Introduction

The importance of analyzing images at many scales arises from the nature of the images themselves. Any analysis procedure that is applied at a single scale will tend to miss information at other scales. The immediate solution to this problem would be to carry out the analysis at all possible scales simultaneously. This is especially true for seismic data where the low frequency components in an image represent the general trend of the image, while the high frequency components add details to this underlying trend. Recently methods have been suggested Abma and Kabir (2005) in which the missing data is reconstructed by adding in frequency components to the missing region with each iteration. While the Fourier transform is a natural way of efficiently separating the various scales in an image, but in the Fourier domain we can no longer recognize the spatial features in their usual form as in the original non-transformed domain. A desirable alternative to the Fourier domain representation is an approach that describes an image at multiple spatial resolutions and also preserves the local spatial structure of the image at each of these multiple scales.

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 $n \rightarrow \infty$, 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.