Color 3D Reverse Engineering

This paper presents a principle and a method of col or 3D laser scanning measurement. Based on the fundamental monochrome 3D measureme nt study, color information capture, color texture mapping, coordinate computati on and other techniques are performed to achieve color 3D measurement. The syste m is designed and composed of a line laser light emitter, one color CCD camera, a motor-driven rotary filter, a circuit card and a computer. Two steps in captu ring object’s images in the measurement process: First, the color CCD camera ta kes the color picture of the object to be used for texture mapping in post proce ssing. Secondly, the monochrome data of the object is taken. The laser light emi tter is started; the light plane and the surface of the object intersect to form an undulate line, which forms an image on the CCD sensor. After the optic- electronics transformation, the electronic signals are captured and send to the computer. A new color 3D measurement model was derived, and a rapid calibra ting method to measure the system parameters was proposed--optical plane equat ion calibrating method. A calibrating drone was designed and built. 3D color sca nning system not only provides an object’s individual point information in the 3D coordinate, in the mean time, it provides the color information of each indiv idual point. This paper also advances a multi-frame auto-merging method, i.e. several frames of color 3D digital images measured are merged quickly according to their curvature characteristics and RGB information. Initial matching and fin e tune of the registration are completely performed by the computer; initial mat ching is via user interface in helping computer to find the transformation matri x. The transformation matrix is found according to the geometric characteristics chosen by hand. After the initial transformation matrix is found, fine tune of the registration is preformed to do the optimum adjustment to achieve a complete color textured 3D model. The system can be broadly used in the fields of produc t design, mold manufacture, multimedia, game development, animation, medical eng ineering, antique digitization, etc.

An Algorithm for Finding Internal Faces Based on the 3D Coordinates

This paper presents a direct method to find the internal faces. It uses four sets of information： get the coordinates of each point on the 3D composite objects; （~）based on the 3D coordinates to calculate all the plane equation of two objects; （~）calculate all the intersecting lines of the two objects and exclude the intersecting lines which are not within the scope of the plane;finding the internal face, the remaining intersecting lines formed the plane is the internal face of the two objects. We proposed a new algorithm for finding internal faces. This algorithm could be adapted to any regular composite objects .The algorithm works well for finding internal faces.

CONVERGENCE OF THE POINT VORTEX METHODS FOR EULER EQUATION ON HALF PLANE

In this paper, we study the point vortex method for 2-D Euler equation of incompressible how on the half plane, and the explicit Euler's scheme is considered with the reflection method handling the boundary condition. Optimal error bounds for this fully discrete scheme are obtained.

ON THE FUNDAMENTAL PROBLEM FOR AN INFINITE ELASTIC PLANE BONDED BY DIFFERENT ANISOTROPIC MATERIALS WITH CRACKS

In this paper, the first fundamental problem for an infinite elastic plane bonded by different anisotropic materials with cracks of arbitrary shape is discussed. The problem is reduced to a certain system of singular integral equations with several undetermined constants, which is proved to be uniquely solvable when these constants are suitably and uniquely chosen.

A Proof of the Jacobian Conjecture on Global Asymptotic Stability

Let f∈C^1(R^2, R^2), f(0)=0. The Jacobian Conjecture states that if for any x∈R^2, the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.