The simplest way to detect a pattern that may appear in an image at any scale is by simple convolution of the target pattern, constructed at various scales, with the image or to convolve a pattern of a fixed size with different versions of the image represented at different resolutions. The immediate bottleneck to this convolution-based method for detecting a pattern is the enormous cost involved in carrying out all the required convolutions. The computer graphics industry has developed a method termed as the image pyramid data structure for efficient scaled convolutions through reduced image representation Burt (1981). This pyramid data structure consists of a stack of copies of the initial image such that both spatial density and resolution decrease as we move from one level of the stack to the next. This data structure can be generated with a highly efficient iterative algorithm that requires fewer computational steps to generate a series of reduced images than are required by the FFT method to compute a single filtered image Burt (1981). Once a fast algorithm is available for generating multi-resolution images in the spatial domain, missing regions of the image can then be filled up also at different scales, starting from the coarest scale and proceeding to more and more finer scales.
In this paper I show how interpolation of seismic data can be carried out using the pyramid structure. The local *n* point operator that is used as the basis function in the pyramid generation process represents a Gaussian distribution in the limit , hence the pyramid structure is termed Gaussian pyramid. I first show how the pyramid structure is generated and then show how interpolation is carried out at different levels of the pyramid to restore missing data.

4/5/2006