A rectangle-of-influence drawing of a plane graph G is a straight-line planar drawing of G such that there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of any edge. In this paper, we show that any given 5-connected plane graph G with five or more vertices on the outer face has a rectangle-of-influence drawing in an integer grid such that W + H ≤ n - 2, where n is the number of vertices in G, W is the width and H is the height of the grid.

The aim of a computer-aided drawing therapy system in this work is to associate drawings which a client makes with the client's mental state in quantitative terms. A case study is conducted on experimental data which contain both pastel drawings and mental state scores obtained from the same client in a psychotherapy program. To perform such association through colors, we translate a drawing to a color feature by measuring its representative colors as primary color rates. A primary color rate of a color is defined from a psychological primary color in a way such that it shows a rate of emotional properties of the psychological primary color which is supposed to affect the color. To obtain several informative colors as representative ones of a drawing, we define two kinds of color: approximate colors extracted by color reduction, and area-averaged colors calculated from the approximate colors. A color analysis method for extracting representative colors from each drawing in a drawing sequence under the same conditions is presented. To estimate how closely a color feature is associated with a concurrent mental state, we propose a method of utilizing machine-learning classification. A practical way of building a classification model through training and validation on a very small dataset is presented. The classification accuracy reached by the model is considered as the degree of association of the color feature with the mental state scores given in the dataset. Experiments were carried out on given clinical data. Several kinds of color feature were compared in terms of the association with the same mental state. As a result, we found out a good color feature with the highest degree of association. Also, primary color rates proved more effective in representing colors in psychological terms than RGB components. The experimentals provide evidence that colors can be associated quantitatively with states of human mind.

In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n-1) × (n-1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. An internally triconnected plane graph G has a convex grid drawing on a 2n × 2n grid if T(G) has exactly four leaves. Furthermore, an internally triconnected plane graph G has a convex grid drawing on a 20n × 16n grid if T(G) has exactly five leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 10n × 5n grid if T(G) has exactly five leaves. We also present a linear-time algorithm to find such a drawing.

Graph layout is a critical component in graph visualization. This paper proposes GRAPHULY, a graph u-nets-based neural network, for end-to-end graph layout generation. GRAPHULY learns the multi-level graph layout process and can generate graph layouts without iterative calculation. We also propose to use Laplacian positional encoding and a multi-level loss fusion strategy to improve the layout learning. We evaluate the model with a random dataset and a graph drawing dataset and showcase the effectiveness and efficiency of GRAPHULY in graph visualization.

A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a linear-time algorithm to find a grid drawing of any given 5-connected plane graph G with five or more vertices on the outer face. The size of the drawing satisfies W + H≤n - 2, where n is the number of vertices in G, W is the width and H is the height of the grid drawing.

The Tokyo subway is one of the most complex subway networks in the world and it is difficult to compute a visually readable metro map using existing layout methods. In this paper, we present a new method that can generate complex metro maps such as the Tokyo subway network. Our method consists of two phases. The first phase generates rough metro maps. It decomposes the metro networks into smaller subgraphs and partially generates rough metro maps. In the second phase, we use a local search technique to improve the aesthetic quality of the rough metro maps. The experimental results including the Tokyo metro map are shown.

In a convex grid drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection, all vertices are put on grid points and all facial cycles are drawn as convex polygons. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n-1)×(n-1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. An internally triconnected plane graph G has a convex grid drawing on a 2n×2n grid if T(G) has exactly four leaves. Furthermore, an internally triconnected plane graph G has a convex grid drawing on a 6n×n2 grid if T(G) has exactly five leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 20n×16n grid if T(G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of O(n2) size.

In this paper, we consider the collaborative editing of two-dimensional (2D) data such as handwritten letters and illustrations. In contrast to the editing of 1D data, which is generally realized by the combination of insertion/deletion of characters, overriding of strokes can have a specific meaning in editing 2D data. In other words, the appearance of the resulting picture depends on the reflection order of strokes to the shared canvas in addition of the absolute coordinate of the strokes. We propose a Peer-to-Peer (P2P) collaborative drawing system consisting of several nodes with replica canvas, in which the consistency among replica canvases is maintained through data channel of WebRTC. The system supports three editing modes concerned with the reflection order of strokes generated by different users. The result of experiments indicates that the proposed system realizes a short latency of around 120 ms, which is a half of a cloud-based system implemented with Firebase Realtime Database. In addition, it realizes a smooth drawing of pictures on remote canvases with a refresh rate of 12 fps.

In this paper, we propose an interactive system for controlling the pressure while cutting paper with a knife. The purpose is to improve the cutting skill of novices learning the art of paper-cutting. Our system supports skill improvement for novices by measuring and evaluating their cutting pressure in real-time. In this study, we use a knife with a blade attached to a stylus with a pressure sensor, which can measure the pressure, coordinates, and cutting time. We have developed a similar support system using a stylus and a tablet device. This system allows the user to experience the pressure of experts through tracing. Paper-cutting is created by cutting paper with a knife. The practice system in this paper provides practice in an environment more akin to the production of paper cutting. In the first experiment, we observed differences in cutting ability by comparing cutting pressures between novices and experts. As a result, we confirmed that novices cut paper at a higher pressure than experts. We developed a practice system that guides the novices on controlling the pressure by providing information on the cutting pressure values of experts. This system shows the difference in pressure between novices and experts using a synchronous display of color and sound. Using these functions, novices learn to adjust their cutting pressure according to that of experts. Determining the right cutting pressure is a critical skill in the art of paper-cutting, and we aim to improve the same with our system. In the second experiment, we tested the effect of the practice system on the knife device. We compared the changes in cutting pressure with and without our system, the practice methods used in the workshop, and the previously developed stylus-based support system. As a result, we confirmed that practicing with the knife device had a better effect on the novice's skill in controlling cutting pressure than other practice methods.

We consider the problem of placing arrows, which indicate the direction of each edge in directed graph drawings, without making them overlap other arrows, vertices and edges as much as possible. The following two methods have been proposed for this problem. One is an exact algorithm for the case in which the position of each arrow is restricted to some discrete points. The other is a heuristic algorithm for the case in which the arrow is allowed to move continuously on each edge. In this paper, we assume that the arrow positions are not restricted to discrete points and propose an exact algorithm for the problem of finding an arrow placement such that (a) the weighted sum of the numbers of overlaps with edges, vertices and other arrows is minimized and (b) the sum of the distances between the arrows and their edges' terminal vertices is minimized as a secondary objective. The proposed method solves this problem by reducing it to a mixed integer linear programming problem. Since this is an exponential time algorithm, we add a simple procedure as preprocessing to reduce the running time. Experimental results show that the proposed method can find a better arrow placement than the previous methods and the procedure for reducing the running time is effective.

Recent research on graph drawing focuses on Right-Angle-Crossing (RAC) drawings of 1-plane graphs, where each edge is drawn as a straight line and two crossing edges only intersect at right angles. We give a transformation from a restricted case of the RAC drawing problem to a problem of finding a straight-line drawing of a maximal plane graph where some angles are required to be acute. For a restricted version of the latter problem, we show necessary and sufficient conditions for such a drawing to exist, and design an O(n2)-time algorithm that given an n-vertex plane graph produces a desired drawing of the graph or reports that none exists.

We propose a method for downsizing line pictures to generate pixel line arts. In our method, topological properties such as connectivity of lines and segments are preserved by allowing slight distortion in the form of objects in input images. When input line pictures are painted with colors, the number of colors is preserved by our method.

In a convex drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an (n-1)×(n-1) grid if either G is triconnected or the triconnected component decomposition tree T(G) of G has two or three leaves, where n is the number of vertices in G. An internally triconnected plane graph G has a convex grid drawing on a 2n×2n grid if T(G) has exactly four leaves. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 6n×n2 grid if T(G) has exactly five leaves. We also present an algorithm to find such a drawing in linear time. This is the first algorithm that finds a convex grid drawing of such a plane graph G in a grid of polynomial size.

A rectangular drawing is a plane drawing of a graph in which every face is a rectangle. Rectangular drawings have an application for floorplans, which may have a huge number of faces, so compact code to store the drawings is desired. The most compact code for rectangular drawings needs at most 4f-4 bits, where f is the number of inner faces of the drawing. The code stores only the graph structure of rectangular drawings, so the length of each edge is not encoded. A grid rectangular drawing is a rectangular drawing in which each vertex has integer coordinates. To store grid rectangular drawings, we need to store some information for lengths or coordinates. One can store a grid rectangular drawing by the code for rectangular drawings and the width and height of each inner face. Such a code needs 4f-4 + f⌈log W⌉ + f⌈log H⌉ + o(f) + o(W) + o(H) bits*, where W and H are the maximum width and the maximum height of inner faces, respectively. In this paper we design a simple and compact code for grid rectangular drawings. The code needs 4f-4 + (f+1)⌈log L⌉ + o(f) + o(L) bits for each grid rectangular drawing, where L is the maximum length of edges in the drawing. Note that L ≤ max{W,H} holds. Our encoding and decoding algorithms run in O(f) time.

With the rapid expansion of vector data model application to digital content such as drawings and digital maps, the security and retrieval for vector data models have become an issue. In this paper, we present a vector data-hashing algorithm for the authentication, copy protection, and indexing of vector data models that are composed of a number of layers in CAD family formats. The proposed hashing algorithm groups polylines in a vector data model and generates group coefficients by the curvatures of the first and second type of polylines. Subsequently, we calculate the feature coefficients by projecting the group coefficients onto a random pattern, and finally generate the binary hash from binarization of the feature coefficients. Based on experimental results using a number of drawings and digital maps, we verified the robustness of the proposed hashing algorithm against various attacks and the uniqueness and security of the random key.

Polymer multimode optical waveguides were fabricated from optically-sensitive hybrid silicone using the ultraviolet laser drawing method. The waveguide loss values were measured as 0.069 dB/cm with a laser diode, 0.069 dB/cm with a vertical-cavity surface-emitting laser, and 0.128 dB/cm with a light-emitting diode. The cross waveguide on a curved waveguide was drawn by overlapped direct laser drawing. The crosstalk and excess loss at the cross angle of 50 in the cross waveguide were measured as 47 dB and 0.5 dB, respectively.

Many factors, such as noise level in the original image and the noise-removal methods that clean the image prior to performing a vectorization, may play an important role in affecting the line detection of raster-to-vector conversion methods. In this paper, we propose an empirical performance evaluation methodology that is coupled with a robust statistical analysis method to study many factors that may affect the quality of line detection. Three factors are studied: noise level, noise-removal method, and the raster-to-vector conversion method. Eleven mechanical engineering drawings, three salt-and-pepper noise levels, six noise-removal methods, and three commercial vectorization methods were used in the experiment. The Vector Recovery Index (VRI) of the detected vectors was the criterion used for the quality of line detection. A repeated measure ANOVA analyzed the VRI scores. The statistical analysis shows that all the studied factors affected the quality of line detection. It also shows that two-way interactions between the studied factors affected line detection.

Hypergraphs drawn in the subset standard are useful to represent group relationships using topographic characteristics such as intersection, exclusion and enclosing. However, they present cluttering when dealing with a moderately high number of nodes (more than 20) and large hyperedges (connecting more than 10 nodes, with three or more overlapping nodes). At this complexity level, a study of the visual encoding of hypergraphs is required in order to reduce cluttering and increase the understanding of larger sets. Here we present a graph model and a visual design that help in the visualization of group relationships represented by hypergraphs. This is done by the use of superimposed visualization layers with different abstraction levels and the help of interaction and navigation through the display.

Noise removal in engineering drawing is an important operation performed before other image analysis tasks. Many algorithms have been developed to remove salt-and-pepper noise from document images. Cleaning algorithms should remove noise while keeping the real part of the image unchanged. Some algorithms have disadvantages in cleaning operation that leads to removing of weak features such as short thin lines. Others leave the image with hairy noise attached to image objects. In this article a noise removal procedure called TrackAndMayDel (TAMD) is developed to enhance the noise removal of salt-and-pepper noise in binary images of engineering drawings. The procedure could be integrated with third party algorithms' logic to enhance their ability to remove noise by investigating the structure of pixels that are part of weak features. It can be integrated with other algorithms as a post-processing step to remove noise remaining in the image such as hairy noise attached with graphical elements. An algorithm is proposed by incorporating TAMD in a third party algorithm. Real scanned images from GREC'03 contest are used in the experiment. The images are corrupted by salt-and-pepper noise at 10%, 15%, and 20% levels. An objective performance measure that correlates with human vision as well as MSE and PSNR are used in this experiment. Performance evaluation of the introduced algorithm shows better-quality images compared to other algorithms.

A subdivision of a rectangle into rectangular faces with horizontal and vertical line segments is called a rectangular drawing or floorplan. It has been an open problem to determine whether there exist a polynomial time algorithm for computing R(n). We affirmatively solve the problem, that is, we introduce an O(n4)-time and O(n3)-space algorithm for R(n). The algorithm is based on a recurrence for R(n), which is the main result of the paper. We also implement our algorithm and computed R(n) for n 3000.