In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given G1 data based on similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

Abstract The $$\kappa $$ κ -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ ϵ κ -curves. $$\kappa $$ κ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ ϵ κ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

We propose a new boundary handling method for smoothed particle hydrodynamics (SPH). Previous approaches required the use of boundary particles to prevent particles from sticking to the boundary. We address this issue by correcting the fundamental equations of SPH with the integration of a kernel function. Our approach is able to directly handle triangle mesh boundaries without the need for boundary particles. We also show how our approach can be integrated into a position-based fluid framework.

This paper describes a method for removing adherent noises from image sequences. In outdoor environments, it is often the case that scenes taken by a camera are deteriorated because of adherent noises such as waterdrops on the surface of the lens-protecting glass of the camera. To solve this problem, our method takes advantage of image sequences captured with a moving camera. The method makes a spatio-temporal image to extract the regions of adherent noises by examining differences of track slopes in cross section images between adherent noises and other objects. Finally, regions of noises are eliminated by replacing with image data corresponding to object regions. Experimental results show the effectiveness of our method.

In this paper, we propose a new method that can remove view-disturbing waterdrops from images by processing images taken with a pan-tilt camera system. In rainy days, it is often the case that images taken by the camera are hard to see because of adherent waterdrops on the surface of the protecting glass of the camera. In our method, an image of a distant prospect is taken at first and another image is taken after changing the direction of eyeshot. The new image is transformed with the projective transformation and compared with the first one to detect the region where waterdrops may exist. We can distinguish which image portion belongs to waterdrops by considering the distance between two waterdrop candidate regions. Finally, the region where waterdrops exist can be eliminated to merge two images. Experimental results show the effectiveness of the proposed method.

In this paper, we propose a new method that can remove view-disturbing noises from images of dynamic scenes. One of the thorny problems in outdoor surveillance by a camera is that adherent noises such as waterdrops or mud blobs on the protecting glass surface lens disturb the view from the camera. Therefore, we propose a method for removing adherent noises from images of dynamic scenes taken by changing the direction of a pan-tilt camera, which is often used for surveillance. Our method is based on the comparison of two images, a reference image and a second image taken by a different camera angle. The latter image is transformed by a projective transformation and subtracted from the reference image to extract the regions of adherent noises and moving objects. The regions of adherent noises in the reference image are identified by examining the shapes and distances of regions existing in the subtracted image. Finally, regions of adherent noises can be eliminated by merging two images. Experimental results show the effectiveness of our proposed method.

We propose an advanced cut-and-paste editing for three-dimensional models. We introduce a new parameterization technique based on constrained B-spline surface/volume fitting. Our cut-and-paste editing is performed on this parameterization. In our method, a parametric volume is generated so that its isoparametric surface fits to the base surface and the volume bounds the detail. Our volume fitting is realized by the optimization of an objective functional with a smoothness factor. We implemented and evaluated our method, and showed that our constrained fitting is effective to edit shapes that may have highly curved areas or handles.

In this paper, we propose a new method for the restoration of deteriorated images by using multiple cameras. In outdoor environment, it is often the case that scenes taken by the cameras are hard to see because of adherent noises on the surface of the lens-protecting glass of the cameras. Our proposed method analyses multiple camera images describing the same scene, and synthesizes an image in which adherent noises are eliminated.

A technique to improve the performance of the shadow-volume method is presented. This technique does not require (1) extensive edge/edge intersection tests and intersection angle computation in shadow polygon construction, or (2) any ray/shadow-polygon intersection tests during scan-conversion. The first task is achieved by constructing ridge edge (RE) loops, an inexact form of silhouette, instead of the silhouette. The RE loops give us the shadow volume without any expensive computation. The second task is achieved by discretizing the shadow volume into angular spans. The angular spans, which correspond to scan lines, are stored in a lookup table. This lookup table enables us to mark the pixels that are in shadow directly, without the need of performing any ray/shadow-polygon intersection tests. In addition, the shadow on an object is determined on a line-by-line basis instead of a pixel-by-pixel basis. The new technique is efficient enough to achieve real time performance, without any special hardware, while being scalable with scene size.

The control polygons of B-spline subdivision curves are usually refined uniformly using a technique called knot-doubling. This uniform refinement approach would perform unnecessary subdivision steps on portions already close to the limit curve enough and, consequently, cause unnecessary (exponential) increase on the number of line segments in the refined polygons. This paper overcomes this problem by proposing a local refinement technique for the control polygons of B-spline subdivision curves. Local refinement is achieved by selectively inserting new knots at midpoints of knot intervals. Efficient adaptive subdivision can be easily realized based on the new technique.

A deformation-based fine tuning technique for parametric curves and surfaces is presented. A curve or surface is deformed by scaling its derivative, instead of manipulating its control points. Since only the norm of the derivative is adjusted, the resulting curve or surface keeps the basic shape of the original profile and curvature distribution. Therefore, the new technique is especially suitable for last minute fine tuning of the design process. Other advantages include: (1) the fine tuning process is a real local method, it can be performed on any portion of a curve or a surface, not just on a set of segments or patches; (2) by allowing a user to drag a scalar function to directly adjust the curvature (and, consequently, fairness) of a curve or surface, the new technique makes the shape design process more intuitive and effective; (3) the new technique is suitable for precise shaping and deforming such as making the curvature of a specific portion twice as big. In many cases, it can achieve results that other methods such as FFD can not; (4) the fine tuning process can also be used for subdivision curves and surfaces. Related techniques and test results are included.

Streamline modeling is a design methodology of fair free form surfaces where the tangent vectors are specified/manipulated to generate/deform them instead of the control points of the traditional surface representations (K.T. Miura et al., 1998). Creations and deformations of complex 3D free form shapes generally require a large amount of labour and cost. Therefore, the authors propose a novel modeling technique based on of fluid flow dynamics in order to create and deform high-quality surfaces more intuitively with a smaller number of parameters based on the streamline modeling. We construct a flow field based on the potential flow, then calculate tangent vectors in the flow field that are required for the streamline modeling. Streamlines are generated by numerically integrating the tangent vectors, then the surface is represented as a set of streamlines. The deformation of the surface is performed by changing of the flow field. We have developed a prototype design system using the new modeling technique.

The paper proposes a new type of free-form curve for fairness. A unit quaternion curve is used to specify the tangent of the curve in order to more directly manipulate its curvature and variation of curvature than is possible for the traditional parametric representations like Bezier and NURBS curves. Since the new curve is represented by can integral form of a unit quaternion curve, it is named unit quaternion integral curve or QI curve for brevity. It is a generalization and an extension of the clothoid into three dimensional space and the norm of its tangent is always equal to 1. Its curvature and variation of curvature are given by rather simple expressions.

The paper proposes a generation system of unit quarternion curves and surfaces to design fair curves and surfaces. Curvature and variation of curvature are the primary factors that determine the quality of fair curves and surfaces; however not so many methods have been explored to directly manipulate them, especially in three dimensional space. We use unit quaternion curves in integral forms to specify the tangent of a curve and the first derivative of a surface in one parameter direction. This enables us to control their curvature and variation of curvature more than is possible with traditional parametric representations like Bezier and NURBS curves and surfaces. Using unit quaternion curves balances the efficiency of the design process and the fairness of the designed curves and surfaces. Our system can generate fair curves and surfaces within a short time and their quality is acceptable for industrial applications as well as CG animation.

By introducing the edge-based blending function, derived from the surface-based function, we can capture the effects of the edges on the shape of the n-sided patch more directly. Designers can manipulate the edge-based blending function, as well as the geometry of the boundary curves, to design and modify the patch shape. We discuss the derivation of the edge-based blending function, the ease of design and the connection of several patches with G/sup 1/ continuity. We demonstrate its validity by modifying rectangular patches according to its changes.

G2 continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G2 continuity to connect two Bézier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the Bézierpatch formulation. Gregory patch is a kind of free-form surface patch which is extended from Bézier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for intergral, rational, and NURBS boundary curves. In this paper, we propose some intergral boundary Gregorytype patches bounded by cubic and quartic curves for G2 continuity.

When prototyping the requirement specifications of a large and complex system, it is very difficult to express them in a single model containing all the features. It is often disastrous if the requirement specifications are prototyped in a single level or single window of a single model. Hence multimodel based hierarchical requirement specifications are desirable in order to fully specify large and complex systems. But multimodel based hierarchical requirement specifications tools are then confronted with verification problems for inter model consistency and syntactic correctness. We introduce a predefined graphical structural syntax checker called Model Hierarchy Definition (MHD) which provides interactive hierarchy syntax checking during modeling the structured requirement specifications and predefined lowest level graphical formalisms called Model Hierarchy Model (MHM), defining the meta rules, the graphical diagramming entity and the hierarchical relationship for MHD embedding in the prototyping models. Having MHD and the MHM in the prototyping environment, one obtains higher quality structured requirement specifications when building hierarchical multimodel based prototypes for complex system analysis, and greatly reduces the prototyping time and costs. The MHD and MHM were implemented to support an integrated graphical prototyping system, which we overview, called the Prototyper's Workbench (PWB) for rapid prototyping and analysis of software requirement specifications of target systems.

G² continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G2 continuity between two Bezier patches has been studied and methods developed to ensure such continuity. However, certain restrictions on the shapes of such patches arise within the Bezier-patch formulation. The Gregory patch is a kind of free-form surface patch which is an extension of the Bezier patch such that cross-boundary first derivatives can be specified without restrictions on the compatibility condition. In this paper, we extend the idea of the Gregory patch and develop a formulation for the C² Gregory patch. The properties of the C² Gregory patch are discussed as well as its connection with a Bezier patch and a G² continuous interpolation method based upon such patches.