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

平滑单变量 (1d) 和双变量 (2d) 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性（跳跃）或二维图像函数中存在不连续性曲线（边缘）时，局部多项式估计等标准平滑方法在不进行修改的情况下不再起作用。 此外，1d 中的不连续点和高度以及 2d 中边缘的位置及其对比度可能具有重大的独立科学意义。目前，对于在模糊条件下观察到的一维回归曲线跳跃的置信区间的构造（即，与点扩散函数的卷积）以及对于二维图像中的边缘的统一置信集，都没有普遍合适的方法。因此，开发和研究这些问题的置信区间和集合是该项目的首要目标。对于二维图像函数，通常会发生额外的模糊，在重建图像本身以及图像中包含的边缘时必须考虑到这一点。因此，边缘估计中去模糊（除去噪之外）的系统处理是该项目的第二个主要目标。应研究边缘估计器的理论最优性特性以及更实用的恢复方法的构建和实现。

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.