Missing data interpolation is a particular case of data
reconstruction, where the input data are already given on a regular
grid. One needs to reconstruct only the missing values in the empty
bins. In general,
is selected as a mask operator
(a diagonal matrix with zeros at locations of missing
data and ones elsewhere), and the problem becomes underdetermined. As
an alternative theory of Nyquist/Shannon sampling theory, compressed
sensing (CS) provides an important theoretical basis for
reconstructing images (Donoho, 2006). Analogous to CS, missing data
interpolation can be generalized to a NP-hard problem
(Amaldi and Kann, 1998) by using inverse generalized
velocity-dependent (VD)-seislet transform
:
An iterative procedure based on shrinkage, also called soft
thresholding, is used by many researchers to solve
equation 10 (Daubechies et al., 2004). However, it is
difficult to find the adjoint of
. In a general case,
forward generalized velocity-dependent (VD)-seislet transform
is an approximate inverse of
; then the
chain
is close to the identity operator
. Therefore, we can obtain an iteration with shaping
regularization (Fomel, 2008) as follows:
Combining equation 9 with the shaping solver
(equation 12), the framework of modified Bregman
iteration is as follows:
This is the analog of “adding back the residual” in the
Rudin-Osher-Fatemi (ROF) model for TV denoising (Osher et al., 2005). By
using a large threshold value, the modified Bregman iteration can
guarantee fast convergence of the objective function
(equation 8) and accurate recovery of the regularized
model
. The final interpolated result can be calculated
by
, where
is
the number of iteration.