Fomel (2009,2007) introduces shaping regularization in inversion problem,
which regularizes the under-determined linear system by mapping the model
to the space of acceptable models.
Consider a linear system given as
is the forward-modeling map,
is the model vector, and
is the data vector.
Tikhonov regularization method amounts to minimize the least square problem bellow
is the regularization operator, and is a scalar parameter.
The solution for equation 28 is:
is the least square approximated of
is the adjoint operator.
If the forward operator
is simply the identity operator, the solution of
equation 29 is the form below:
which can be viewed as a smoothing process. If we let:
Substituting equation 32 into equation 29
yields a solution by shaping regularization:
The forward operator
may has physical units that require scaling. Introducing
, equation 33 be written as:
with square and invertible
. Equation 34 can be written as:
The conjugate gradient algorithm can be used for the solution of the equation 35.