Theory and practice of interpolation in the pyramid domain

Transformation artifacts

Going back and forth between the $(\omega ,x)$ and $(\omega ,u)$ domains will leave artifacts in $(t,x)$. There are two main reasons for this. First, the linear interpolation operator we use to transform one domain to the other is not unitary. Second, the pyramid transform (forward and adjoint operators) compresses and stretches the horizontal axis in ways that can affect the reconstruction of the frequency content, especially for the low frequencies [remember that a trace in $(\omega ,x)$-space is mapped into a radial trace in $(\omega ,u)$-space]. We illustrate this effect in Figure 2a. This Figure is the result of the following operation:

$${\bf\tilde d} = {\bf LL'd}$$ (10)

where ${\bf d}$ is the Fourier transform of Figure 1a and ${\bf\tilde d}$ contains the reconstructed data. In this example, because the $u$ axis is too coarse, the information of the slowest event (ringiest on the $u$ axis) disappears.

In order to mitigate these effects, we propose making the $u$ axis very dense. Theoretically, we could derive the maximum bin size $\delta u$ to accommodate the slowest event. From simple Fourier analysis, we can establish that

$$\delta u \leq \frac{1}{2 p_{max}},$$ (11)

where $p_{max}$ is the slope of the slowest event. This relation is a necessary, but not sufficient, condition for $\delta u$ since some of the artifacts are also due to the linear interpolation itself (i.e., the linear interpolation operator is not unitary). Therefore in practice, smaller $\delta u$'s than the one in equation 11 are necessary. Having very fine sampling in $u$ will help attenuate most of the transformation errors seen in Figure 2. Figure 3 shows ${\bf LL'd}$ for the same dataset, but with a sampling 12 times finer on the $u$ axis than on Figure 2b. Now, Figure 3a shows the two events with some remaining artifacts due to the linear interpolation operator only. In the next section, we introduce an algorithm that will both remove these remaining artifacts and also allow us to interpolate missing data.

Theory and practice of interpolation in the pyramid domain