Mathematical Sciences: Inverse Estimation Problems

科学和工程中的许多问题都有一个共同的形式，即必须从对已知信号变换的模糊观察中恢复未知的“信号”。从数学上讲，这需要求解一个（线性）方程组，通常在无限维空间中，其中的数据仅足以构建变换后信号的噪声近似值。解决方案取决于操作符的反转。在希尔伯特空间中研究这个反演问题，可以利用谱分析和谐波分析的技术，为构造解提供一个统一的方法。然后可以检查误差分析和交叉验证等统计方面的问题。这类问题的一个著名例子是医学实践中使用的计算机断层扫描技术，用于恢复人体内部结构的图像。因为在实际操作中，测量结果会因误差而变得模糊，而且因为只有有限的x射线照射是可能的，所以对“信号”的测量（在这种情况下是x射线通过人体的衰减）确实是不完美的。不过，非常精细的图像是可以重建的，而且非常有用。其他著名的应用是图像重建和各种反卷积问题。

Many problems in science and engineering share the common form that an unknown "signal" has to be restored from blurred observations on a known transformation of the signal. Mathematically this requires solving a (linear) system of equations, typically in an infinite dimensional space, where the data only suffice to construct a noisy approximation of the transformed signal. Solution depends upon the inversion of the operator. Studying this inversion problem in a Hilbert space setting can provide a unifying approach for constructing solutions, by exploiting techniques from spectral and harmonic analysis. Then statistical aspects such as error analysis and cross-validation can be examined. A famous example of this kind of problem is the technique of computerized tomography used in medical practice to recover an image of internal structures in the body. Because in practice, measurements are blurred by errors, and because only a limited x- ray exposure is possible, measurement of the "signal" (in this case the attenuation of the x-rays passing through the body) is imperfect indeed. Still, very refined images can be reconstructed and are very useful. Other well-known applications are image reconstruction and all kinds of deconvolution problems.