Interpolating beyond spatial Nyquist is achieved by estimating the Fourier spectrum using matching pursuit. This is an iterative method where, for each iteration, a discrete Fourier transform of the data is first computed. Then, the Fourier component with maximum energy is selected.

This component is added to the estimated spectrum. Additionally, an inverse Fourier transform of the selected Fourier component is computed. Finally, the selected Fourier component from the input data is subtracted.

This approach makes use of ideas from compressive sensing theory, and provides reconstruction of data beyond what was thought possible according to classical sampling theory.

• Interpolation of additional source lines, detector lines, or both.
• Regulations of source lines, detector lines, or both.
• Regularization of midpoint locations.
• Regularization of offsets..

• Provides output that can be any specified location, either regular or irregular.
• Improves handling of aliased data by using priors.
• Handles regularly sampled aliased input data.
• Is robust in the presence of noise and large gaps.
• Provides excellent amplitude preservation.

• Is a multidimensional implementation, up to 5D (four spatial coordinates and time).
• Is not limited to 2× beyond aliasing interpolation.
• Is applied in overlapping spatial/temporal windows.
• Uses the exact locations of the input data.

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