Theory of missing data interpolation

Claerbout (1992) formulates the basic principle of missing data interpolation as follows:

A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy.

Mathematically, this principle can be expressed by the simple equation  

$$D m \approx 0\;,$$ (1)

where m is the data vector, and D is the specified filter. The approximate equality sign means that equation (1) is solved by minimizing the squared norm (the power) of its left side. Additionally, the known data values must be preserved in the optimization scheme. Introducing the mask operator K, which can be considered as a diagonal matrix with zeros on the missing data locations and ones elsewhere, we can rewrite equation (1) in the more rigorous form  

$$D (I-K) m \approx - D K m = -D m_k\;,$$ (2)

in which I is the identity operator, and m_k is the known portion of the data. It is important to note that equation (2) corresponds to the limiting case of the regularized linear system  

$$\left\{\begin{array} {l} K m = m_k\;, \ \lambda D m \approx 0\end{array}\right.$$ (3)

for the scaling coefficient $\lambda$ approaching zero. This means that we put far more weight on the first equation in (3) and use the second equation only to constrain the null space of the solution. Applying the general theory of data-space regularization Fomel (1997), one can immediately transform system (3) to the equation  

$$K P x \approx m_k\;,$$ (4)

where P is a preconditioning operator, and x is a new variable, connected with m by the simple relationship

$$m = P x\;.$$

In theory, equations (4) and (2) have exactly the same solutions if the following condition is satisfied:  

$$P P^T = (D^T D)^{-1}\;,$$ (5)

where we need to assume the self-adjoint operator D^T D to be invertible. If D is represented by a discrete convolution, the natural choice for P is the corresponding deconvolution operator:  

$$P = D^{-1}\;.$$ (6)

The helix transform provides a constructive way of implementing multidimensional deconvolution by one-dimensional recursive filtering.