Extracting geometric models from medieval moulding profiles for case-based reasoning

This paper presents an algorithm for approximating planar curves with conjugated circular arcs and straight-line segments. Approximation of digital planar curves using line segments and analytic curve pieces is a problem of fundamental importance in image analysis and pattern recognition. A specific variation of this problem is approximation by conjugated arcs and line segments, which also arises in robotics, particularly in motion planning. We propose an algorithmic solution to this problem, designed for the automatic generation of programs for robotic manipulators in the AS language. The originality of the method lies in employing a special representation of planar curves that is invariant under translation, rotation, and scaling. Such a representation enables the development of an optimal algorithm that automatically determines the number and placement of conjugated arcs and line segments approximating a given curve with a prescribed accuracy. The simplicity, flexibility with respect to input data, and low computational requirements of the proposed method make it suitable not only for image recognition tasks but also for robotic trajectory planning.

${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point $\leftrightarrow$ line duality. The result is a ``line-curve'' which is seen as the envelope of it's tangents. Usually this gives an unclear image and is at the heart of the ``over-plotting'' problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation which provides directly the ``point-curve'' representation of a curve. Earlier this was applied to conics and their generalizations. Here the representation, also called dual, is extended to all planar algebraic curves. Specifically, it is shown that the dual of an algebraic curve of degree $n$ is an algebraic of degree at most $n(n - 1)$ in the absence of singular points. The result that conics map into conics follows as an easy special case. An algorithm, based on algebraic geometry using resultants and homogeneous polynomials, is obtained which constructs the dual image of the curve. This approach has potential generalizations to multi-dimensional algebraic surfaces and their approximation. The ``trade-off'' price then for obtaining {\em planar} representation of multidimensional algebraic curves and hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in $\|$-coords.

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

Determinantal representations of hyperbolic plane curves: An elementary approach

A dynamic programming approach for fitting digital planar curves with line segments and circular arcs

Two-dimensional pin array displays enable access to tactile graphics that are important for the education of students with visual impairments. Due to their prohibitive cost and limited access, there is limited research within HCI and the rules to design graphics on these low-resolution tactile displays are unclear. In this paper, eight tactile readers with visual impairments qualitatively evaluate the implementation of Pixel Art to create tactile graphical primitives on a pin array display. Every pin of the pin array is assumed to be a pixel on a pixel grid. Our findings suggest that Pixel Art tactile graphics on a pin array are clear and comprehensible to tactile readers, positively confirming its use to design basic tactile shapes and line segments. The guidelines provide a consistent framework to create tactile media which implies that they can be used to downsize basic shapes for refreshable pin-array displays.

The extraction of component line segments and circular arcs from freehand strokes along with their relations is a prerequisite for sketch understanding. Existing approaches usually take three stages to segment a stroke: first identifying segmentation points, then classifying the substroke between each pair of adjacent segmentation points, and, finally, obtaining graphical representations of substrokes by fitting graphical primitives to them. Since a stroke inevitably contains noises, the first stage may produce wrong or inaccurate segmentation points, resulting in the wrong substroke classification in the second stage and inaccurately fitted parameters in the third stage. To overcome the noise sensitivity of the three-stage method, the segmental homogeneity feature is emphasized in this paper. We propose a novel approach, which first extracts graphical primitives from a stroke by a connected segment growing from a seed-segment and then utilizes relationships between the primitives to refine their control parameters. We have conducted experiments using real-life strokes and compared the proposed approach with others. Experimental results demonstrate that the proposed approach is effective and robust.

This paper lays the theoretical foundations to path openings and closings. The traditional morphological filter used for the analysis of linear structures in images is the union of openings (or the intersection of closings) by linear segments. However structures in images are rarely strictly straight, and as a result a more flexible approach is needed. An extension to the idea of using straight line segments as structuring elements is to use constrained paths, i.e. discrete, one-pixel thick successions of pixels oriented in a particular direction, but in general forming curved lines rather than perfectly straight lines. However the number of such paths is prohibitive and the resulting algorithm by simple composition is inefficient. In this paper we propose a way to compute openings and closings over large numbers of constrained, oriented paths in an efficient manner, suitable for building filters with applications to the analysis of oriented features, such as for example texture.

This paper describes a new training method for a continuous mapping and/or pattern classification neural network that performs local sample-density smoothing. A conventional training method uses point-to-point mapping from an input space to an output space. Even though the mapping may be precise at two given training sample points, there are no guarantees of mapping accuracy at points on a line segment connecting the sample points. This paper first discusses a theory for formulating line-to-line mapping. The theory is called interpolation training. This paper then expands the theory to k-nearest neighbor interpolation. The k-neighbor interpolation training (KNIT) method connects an input sample training point to its k-neighbor points via k line segments. Then, the method maps these k line segments in the input space for each training sample to linear line segments in the output space that interpolate between training output values. Thus, a web structure made by connecting input samples is mapped into the same structure in an output space. The KNIT method reduces the over learning problem caused by point-to-point training by smoothing input/output functions. Simulation tasks show that KNIT improves vowel recognition on a small speech database.

Recognition of geometric primitives such as line and arc helps in automatic conversion of line drawing document images into electronic form. A large number of raster to vector methods can be found in the literature. A line and arc segmentation contest was held in conjunction with the tenth IAPR International Workshop on Graphics Recognition (GREC 2013) for comparing performance of different methods on a uniform platform. The contest was broken down into two challenges: arc segmentation and line segmentation. The dataset includes engineering drawings (for arc segmentation challenge) and cadastral maps (for line segmentation challenge). Jianping Wu’s method got the highest score (0.541), hence the winner of the Arc Segmentation Contest. Liu Wenyin’s method, the only method participated in the line segmentation contest achieved 66 % segmentation accuracy.

Presents a new method for the separation of merged text/form-structure components in forms. The technique described uses a modified version of the Hough transform to detect the structure of the form. The closed contours of the connected components are approximated by piecewise linear line segments. The parameters of the Hesse normal form of each line segment serve as input for the Hough transform. Compared to the vectorized boundary of characters, the lines of the form structure consist of appreciable more line segments with the same orientation and distance. So, the problem of the form structure detection in the database of line segments can be reduced to the detection of local peaks in the Hough space. Subsequent processing steps reconstruct the remaining contour fragments to characters. >

A bicycle path is a pair of trajectories in Rn , the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for n⩾3 each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler.

Design of nonlinear spring for a designated load path Dein Shaw1 and Chih Ren Huang1 Summary The purpose of this research is to propose a design method for displacement path of a curved nonlinear spring (As shown in Fig. 1). The design method is divided the spring into several nonlinear linear spring segments of coordinates to be determined connecting points (nodal points). The relation between external load and displacement is defined by the request of user. The relation between load and displacement of each line segment of the spring is presented by large deformation theorem. The prior known variables are moment of inertia, cross section area, material property and load path. Those variables can be input to the stiffness equations to get equation of coordinate of the connecting points. After solving the coordinate equation, the coordinate of nodal points obtained. A nonlinear spring is manufactured according to the design coordinates. The large displacement path and rotation angle are measured. Finally, the experiment results are compared with simulation. The verifying results show the feasibility of design method.

Multiscale Representation and Matching of Curves Using Codons

In the paper, we detect 2D surfaces and recognize 3D structures in images, and generate occluded part of three-dimensional structure. We detect objects vanish points and geometric segmentation lines, and generate the vector and normal plane. We measure the objects boundaries which have been detected. In the process, we have found the best linear segmentation positions, generated buildings, and other 3D models in room. We propose some effective assumptions to recognize the object model from images by geometric reasoning. At the same time, we put forward structure prediction technology combined with the volume reasoning by parameter representation of spatial objects. We detect scene model relationships of each other by combining image-rich appearance with geometric features. We proposed image geometric modeling grammar framework according to previous discriminative classifier. It is used to represent the physical structure of the visual component. The framework has broken the traditional probability texture context grammar tree model. We proposed spatial context theory and model generation rules. Finally, on the public existing data sets, we proved that we only use structure prediction and linear segmentation of scene to recover 3D structure by a lot of experiments, comparing to restoration algorithm using whole image structure, ours method can produce more credible scene entity models and space constraint relationships.