In this paper, we study the decomposition of sets of consecutive integers, i.e., how to express the set {0, 1, . . . ,mn−1} as the Minkovski sum of two sets, one with m terms and the other with n terms. First, we translate the problem into the language of polynomials. Then we use the properties of cyclotomic polynomials to determine the structure of all the solutions. Then, we deduce formulas for the number of such expressions. Finally, we give an algorithm to calculate the number of such expressions and analyze its computational complexity.

In virtual reality/augmented reality (VR/AR) applications, especially medical image visualization, it is highly desirable to magnify region of interests. In this work, we propose a novel method for virtual magnifying glass for visualizing volumetric data based on Optimal Mass Transportation. An optimal mass transportation map deforms a volume to itself, transforms the source measure (volumetric element) to the target measure with the minimal transportation cost. Solving the optimal mass transportation problem is equivalent to a convex optimization, and can be converted to computing power Voronoi diagrams in classical computational geometry. The proposed method allows the user to accurately control the target measure, and select multiple regions of interests with irregular shapes. We demonstrate the effectiveness and efficiency of our method with several volume data sets from medical applications.

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