We consider the problem of recovering characteristic functions $u:=\chi_\Omega$ from cell-average data on a coarse grid, and where $\Omega$ is a compact set of $\mathbb{R}^d$. This task arises in very different contexts such as image processing, inverse problems, and the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface $\Gamma:=\partial\Omega$ by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes, the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

翻译：我们考虑从粗网格上的单元平均数据恢复特征函数 $u:=\chi_\Omega$ 的问题，其中 $\Omega$ 是 $\mathbb{R}^d$ 中的紧集。该任务出现在诸如图像处理、反问题以及有限体积格式中界面的精确处理等截然不同的背景下。已知线性恢复方法性能较差，而基于通过几何上更简单的界面对跳跃界面 $\Gamma:=\partial\Omega$ 进行局部重建的非线性策略可能带来显著改进。我们研究了两类主要的局部重建方案：第一类基于非线性最小二乘拟合，第二类基于显式计算拟合数据的多项式形状曲线，后者能实现更简单的数值计算和更高的几何拟合阶数。对于每种方案，我们推导了一个通用的理论框架，使得恢复误差能够被最佳逼近误差控制在一个固定乘法常数范围内。二维数值试验验证了这些策略预期的逼近阶数。我们还讨论了几种扩展情形，特别是带角点的分段光滑界面的处理。