where is the regularization operator, and is a scalar parameter. The solution for equation 28 is:

Where 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:

or

Substituting equation 32 into equation 29 yields a solution by shaping regularization:

The forward operator may has physical units that require scaling. Introducing scaling into , equation 33 be written as:

If with square and invertible . Equation 34 can be written as:

The conjugate gradient algorithm can be used for the solution of the equation 35.

2020-07-18