基于动力学的投影重建：非随机情形
Reconstruction from projections using dynamics: Non-Stochastic Case
摘要：
从单个二维投影确定三维密度场的问题在没有附加假设的情况下是完全欠定的。虽然参数化反演通常用于解决这个问题，我们提出了理论结果沿着不同的路线消除不确定性。假设我们有一系列N张射线照片，测量X射线或质子在一个时间序列上通过一个不断变化的物体的衰减。再假设我们知道我们用射线照相术探测的物体的演化的决定性动力学规律。这些定律可用于将射线照片组合成一个“超级”测量，然后可以将其反转为对象序列。现在假设对象是$R^n$中的点，射线照片是$R^d$中的点。可以预期，我们所能做的最好的事情是当N = $\lceil n/d \rceil$时得到可逆性。最坏的情况是我们永远不会得到可逆性。我们仔细研究线性动力学的两种情况。我们表明，动力学规律，使这个最佳的短可逆时间是通用的。然后，我们用数值例子说明，并提出了一个猜想的非线性情况下。
英文摘要：
The problem of determining three-dimensional density fields from single two-dimensional projections is hopelessly underdetermined without additional assumptions. While parameterized inversions are typically used to solve this problem, we present theoretical results along a different route to the elimination of indeterminacy. Suppose that we have a series of N radiographs measuring X-ray or proton attenuation through an evolving object at a sequence of times. Suppose also that we know the deterministic dynamical laws governing the evolution of the object we are probing with radiography. The these laws can be used to combine the radiographs into one "super"-measurement which can then be inverted to the object sequence. Now suppose the objects are points in $R^n$ and the radiographs are points in $R^d$. It would be expected that the best we could do would be to get invertability when N = $\lceil n/d \rceil$. The worst case is that we never get invertability. We look at both cases carefully for linear dynamics. We show that dynamical laws giving this optimally short invertibility time are generic. We then illustrate with numerical examples and present a conjecture for the nonlinear case.
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