Image Processing Based on Spline Representation

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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.