The Radon transform on a family of curves in the plane. II
The Radon transform on a family of curves in the plane. II
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平面上一系列曲线的氡变换。
DOI：
10.1090/s0002-9939-1981-0624923-1
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发表时间：
1981
期刊：
影响因子：
2.1
通讯作者：
A. Cormack
中科院分区：
文献类型：
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作者：
A. Cormack
Radon问题，即从平面内沿着直线的积分中恢复函数，由于其许多实际应用，特别是在医学上，近年来受到了相当大的关注。直线之所以重要，是因为在许多应用中，探测剂，光子，带电粒子和声子，遵守牛顿第一定律，除非它们发生相互作用，否则它们会以直线运动。很容易设想探测代理不遵循直线的情况，因此Radon问题需要推广到其他曲线。这样的推广已经对某些椭圆和圆[5]，[7]以及通过原点的圆[1]，[3]进行了。当某些傅立叶展开，这些最后的圆的治疗是如此类似的直线使用相同的扩展[1]，[2]，[6]，它似乎希望看到他们是否都可以是一个更一般的一组曲线在平面上的特殊情况。本注记的目的是把平面上的一族曲线当作特殊情况来处理，这些曲线确实包含这些曲线。设（r，θ）和（p，q）为平面中的极坐标，并考虑由下式给出的曲线：
Radon's problem, the recovery of a function from its integrals along straight lines in the plane, has received considerable attention in recent years because of its many practical applications, notably in medicine. The reason that straight lines are significant is that the probing agents in many applications, photons, charged particles, and phonons, obey Newton's first law and travel in straight lines unless they undergo interactions. It is easy to conceive of situations in which the probing agents do not follow straight lines, so Radon's problem needs to be generalized to other curves. Such generalizations have been made for certain ellipses and circles [5], [7], and for circles through the origin [1], [3]. When certain Fourier expansions are made, the treatment of these last circles is so similar to the treatment of straight lines using the same expansions [1], [2], [6] that it seemed desirable to see whether they could both be special cases of a more general set of curves in the plane. The purpose of this note is to treat a family of curves in the plane which does contain these as special cases. Let (r, 9) and (p, q) be polar coordinates in the plane and consider the curves given by