Image Processing Based on Spline Representation

本研究的目的是将我们的鲁棒几何计算方法应用于图像处理。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大量的计算时间。本研究旨在以基于样条的技术取代这些技术，从而节省空间和时间。在这一研究期间，我们获得了以下结果。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，我们可以在任何尺度下扩展和缩小图像数据。其次，将闵可夫斯基代数推广到一个更大的代数中，其中任何元素都有其逆。这种扩展保证了闵可夫斯基加法和减法的稳定性，因此我们可以在不担心闪光元素的情况下操纵数字。第三，构建并实现了平面上的Voronoi图、球面上的Voronoi图、三维空间上的Voronoi图、一般图形上的Voronoi图等鲁棒几何算法。第四，找到了用模算法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的角度有效地表示和处理图像和图形信息。

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.