Despite recent developments in improved acquisition, seismic data often remains undersampled along source and receiver coordinates, resulting in incomplete data for key applications such as migration and multiple prediction. We interpret the missing-trace interpolation problem in the context of matrix completion and outline three practical principles for using low-rank optimization techniques to recover seismic data. Specifically, we strive for recovery scenarios wherein the original signal is low rank and the subsampling scheme increases the singular values of the matrix. We employ an optimization program that restores this low rank structure to recover the full volume. Omitting one or more of these principles can lead to poor interpolation results, as we show experimentally. In light of this theory, we compensate for the high-rank behavior of data in the source-receiver domain by employing the midpoint-offset transformation for 2D data and a source-receiver permutation for 3D data to reduce the overall singular values. Simultaneously, in order to work with computationally feasible algorithms for large scale data, we use a factorization-based approach to matrix completion, which significantly speeds up the computations compared to repeated singular value decompositions without reducing the recovery quality. In the context of our theory and experiments, we also show that windowing the data too aggressively can have adverse effects on the recovery quality. To overcome this problem, we carry out our interpolations for each frequency independently while working with the entire frequency slice. The result is a computationally efficient, theoretically motivated framework for interpolating missing-trace data. Our tests on realistic two-and three-dimensional seismic data sets show that our method compares favorably, both in terms of computational speed and recovery quality, to existing curvelet-based and tensor-based techniques.

In this work, we propose a framework for multimodal data fusion at decision level under a multilayer hierarchical ensemble learning architecture. The architecture provides a generative discriminative model for probability density estimations and decreases the entropy of the data throughout the vector spaces. The architecture is implemented for human motion detection problem, where the motion analysis problem is formulated as a multi-class classification problem on audio-visual data. The vector space transformations are analyzed by the investigation of probability density and entropy transitions of data across the levels. The architecture provides an efficient sensor fusion framework for the robotics research, object classification, target detection and tracking applications.