Shape Deformation Method Based On Iterated Function Systems Sub-study

This thesis studies the approach of the fractals morphing based on the iterated function systems (IFS). In general, morphing can be defined as a gradual, smooth and natural transformation from one key shape into another and also can be split into two more or less distinct sub problems, namely, feature correspondence and interpolation trajectory. Morphing which is included graphics, image and fractals metamorphosis constitutes a wide and important area of computer graphics activity. With the rapid development of computer graphics and the technology of hardware, apart from its theoretical interest, morphing has the practical use ranging from computer animation, through virtual reality, to the industrial design and entertainment:The methods of the fractals morphing have been offered for graphics and images with the features of the fractals. The main content of this thesis is to be discussed the approaches of morphing the two-dimensional and three-dimensional fractals attractors based on the iterated function systems.First of all, we introduce the traditional methods of morphing fractals. Furthermore, we point out the necessity of studying the fractals, morphing.Then in the aspect of the two-dimensional fractals' morphing based on the iterated function systems, at first the IFSP canonical form with original probability vector is defined. Then we pick up the parameters corresponding to dynamics of the transformations with the singular value decomposition. And at last we summarize the variety rule of the IFSs with parameters combining with the actions of the affine transformations. Whereas feature correspondence has been established artificially in traditional methods, we present a new method to realize the feature correspondence of the two-dimensional IFS fully automatically. Firstly the similar function of the IFS affine transformations is constructed, and then the optimum path with restricted condition in the IFS fuzzy similar graph is searched. Using the path that maximized the membership function in the IFS fuzzy set, we can establish the feature correspondence between two iterated function systems. Considering the reality and universality of the applications on the natural color fractals, we still construct a modelof the two-dimensional true-color IFS fractals and realize the morphing process by the affine transformations' corresponding, canonizing, matching and interpolation with the polar decomposition.Finally we generalize the morphing method to the three-dimensional space. Firstly we present two ways to establish the feature correspondence artificially, namely, the affine transformations act on unit cuboids and directly act on the attractors. Then we discuss the IFSP canonical form in the three-dimensional space and lastly achieve the fractals attractors' morphing using the polar decomposition. We substitute the quaternions interpolation for the interpolation of the rotation transformations and the fix-points in the three-dimensional space and it will shorten the time of the morphing effectively and deepen the theory foundation of the morphing ulteriorly.