Convolutive Sparse Decomposition in Inverse Problems: A Sensor Space Approach

We propose a novel approach that applies convolutional dictionary learning (CDL- decomposition directly in the sensor domain before solving the inverse problem, contrasting it with the classical approach where the inverse problem is solved first. This method enhances computational efficiency and improves spatial localization accuracy, especially when he sources activity is sparse. Theoretically, we establish a formal equivalence between CDL decomposition in the source and sensor domains, laying the groundwork for a practical transition between these two approaches. Using synthetic data, we compare two approaches: (1) resolving the inverse problem first and then performing CDL decomposition, and (2) initially decomposing in the sensor domain, followed by solving the inverse problem on the spatial dictionary. Our results show that the second approach offers improved reconstruction quality and reduced computational costs, making it a promising strategy for addressing complex inverse problems.