The well-known iterative shrinkage-thresholding (IST) algorithm is used for solving equation 3 with a sparsity-promoting constraint:
is the transformed domain data such that
is a tight frame such that
(e.g. Fourier transform),
is a nonlinear thresholding operator,
denotes adjoint. Considering that
, combined with equation 7, we get:
which is equal to equation 6 with
taken as an identity operator.
Thus, we prove that the IST and shaping regularization are actually mathematically equivalent.
Seismic data interpolation using nonlinear shaping regularization