We investigate a class of recovery problems for which observations are a noisy combination of continuous and step functions. These problems can be seen as non-injective instances of non-linear ICA with direct applications to image decontamination for magnetic resonance imaging. Alternately, the problem can be viewed as clustering in the presence of structured (smooth) contaminant. We show that a global topological property (graph connectivity) interacts with a local property (the degree of smoothness of the continuous component) to determine conditions under which the components are identifiable. Additionally, a practical estimation algorithm is provided for the case when the contaminant lies in a reproducing kernel Hilbert space of continuous functions. Algorithm effectiveness is demonstrated through a series of simulations and real-world studies.

翻译：我们研究一类观测数据为连续函数与阶跃函数噪声组合的恢复问题。此类问题可视为非单射非线性独立成分分析的特例，可直接应用于磁共振成像中的图像去噪。从另一角度看，该问题也可理解为存在结构化（平滑）污染物的聚类问题。研究表明，全局拓扑性质（图连通性）与局部性质（连续分量的平滑度）相互作用，共同决定了成分可辨识的条件。此外，针对污染物位于连续函数再生核希尔伯特空间中的情形，我们提出了一种实用估计算法。通过一系列仿真实验和真实案例研究，验证了该算法的有效性。