A new algorithm for two-dimensional line clipping via geometric transformation

Logarithmic arithmetic (LA) is a very fast computational method for real numbers. And its computation precision is much better than a floating point arithmetic of equivalent word length and range. This paper shows a method of fixed point number computations by LA—just to do numeric conversion before and after LA computations. It is used to handle discrete coordinate addresses and pixel intensity data of digital images. The geometrical transformation is a typical application of the method. Linear (affine) and non-linear transformations with three interpolations of "nearest neighbor," "bi-linear" and "cubic convolution" in LA are demonstrated. It is the processing of coordinate addresses and pixel intensities of fixed point numbers. Experiments by 16-bit personal computer program showed that quality and speed are surprisingly high. The latter is comparable to that by using the floating point hardware chip. Some other examples of the applications are shown—curve drawing, three dimensional computer graphics and fractal image generation—all excellent.

In this article, the usage of matrices and matrix operations in computer graphics is shown. A brief overview of geometric transformations in computer graphics is given. The "Matrix - Computer Graphics" application software is created for the representation and easier understanding of relations between geometric transformations and matrix calculus.

Free space detection is very important for vehicle navigation and safe driving. 3-D line segment reconstruction of a street is important for the free space detection because a street-view includes many line segments. For the free space detection, we propose a method for reconstructing 3-D line segments in a streetscape using a monocular in-vehicle camera. The 3-D reconstruction of the line segments is achieved by using each three images from an image sequence. Once accurate camera poses of these images are obtained, one of the remaining crucial problems is to match the line segments between the images correctly. A strategy for finding correspondence of the line segments is as follows: First, the correspondences of line segment candidates are searched by using a two-view constraint. However, the two-view constraint has difficulty on determining an unique correspondence geometrically. Therefore, the candidates of the line segment correspondences are reduced using a three-view constraint. In order to improve the accuracy, the proposed method exploits a color feature of the line segment and a preliminary knowledge of the vehicle motion. Finally, the line segments are reconstructed using the correspondences. From an experimental result, we confirmed the effectiveness of the proposed method. Application to the free space detection demonstrated the usefulness of the reconstructed line segments.

A statistical simulation model for positional error of line features in Geographic Information Systems (GIS)

Airports are one of the most important traffic facilities; thus, airport detection is of great significance in economic and military construction. This letter proposes a novel method for airport detection, with the entire algorithm based on line segment classification and texture classification. First, a fast line segment detector is applied to extract the line segments in images and compute the features of these line segments. Then, the line segments are discriminated by a trained runway line classifier, and the regions of interest (ROIs) are extracted from the line segments, which are classified as runway lines. Finally, whether the ROI is actually an airport is determined by analyzing the classification results of the image blocks. This method is unique in terms of the computing of line segment features and line segment classification. Experimental results demonstrate the effectiveness and robustness of the proposed method.

Both in Image Processing and in Computer Graphics digital straight line segments play a role as the simplest way to connect pixels. But while in Computer Graphics the emphasis is on the generation of these lines, given the parameters intercept and slope, the focus in Image Processing is on measuring the parameters, given the lines. Aspects of this analysis are the so-called linearity conditions (specifying the necessary and sufficient conditions for a set of points to be a digitized straight line segment) and the accuracy of the representation (specifying how accurately a given digitized straight line segment describes the underlying continuous reality). Both aspects have been investigated in depth [1]–[9]. The question of accuracy has recently been solved completely, when mathematical expressions were given for the set of all continuous line segments that could have generated a given digital straight line segment [8]. These expressions are in fact the formulas for the inversion of the digitization process. As such, they are also interesting for Computer Graphics.

Geometric transformations are most commonly represented as square matrices in computer graphics. Following simple geometric arguments we derive a natural and geometrically meaningful definition of scalar multiples and a commutative addition of transformations based on the matrix representation, given that the matrices have no negative real eigenvalues. Together, these operations allow the linear combination of transformations. This provides the ability to create weighted combination of transformations, interpolate between transformations, and to construct or use arbitrary transformations in a structure similar to a basis of a vector space. These basic techniques are useful for synthesis and analysis of motions or animations. Animations through a set of key transformations are generated using standard techniques such as subdivision curves. For analysis and progressive compression a PCA can be applied to sequences of transformations. We describe an implementation of the techniques that enables an easy-to-use and transparent way of dealing with geometric transformations in graphics software. We compare and relate our approach to other techniques such as matrix decomposition and quaternion interpolation.

After the long period of mandatory distance learning, either synchronous or asynchronous, caused by the COVID-19 pandemic, the existence of learning tools to support students in similar scenarios is welcomed in many curriculum subjects. Geometric transformations are relevant in mathematics and an omnipresent topic in the computer graphics curriculum that needs plenty of visual tools and learning-by-doing environments. In this work, we propose a framework based on an extended Translation-Rotation-Scale (TRS) pattern to generate composite transformations to support a specific style of learning exercises that we call “before and after”. The pattern, together with specific constraints to reduce the domain into a finite set of cases, allows the automatic exercises generation, evaluation, and real-time feedback. Two pieces of software, called GTVisualizer and GTCards, were developed to empirically test the proposed framework embedding it in a gaming environment. The first tool supports the initial instruction and introduces the formal matrix-based methodology to specify composite transformations; the second one provides a ludic environment where students can drill and reinforce the subject by playing. Our initial results suggest that these tools are suitable for learning geometric transformations; they helped students in simulated self-study mode achieve comparable results to those students receiving the regular lecture materials and explanations in the classroom. Moreover, the usage of the tools appears to favor a similar increasing effect on students’ visual-spatial abilities, similar to the obtained by taking an in-person course on computer graphics, as measured with the Purdue Spatial Visualization Test.

A segment tree is a versatile tree-based data structure over intervals or line segments efficiently supporting several computational operations such as stabbing query, segment arrangement, and planar point location, both theoretically and practically. Polygon clipping is a basic operation in domains such as Computer Graphics, Computer-aided Design, and Geographic Information Science (GIS). Given two polygons with n vertices, polygon clipping algorithms find the geometric intersection or union in <Formula format="inline"><TexMath><?TeX $\mathcal {O}(n^2)$?></TexMath><AltText>Math 1</AltText><File name="icpp24-103-inline1" type="svg"/></Formula> time using Foster’s all-to-all edge intersection testing and <Formula format="inline"><TexMath><?TeX $\mathcal {O}((n+k)\log n)$?></TexMath><AltText>Math 2</AltText><File name="icpp24-103-inline2" type="svg"/></Formula> time using Vatti’s sweep line-based method, where k is the number of intersections. No known segment tree implementation, including the CGAL library, supports intersection finding or polygon clipping. We extended the segment tree leveraging Chaselle’s PRAM-model augmentation, parallelized the construction of our augmented segment tree, and employed it to find line segment intersections for polygon clipping while handling degenerate cases. Augmented segment tree eliminates 99% of non-intersecting edge pairs compared to 63% by the state-of-the-art filtering based on common minimum bounding rectangle method employed in Foster’s GPU-based implementation. This, coupled with Ω(nlog n) work on a single CPU core, beats Foster’s GPU performance with <Formula format="inline"><TexMath><?TeX $\mathcal {O}(n^2)$?></TexMath><AltText>Math 3</AltText><File name="icpp24-103-inline3" type="svg"/></Formula> work. Our OpenMP directive based multi-core implementation achieves up to 4X relative speedup for clipping two polygons with 182K vertices and 5X speedup for five polygons with 398K vertices. We also offloaded the parallel kernels to a GPU using OpenACC achieving performance competitive with Foster’s GPU implementation. Our profiling indicates limitations of the compiler directives and potential for superior performance by employing pthread/cuda libraries.

The data obtained from rock mass discontinuity traces are used as significant parameters for rock engineering projects. The methodology of documenting discontinuity orientations by photogrammetry has the advantages of non-contact, fast, safe, and high precision measurement. The discontinuity trace detection and recognition is the key technology. In this paper, a new discontinuity trace detection methodology is presented based on line detection and segmentation traces' semi-automated linking. This methodology considers a rock mass exposure digital image as a discrete surface, the elevation of which is given by pixel brightness levels. The discontinuity trace line feature is automatically detected by using a line detection algorithm based on Hessian matrix to extract the curvilinear structure of the digital image. The discontinuity trace map is constructed by selecting the seed pixel in the line segment map and linking the line segments interpretatively. The experimental results show that this paper's methodology can be used in processing digital images of rock exposures from underground caverns and open environments. The trace map constructed by the new methodology is mainly consistent with a map drawn manually. These results demonstrate the effectiveness of the proposed methodology.

Extraction of lane markings is still a challenge due to the poor quality of lane markings and interferences from external circumstances. In this paper, we utilize line segments as low-level features to detect lane markings on structured road scenes. Our novel algorithm can be highlighted in four items as follows. Firstly, a road surface region is reasonably located using adaptive segmentation method, and then most of the interferences from the external circumstances are eliminated by Laplacian filter. Secondly, the line segments detected by the line segment detector are applied to represent the structural information of lanes. Thirdly, non-lane candidate markings are removed through orientation and vanishing point constraints. Finally, the lanes are accurately extracted from the remaining candidate lane marking. Experimental results on complex structured road scenarios in urban streets are shown and the effectiveness and robustness of our novel algorithm are underpinned by the experimental results.

This paper presents a method for the detection of roads in high resolution Synthetic Aperture Radar (SAR) images using an Ant Colony Algorithm (ACA). Roads in a high resolution SAR image can be modeled as continuously straight line segments of roadsides that possess width. In our method, line segments which represent the candidate positions for roadsides are first extracted from the image using a line segments extractor, and next the roadsides are accurately detected by grouping those line segments. For this purpose, we develop a method based on an ACA. We combine perceptual grouping factors with it and try to reduce its overall computational cost by a region growing method. In this process, a selected initial seed is grown into a finally grouped segment by the iterated ACA process, which considers segments only in a search region. Finally to detect roadsides as smooth curves, we introduce the photometric constraints in ant colony algorithm as external energy in a modified snake model to extract geometric roadsides model. We applied our method to some parts of TerraSAR-x images that have a resolution of about 1 m. The experimental results show that our method can accurately detect roadsides from high resolution SAR images.

As one of the most common features in the man-made environments, straight lines play an important role in many applications. In this paper, we present a new framework to extract line segments from large-scale point clouds. The proposed method is fast to produce results, easy for implementation and understanding, and suitable for various point cloud data. The key idea is to segment the input point cloud into a collection of facets efficiently. These facets provide sufficient information for determining linear features in the local planar region and make line segment extraction become relatively convenient. Moreover, we introduce the concept “number of false alarms” into 3-D point cloud context to filter the false positive line segment detections. We test our approach on various types of point clouds acquired from different ways. We also compared the proposed method with several other methods and provide both quantitative and visual comparison results. The experimental results show that our algorithm is efficient and effective, and produce more accurate and complete line segments than the comparative methods. To further verify the accuracy of the line segments extracted by the proposed method, we also present a line-based registration framework, which employs these line segments on point clouds registration.

Mean shift based clustering of Hough domain for fast line segment detection

A polyhedral object in 3-dimensional space is often well represented by a set of points and line segments that act asits features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of theobject, relevant for some task, are clearly visible. Such a view is often called a non-degenerate view or projection. Inthis paper we are concerned with computing non-degenerate orthogonal and perspective projections of sets of pointsand line segments (objects) in 3-dimensional space. We outline the areas in which such problems arize, discuss recentresearch on the computational complexity of these problems, illustrate the fundamental ideas used in the design ofalgorithms for computing non-degenerate projections, and provide pointers to the literature where the results can befound.Keywords: Data visualization, nice viewpoints, knot theory, computer vision, graph drawing, computer graphics,degeneracies, general position, orthogonal projections, perspective projections, robustness of algorithms, computa-tional geometry.