Image Processing Based on Spline Representation

本研究的目的是应用我们的方法，鲁棒几何计算的图像处理。传统的图像处理技术是基于数字图像的像素结构，因此需要巨大的存储空间和巨大的计算时间。本研究的目的是以样条技术取代这些技术，以节省空间与时间。首先，构造了一种用样条曲面表示数字彩色图像的方法，由于样条曲面的灵活性，我们可以在任意尺度上扩展和收缩图像数据。其次，将Minkowski代数推广到任意元素都有其逆元的更大代数。这种扩展保证了闵可夫斯基加法和减法的稳定性，因此我们可以操纵图形而不必担心flase元素。第三，构造并实现了平面上的Voronoi图、球面上的Voronoi图、三维空间中的Voronoi图、一般图形的Voronoi图等的鲁棒几何算法。利用模运算的方法，找到了判断大整数符号的几种有效方法，这些结果使我们能够从时间和空间两个方面有效地表示和处理图像和图形信息。空间的观点。

The purpose of this research was to apply our method for robust geometric computation to image processing. Conventionally image processing techniques are based on the pixel structure of the digital images, and consequently require huge space of memory and huge time of computation. This research aims at replacing these techniques by spline-based techniques and thus saving space and time.In this research period, we obtained the following results. First, a method was constructed for representing digital color images by spline surfaces ; because of the flexibility of spline surfaces, we can expand and shrink image data in any scales. Secondly, the Minkowski algebra was extended to a larger algebra in which any element has its inverse. This extension guarantees the stability of the Minkowski addition and substraction, and therefore we can manipulate figures without worrying about flase elements. Thirdly, robust geometric algorithms were constructed and implemented for Voronoi diagram on the plane, Voronoi diagram on the sphere, Voronoi diagram in the three-dimensional space, Voronoi diagrams for general figures, etc. Fourthly, several efficient methods were found for judging the sign of a large integer by modular arithmetic.Those results altogether enable us to represent and manipulate image and figure information efficiently from both time and space points of view.