Optimal estimation and confidence sets for discontinuities in noisy, blurred regression functions

光滑一元(一维)和二元(二维)回归函数的非参数估计是一个非常成熟的理论。然而，在一维回归中存在不连续(跳跃)，或者在二维图像函数中存在不连续曲线(边缘)时，标准的平滑方法，如局部多项式估计，如果没有修改就不再起作用。此外，一维的间断点和高度以及二维的边缘位置和对比度可能具有重大的独立科学意义。目前，对于在模糊情况下观察到的一维回归曲线中的跳跃的置信度区间的构造，即与点扩展函数的卷积，或者对于二维图像中边缘的均匀置信集，都没有普遍合适的方法可用。因此，开发和研究这些问题中的可信度区间和集合是该项目的第一个主要目标。对于2D图像函数，在重建图像本身以及包含在图像中的边缘时必须考虑额外的模糊。因此，在边缘估计中系统地处理去模糊(除去噪之外)是该项目的第二个主要目标。既要研究边缘估计器的理论最优性，也要研究更实用的恢复方法的构造和实现。

Nonparametric estimation of smooth univariate (1d) and bivariate (2d) regression functions is a very well-developed theory. However, in the presence of a discontinuity (jump) in a 1d regression, or of a discontinuity curve (edge) in a 2d image function, standard smoothing methods like local polynomial estimation do no longer work without modifications. Further, the discontinuity point and height in 1d as well as the location of the edge and its contrast in 2d may be of substantial independent scientific interest. Currently, no generally appropriate methods are available neither for the construction of confidence intervals for jumps in 1d regression curves which are observed under blurring, that is, convolution with a point-spread function, nor for uniform confidence sets for edges in 2d images. Developing and investigating confidence intervals and sets in these problems is therefore the first main aim of the project. For 2d image functions, often an additional blurring occurs which must be taken into account in reconstructions of the image itself as well as of edges contained in the image. The systematic treatment of deblurring (in addition to denoising) in edge estimation is therefore the second main aim of the project. Both the theoretical optimality properties of edge estimators and the construction and implementation of more practical recovering methods shall be investigated.