Theory and practice of interpolation in the pyramid domain

The pyramid transform

To begin introduce the forward transformation from the pyramid domain $(\omega ,u)$ to the Fourier space $(\omega ,x)$ as follows:

$$d(\omega,x)=m(\omega,u=\omega\cdot x).$$ (2)

The adjoint transformation is defined in equation (1). These definitions can be extended in 3-D easily for the forward case:

$$d(\omega,x,y)=m(\omega,u=\omega\cdot x,v=\omega\cdot y),$$ (3)

and for the adjoint case:

$$m(\omega,u,v)=d(\omega,x= u /\omega,y=v/\omega),$$ (4)

where $y$ is a spatial axis in the crossline direction (offset of mid-point position) and $v$ the dual of $y$ in the pyramid domain. Equations (2) and (3) can be rewritten in a more compact form:

$${\bf d}={\bf Lm},$$ (5)

where $\bf {L}$ is the pyramid transform. Similarly, equations (1) and (4) can be rewritten as

$${\bf m}={\bf L'd},$$ (6)

where $\bf {L'}$ is the adjoint of ${\bf L}$. Note that the remapping between $x$ and $u$ (plus $y$ and $v$ in 3-D) requires an interpolation operator. A linear interpolation process looping over the data space $d(\omega,x)$ is applied in all our results.

Theory and practice of interpolation in the pyramid domain