Parametric Convolutional Dictionary Learning and its Applications to Seismic Data Processing

Convolutional dictionary learning (CDL) can represent signals and images via the superposition of components given by the convolution of sparse coefficients (features) and the elements of a dictionary (filters). The filters represent universal signals that can model different images, whereas the coefficients are intrinsic to one particular image. Estimating the coefficients and the filters from a set of observed signals is similar to a blind deconvolution problem where we aim to simultaneously represent a signal via the convolution of two unknown signals. Classical CDL provides data-dependent filters that, in the seismic data processing case, might not have a solid resemblance to typical waveforms that one observes in seismic records. This limits the dictionary's representation and discriminability, thus suffering from suboptimal denoising or reconstruction results. To address this challenge, we propose a new CDL algorithm. The proposed approach introduces a parametric constraint to enforce simplicity on the filters, guiding the learning process toward a more efficient and structured representation of the data. Specifically, we restrict each filter to include one single waveform parametrizable via a second-order traveltime curve and a seismic wavelet. The learned dictionary comprises linear and parabolic events that adapt adequately to observed seismic waveforms and resemble local Radon transform basis functions. The alternating direction method of multipliers (ADMMs) is adopted to solve the proposed parametric convolutional learning problem. The experimental results demonstrate that the proposed method achieves superior reconstruction results compared to the existing convolutional and patch-based dictionary learning methods.