从直线上的投影长度重构曲线
Reconstructing curves from lengths of projections onto lines
摘要：
在本文中，我们解决的问题，重建曲线的长度，其投影到线。我们首先注意到，曲线本身并不是由这些测量唯一确定的。然而，我们发现，曲线确定射影空间上的一个措施，作为一个功能的博雷尔子集的射影空间，返回的长度曲线平行的元素集。我们证明了投影长度数据可以表示为该测度在射影空间上的余弦变换。余弦变换是球面上的一种研究得很好的积分变换，已知球面是单射的。我们的结论是，测量的长度数据唯一地确定射影空间上的相关措施。然后，我们的特征曲线类，产生一个共同的措施，开始的情况下，分段线性曲线，然后通过限制，以获得更一般的曲线的结果。
英文摘要：
In this paper, we address the problem of reconstructing a curve from the lengths of its projections onto lines. We first note that the curve itself is not uniquely determined from these measurements. However, we find that a curve determines a measure on projective space which, as a function on Borel subsets of projective space, returns the length of curve parallel to elements of the set. We show that the projected length data can be expressed as the cosine transform of this measure on projective space. The cosine transform is a well studied integral transform on the sphere which is known to be injective. We conclude that the measured length data uniquely determines the associated measure on projective space. We then characterize the class of curves that produce a common measure by starting with the case of piecewise linear curves and then passing to limits to obtain results for more general curves.
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