Mathematical Sciences: Statistical Methods for Incomplete and Biased Data with New Applications

在具有线性失真的输入-输出系统中，从模糊输出中恢复输入信号的问题在科学和技术中是普遍存在的。当模糊输出没有被统计噪声破坏时，问题是完全确定的，并且相当于具有正参数的线性系统的数学逆，受制于解的正性约束。通过证明所有这种具有正性约束的线性反问题都可以解释为统计估计问题，可以开发出寻找问题的最大似然解的极其简单的算法。接下来将扩展到特定问题，如噪声退化、模糊参数的估计(点扩展函数的参数)和正则化技术。这种方法将被应用于新的问题，包括恢复因被摄体和相机镜头之间的相对运动而模糊的图像。信号(声音、图像等)的模糊在科学和技术中很常见，例如，观测信号的系统失真变成了数学家所说的具有正性约束的线性逆问题。这项研究将开发出广泛适用于此类问题的解决方法。这种方法的基础是利用统计原理，并利用这种线性逆问题与已有方法的不完整数据的统计估计问题之间的联系。

The problem of recovering an input signal from a blurred output, in an input-output system with linear distortion, is ubiquitous in science and technology. When the blurred output is not corrupted by statistical noise, the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters, subject to positivity con- straints on the solution. By showing that all such linear inverse problems with positivity constraints can be interpreted as statistical estimation problems, extremely simple algorithms can be developed for finding the problems' maximum likelihood solutions. Extensions to specific problems like noise degrada- tion, estimation of blurring parameters (parameters of the point spread function) and regularization techniques will follow. The methodology will be applied to new problems including restoration of images blurred by relative motion between the subject and the camera lens. The blurring of a signal (sound, image, etc.) is common in science and technology, for example the systematic distort observed signal becomes what mathematicians call linear inverse problems with positivity constraints. This research will develop methods of solution which are broadly applicable to such problems. The basis for this approach is to draw on statistical principles and exploit the connection between such linear inverse problems and statistical estimation problems for incomplete data where methods already exist.