Construction Method Of 3D Birational Mapping Based On Barycentric Coordinates

Image morphing technology is the basis for the development of computer graphics and digital image technology,which is widely used on television,advertising,medicine.Image transformation operation involves mapping problem between different regions.Typically,people do not consider the inverse mapping.However,in order to facilitate the subsequent application of deformation technology,we need to study the inverse mapping.In particular,if the map is a bi-rational mapping.That is,the mapping and its inverse mapping are both rational.Computing based on rational map is convenient and efficient.T.W.Sederberg presented a method to construct a birational map on quadrilaterals.The specific method is to attach a weight to each control point.When these weights satisfy one special equation,the mapping is a birational map.T.W.Sederberg then showed that generic ratioanl maps of degree 1?n can be made birational by a suitable assignment of weights to the Bézier or B-spline control points.Birational quadriateral maps and birational maps of degree 1?n were in 2-dimensional space.Based on the knowledge of three dimensional generalized barycentric coordinates,the results of birational mapping on planar quadrilateral are generalized to 3-dimensional convex hexahedron.That is,by assigning a suitable weight to every vertex of a convex hexahedron,a three dimensional birational mapping is achieved.In addition,an example is given to illustrate the correctness of our method.