The three-dimensional extension of t-x prediction

Extending the t-x prediction to three dimensions is a simple matter of defining the 3-dimensional convolution and its adjoint, that is, its conjugate transpose. Instead of dividing the input into 2-dimensional windows, the input volume is broken into 3-dimensional sub-volumes. Each of these sub-volumes is used in a separate least-squares problem for calculating a 3-dimensional filter, which is a straightforward extension of the calculation of a 2-dimensional t-x filter. Figure  shows the configuration of the 3-dimensional filter I used. This configuration is a modification of the more general 3-dimensional filter shown in Claerbout 1992a, page 198. While the 3-dimensional filter might be made symmetrical in space by constraining the filter coefficients across the central element to be equalGulunay et al. (1993), I simply applied the filter in multiple directions and averaged the results.

 

DDDfilter
Figure 4 A three-dimensional lateral prediction filter. The darkened cube is the output position.