Quaternion-based sparse tight frame for multicomponent signal recovery

Abstract

Multicomponent noise attenuation often presents more severe processing challenges than scalar data owing to the uncorrelated random noise in each component. Meanwhile, weak signals merged in the noise are easier to degrade using the scalar processing workflows while ignoring their possible supplement from other components. For seismic data preprocessing, transform-based approaches have achieved improved performance on mitigating noise while preserving the signal of interest, especially when using an adaptive basis trained by dictionary-learning methods. We have developed a quaternion-based sparse tight frame (QSTF) with the help of quaternion matrix and tight frame analyses, which can be used to process the vector-valued multicomponent data by following a vectorial processing workflow. The QSTF is conveniently trained through iterative sparsity-based regularization and quaternion singular-value decomposition. In the quaternion-based sparse domain, multicomponent signals are orthogonally represented, which preserve the nonlinear relationships among multicomponent data to a greater extent as compared with the scalar approaches. We test the performance of our method on synthetic and field multicomponent data, in which component-wise, concatenated, and long-vector models of multicomponent data are used as comparisons. Our results indicate that more features, specifically the weak signals merged in the noise, are better recovered using our method than others.

Admissibility and Frame Homotopy for Quaternionic Frames

“…Moreover, quaternions have become an increasingly popular tool for representing signals in data-driven applications. For example, quaternion-valued signals have recently been used to encode RGB images [18,21], measurements in industrial machinery [60], and multicomponent seismic measurements [62]; see also the special issue of Signal Processing on Hypercomplex Signal Processing [4]. This motivates the extension of signal processing techniques for classical (i.e., real-or complex-valued) signals to quaternionic signals.…”