblycombined with information about camera parameters is useful to obtain the euclidean reconstruction. We propose anelegant formalism to incorporate such information, in which camera parameters are dual to parallelepiped parameters,i.e. any knowledge about one entity provides constraints on the parameters of the others. Consequently, an image aparallelepiped with known Euclidean structure allows to compute the intrinsic camera parameters, and reciprocally, acalibrated image of a parallelepiped allows to recover its euclidean shape (up to size). On the conceptual level, thisduality can be seen as an alternative way to understand camera calibration: usually, calibration is considered to beequivalent to localizing the absolute conic or quadric in an image, whereas here we show that other primitives, such ascanonic parallelepipeds, can be used as well. While the main contributions of this work concern the estimation ofcamera and parallelepiped parameters. The complete system allows both calibration and 3D model acquisition from asmall number of arbitrary images with a reasonable amount of user interaction