Pyramid expansion

An approximate reverse of the process of generating a pyramid structure is termed as pyramid expansion. The main function of the expansion algorithm is to enhance a pyramid level of size M+1 by N+1 to size 2M+1 by 2N+1 by interpolating new sample values between those at the current level. Thus the expansion process applied l times to an image at pyramid level j would yield an image $\bold g_{j,l}$ which is of the same size as an image at the pyramid level j-l that is $\bold g_{j-l}$. Formally the expansion operation can be defined as:

 

$$\bold g_{\rm {l,n}}(i,j) = 4 \sum_{n=-2}^2\sum_{m=-2}^2\bold w(m,n) \bold g_{\rm {l,n-1}}(\frac{i-m}{2},\frac{j-n}{2}).$$ (5)

 

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Figure 1 Three levels of the Gaussian pyramid formation (top panel) and expansion (bottom panel).In the bottom panel each level is expanded to the next lower level.

 

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Figure 2 Four levels of the Gaussian pyramid formation (top panel) and expansion (bottom panel).In the bottom panel each level is expanded to the next lower level.

 

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Figure 3 Four levels of the Gaussian pyramid formation for data with holes.

 

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Figure 4 Filling up the holes at each level of the pyramid starting from the topmost pyramid level (upper left).

Notice that during the sum only values for which the indices are integer are included as contribution to the next higher level. A simple example consisting of two shot gathers is used to demonstrate the pyramid generation and expansion operation in Figure . The top panel of the figure shows three levels of the pyramid while the bottom panel shows the process of expansion from one level to the next. The next example (Figure ) consists of a stacked seismic. Again the top panel of the figure shows the levels of the Gaussian pyramid while the bottom panel shows the expansion. Each of the images in the top panels of Figures and have been obtained by applying the same 5-by-5 weighting function. Notice that as the pyramid levels stack up the sample density along each of the two axes reduces by a factor of 2. During the expansion process (bottom panels of figures and ) on the other hand the number of sample point increase by a factor of 2 along each axis.