逆问题中的贪心技术
Greedy techniques for inverse problems
AI总结：
本文提出了一种用于逆问题中优化间接测量选择的贪心框架，结合核插值与外推技术，通过显式误差界实现高质量重建。
摘要：
逆成像问题依赖于有限且间接的测量，使得重建高度依赖于正则化和样本位置。我们引入了一种新的贪心框架，用于在算子余域中最优选择间接测量。我们的方法采用两步方案，结合基于核的插值和外推。在此框架中，贪心方案可以基于残差，即根据特定目标函数的当前近似误差选择点，或基于误差，即使用先验误差指标独立于残差来选择点。对于后者，我们推导出显式的误差界，量化了通过插值和外推传播的近似误差。数值应用表明，所提出的贪心采样策略仅使用少量可用测量即可实现高质量的重建。
英文摘要：
Inverse imaging problems rely on limited and indirect measurements, making reconstruction highly dependent on both regularization and sample locations. We introduce a novel greedy framework for the optimal selection of indirect measurements in the operator codomain, specifically tailored to inverse problems. Our approach employs a two-step scheme combining kernel-based interpolation and extrapolation. Within this framework, greedy schemes can be residual-based, where points are selected according to the current approximation error for a specific target function, or error-based, where points are chosen using a priori error indicators independent of the residual. For the latter, we derive explicit error bounds that quantify the propagation of approximation errors through both interpolation and extrapolation. Numerical applications to solar hard X-ray imaging demonstrate that the proposed greedy sampling strategy achieves high-quality reconstructions using only a few available measurements.
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