No Arabic abstract
An algorithm to generate the locus of a circle using the intersection points of straight lines is proposed. The pixels on the circle are plotted independent of one another and the operations involved in finding the locus of the circle from the intersection of straight lines are parallelizable. Integer only arithmetic and algorithmic optimizations are used for speedup. The proposed algorithm makes use of an envelope to form a parabolic arc which is consequent transformed into a circle. The use of parabolic arcs for the transformation results in higher pixel errors as the radius of the circle to be drawn increases. At its current state, the algorithm presented may be suitable only for generating circles for string art.
Fractal image generation algorithms exhibit extreme parallelizability. Using general purpose graphics processing unit (GPU) programming to implement escape-time algorithms for Julia sets of functions,parallel methods generate visually attractive fractal images much faster than traditional methods. Vastly improved speeds are achieved using this method of computation, which allow real-time generation and display of images. A comparison is made between sequential and parallel implementations of the algorithm. An application created by the authors demonstrates using the increased speed to create dynamic imaging of fractals where the user may explore paths of parameter values corresponding to a given functions Mandelbrot set. Examples are given of artistic and mathematical insights gained by experiencing fractals interactively and from the ability to sample the parameter space quickly and comprehensively.
Generalized Pythagoras trees were developed for visualizing hierarchical data, producing organic, fractal-like representations. However, the drawback of the original layout algorithm is visual overlap of tree branches. To avoid such overlap, we introduce an adapted drawing algorithm using ellipses instead of circles to recursively place tree nodes representing the subhierarchies. Our technique is demonstrated by resolving overlap in diverse real-world and generated datasets, while comparing the results to the original approach.
Complex 3D curves can be created by directly drawing mid-air in immersive environments (Augmented and Virtual Realities). Drawing mid-air strokes precisely on the surface of a 3D virtual object, however, is difficult; necessitating a projection of the mid-air stroke onto the user intended surface curve. We present the first detailed investigation of the fundamental problem of 3D stroke projection in VR. An assessment of the design requirements of real-time drawing of curves on 3D objects in VR is followed by the definition and classification of multiple techniques for 3D stroke projection. We analyze the advantages and shortcomings of these approaches both theoretically and via practical pilot testing. We then formally evaluate the two most promising techniques spraycan and mimicry with 20 users in VR. The study shows a strong qualitative and quantitative user preference for our novel stroke mimicry projection algorithm. We further illustrate the effectiveness and utility of stroke mimicry, to draw complex 3D curves on surfaces for various artistic and functional design applications.
This paper presents a novel line-aware rectification network (LaRecNet) to address the problem of fisheye distortion rectification based on the classical observation that straight lines in 3D space should be still straight in image planes. Specifically, the proposed LaRecNet contains three sequential modules to (1) learn the distorted straight lines from fisheye images; (2) estimate the distortion parameters from the learned heatmaps and the image appearance; and (3) rectify the input images via a proposed differentiable rectification layer. To better train and evaluate the proposed model, we create a synthetic line-rich fisheye (SLF) dataset that contains the distortion parameters and well-annotated distorted straight lines of fisheye images. The proposed method enables us to simultaneously calibrate the geometric distortion parameters and rectify fisheye images. Extensive experiments demonstrate that our model achieves state-of-the-art performance in terms of both geometric accuracy and image quality on several evaluation metrics. In particular, the images rectified by LaRecNet achieve an average reprojection error of 0.33 pixels on the SLF dataset and produce the highest peak signal-to-noise ratio (PSNR) and structure similarity index (SSIM) compared with the groundtruth.
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.