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性（跳跃），或者在二维图像函数中存在不连续性曲线（边缘）的情况下，标准平滑方法（如局部多项式估计）在没有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中的边缘的位置及其对比度可能具有实质性的独立科学意义。目前，没有一般合适的方法是既不为建设的置信区间跳跃在1D回归曲线下观察到的模糊，即，卷积与点扩散函数，也不为均匀的置信集的边缘在2D图像。因此，开发和研究这些问题的置信区间和置信集是该项目的第一个主要目标。对于2d图像函数，经常出现附加的模糊，这在图像本身以及图像中包含的边缘的重建中必须考虑。因此，在边缘估计中系统地处理去模糊（除去噪外）是该项目的第二个主要目标。边缘估计器的理论最优性质和更实用的恢复方法的构造和实现都将被研究。

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.