The Study Of Several Problems In Graphics And Image Processing

The key subject in computer graphics is to use a mathematical model tocharacterize image problems, and then represents its corresponding3D models,e.g. computer aided design, movies and games and so on. Correspondingly,image processing focuses on improving existing images’ quality to reduce noise,enhance contrast or others. With the development of digital medical§computergraphics and image processing is fusing, and many significant problem aregenerated, and it is difcult and not necessary to distinct them. With the rapiddevelopment of3D scanner devices and mobile internet, graphics and imageprocessing faces new challenge with respect to high level, multi-modal and bigdata etc. How to use a more convenient method, and use exist meshes andimages to realize information reuse for graphics and images, which should havean extensive applications and industrialization future in computer animation,image inpainting, virtual reality and industrial design and so on.In this thesis we focus on several important subjects.First we construct a modeling system and considers its theory basis. Apart-based assembly3D modeling system is designed in the third chapter, andits input is user’s sketch, and a part model is searched and assembled by thesketch. The system can analysis the sketch’s feature, and retrieval the most similar part model in time, and the part is placed in the original model to re-place the old part model, and a new model is designed. A solid theory basis isprovided in chapter5for interpolating surface boundary curves: the dynamicprogramming solution for developable surface interpolation problem. The pro-posed method can be used to interpolate boundary curves of multi-connectionregions, and it is also generalized to variational discrete developable surface in-terpolation problem. In particular, we also derive a G1continuity condition tosmoothly interpolate the converted quad strips using Coons patches, and thecondition is mathematically simple and beautiful and easy to be implemented.A lot of industrial model results are presented in this thesis, which also statethe reasonability of the proposed methods.Second we provide theory basis for architectural geometry. In architec-tural design, quad mesh is extensively used due to its steady forced structure,but the study of quad mesh parameterization is rare. We attempt to solvethis problem in chapter4, and a planar-like and a spherical-like quad meshparameterization methods are proposed. For planar-like quad mesh parame-terization, we design a conformal energy functional, and the minimal solutionof the functional can be used to flat quad meshes on a planar domain withminimal angle distortion. For spherical-like quad mesh parameterization, weconstruct a surface geometrical flow (GPDE, geometrical partial diferentialequation) which can conformally map quad meshes onto spherical domains.A mathematical model is proposed to quantitatively characterize the param-eterization results, and comparisons are presented in detail with respect totriangular mesh parameterization results.Finally, we investigate the intrinsic image restoration problem in imageprocessing. Classical logarithm total variational model is generalized, and thegeneralized version can be used to deal with a single image or a sequence ofimages. Using the ADMM optimization technique, the generalized one is solvedby a iterative scheme. The new model not only preserves the advantage of edgepreserving, but also reduces the ringing-efect and obtain a stable solution. Based on generalized one, plenty of image applications are presented, includingface recognition, face relighting and l1-sparse constraint image recoloring andso on. These results shown in this thesis are state-of-art which also illustratethat the proposed model has its real significance.