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## Laplacian Pyramid

The Laplacian pyramid Burt and Adelson (1983) is a sequence of error images such that each error image is the difference between two levels of the Gaussian pyramid, that is:
 (7)
where is the image at pyramid level j+1 expanded to size of the image at level j. Thus it is immediately clear tht the Gaussian pyramid formation and expansion process is exact, in the sense that the original image is fully recoverable, as :
 (8)
The way to do this is to first expand the top pyramid level, , and then add the expanded version to Ln-1 to form . This process is repeated for each level until we reach the base of the pyramid where the original image is fully recovered. Since the top of the pyramid does not have an error image we can treat the image at the top of the pyramid as the error image :
 (9)
Notice that the value of each node of the Laplacian pyramid is the difference between the convolutions of two equivalent weighting functions hl and hl+1. This operation is similar to convolving the image with an appropriately scaled Laplacian weighting function and hence the name Laplacian pyramid. But the cost involved with this operation would be substantially more than constructing error images as a difference between two pyramid levels. The Laplacian pyramid can be treated as a set of band-pass filtered versions of the original image just as the Gaussian pyramid represents low-pass filtered versions of the original image. In the next section I use the concepts of both Gaussian pyramids and Laplacian pyramids and the fact that the pyramid forming process is exactly reversible to show how interpolation of missing data can be done using the pyramid scheme.

Next: Gaussian Pyramid Interpolation Up: Gaussian Pyramid Generation Previous: Mathematical basis of Pyramid
Stanford Exploration Project
4/5/2006