Convolution Curve Research Articles

Sequence of Gn LN polynomial curves approximating circular arcs

In this paper we derive a sequence of linear normal (LN) curves b 2 n of degree 2 n which are G n endpoint interpolations of a circular arc and have approximation order 2 n + 2 . This is an extension of the circle approximation method by LN Bézier curves given in Ahn and Hoffmann (2014) to all even degrees. We also extend the circle approximation to an ellipse approximation by G n LN curves of degree 2 n . An upper bound of the Hausdorff distance between the ellipse and its LN approximation is obtained. We illustrate our results through an LN approximation of convolution curves of ellipses and a spline curve.

Detailed Consideration of Graphical Calculation of Min-Plus Convolution in Deterministic Network Calculus

The convolution operation used in deterministic network calculus differs from its counterpart known from the classic systems theory. A reason for this lies in the fact that the former is defined in terms of the so-called min-plus algebra. Therefore, it is oft difficult to realize how it really works. In these cases, its graphical interpretation can be very helpful. This paper is devoted to a topic of construction of the min-plus convolution curve. This is done here in a systematic way to avoid arriving at non-transparent figures that are presented in publications. Contrary to this, our procedure is very transparent and removes shortcomings of constructions known in the literature. Some examples illustrate its usefulness.

Geometric shape analysis for convolution curve of two compatible quadratic Bézier curves

In this paper we consider the local and global geometric properties of the convolution curve of two compatible quadratic Bézier curves. We characterize all shapes of convolution curves using the ratios of lengths of the corresponding control polygon of two Bézier curves. Especially we show that there are only three cases in the classification of local shapes with respect to the tangent direction and sign of curvature at each endpoint of the convolution curve. This special property can be extended to the convolution curve of two compatible Bézier curves of any degree n. We also classify all cases of global shapes of the convolution curves using the local shapes. The geometric properties of convolution curves are also presented when the ratio is critical point. Some examples are given to illustrate our characterization.

Approximate convolution with pairs of cubic Bézier LN curves

In this paper we present an approximation method for the convolution of two planar curves using pairs of compatible cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves with the same linear normal map to exist. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustration, we apply our method to the approximation of a font where the letters are constructed as the Minkowski sum of two planar curves. We also present numerical results using our approximation method for offset curves and compare our method to previous results.

Transmission of complex sounding signals in human respiratory system.

A complexity of respiratory path structure has already caused assumptions on existence of several ways of sound transmission from mouth to chest wall. The objective is a study of these mechanisms. The studied sample included 25 healthy subjects. They breathed with room air, helium-oxygen, and krypton-oxygen gas mixtures which had various sound velocities. Phase manipulated and frequency sweep signals were injected into mouth. Signals transmitted to chest wall sites (bottom part of trachea and basal area of right lung) were recorded by accelerometers. A convolution procedure was carried out. At least two signal arrivals are recognized in each convolution curve recorded above basal area for any gas mixture. One main arrival is recognized above trachea, which times are statistically dependent on gas mixture for all signals. The first arrival times above basal area of lung are also dependent on gas mixture. However, the second arrival times above basal area of lung are independent of gas mixture. Thus the arrival above trachea and the first arrival above basal area of lung, being dependent on filling gas sound velocity, are transmitted through airways lumen at least in part of their path. Alternatively the second arrival recorded above basal area of lung, being independent of filling gas sound velocity, seems to be transmitted through lung parenchyma. [The study was supported by RFBR grant 09-08-00105.]

TWO APPROXIMATE MINKOWSKI SUM ALGORITHMS

We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm.

Finding the best conic approximation to the convolution curve of two compatible conics based on Hausdorff distance

We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bézier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bézier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.

3D human model and joint parameter estimation from monocular image

In this paper we present a novel class of human model described by convolution surface attached to articulated kinematics skeletons. The human pose can be estimated from silhouette in monocular images. The contribution of this paper consists of three points: First, human model of convolution surface is presented and its shape is deformable when changing polynomial parameters and radius parameters. Second, convolution surface and curve correspondence theorem is presented to give a map between 3D pose and 2D contour. Third, we model the human silhouette with convolution curve in order to estimate joint parameters from monocular images and we also give an effective constraint function. Evaluation of this approach is performed on some video frames about a walking man. The experiment result shows that our method works well without self-occlusion.

GEOMETRIC CONIC SPLINE APPROXIMATION IN CAGD

We characterize the best geometric conic approximation to regular plane curve and verify its uniqueness. Our characterization for the best geometric conic approximation can be applied to degree reduction, offset curve approximation or convolution curve approximation which are very frequently occurred in CAGD (Computer Aided Geometric Design). We also present the numerical results for these applications.

Linear time computation of feasible regions for robust compensators

AbstractWe introduce an application of computational geometry, including figures of merit standard in the analysis of algorithms, to the design of robust control systems. With respect to system transfer function magnitude, we show how to compute feasible regions for compensators whose plant transfer function is the ratio of uncertain interval polynomials. Our solution sweeps the Minkowski quotient set of the corresponding Kharitonov rectangles. Enumerating the winding numbers of Minkowski sum convolution curves, we obtain optimal, linear time algorithms that eliminate three factors from the execution inefficiency of traditional gridding approaches. We illustrate with examples pertinent to quantitative feedback theory (QFT). Copyright © 2001 John Wiley & Sons, Ltd.

Polynomial/Rational Approximation of Minkowski Sum Boundary Curves

Given two planar curves, their convolution curve is defined as the set of all vector sums generated by all pairs of curve points which have the same curve normal direction. The Minkowski sum of two planar objects is closely related to the convolution curve of the two object boundary curves. That is, the convolution curve is a superset of the Minkowski sum boundary. By eliminating all redundant parts in the convolution curve, one can generate the Minkowski sum boundary. The Minkowski sum can be used in various important geometric computations, especially for collision detection among planar curved objects. Unfortunately, the convolution curve of two rational curves is not rational, in general. Therefore, in practice, one needs to approximate the convolution curves with polynomial/rational curves. Conventional approximation methods of convolution curves typically use piecewise linear approximations, which is not acceptable in many CAD systems due to data proliferation. In this paper, we generalize conventional approximation techniques of offset curves and develop several new methods for approximating convolution curves. Moreover, we introduce efficient methods to estimate the error in convolution curve approximation. This paper also discusses various other important issues in the boundary construction of the Minkowski sum.

A method of obtaining a highly parallel and monochromatic X-ray beam by successive diffraction

A beam of Cu Kα1 with an angular spread 1.0 × 10−5 radian and a wavelength spread (δλ/λ) = 0.8 × 10−5 was obtained by a silicon monochromator–collimator system. It consisted of two crystal elements in which the 620 and 333 reflections occurred successively. The two elements, originally monolithic, were cut after they had been glued to two metal blocks connected by a leaf spring. In order to satisfy the diffraction conditions in the two elements, one element was rotated about 10′ with respect to the other by bending the leaf spring. Using this beam, rocking curves of germanium single crystals were measured for the 220, 440 and 222 reflections. The half-value widths of the experimental curves were found to be 12.93± 0.01′′ for the 220 and 5.88±0.01′′ for 440 which are in good agreement with those of calculated convolution curves, 12.94 and 5.88′′ respectively. The integrated intensity of 222 was 1.59±0.1 × 10−8. From this value the structure factor of 222 was determined to be 1.05±0.08.