Conclusion

In this paper I have outlined a method for interpolation of missing data at multiple scales. The Gaussian pyramid structure has been used to represent data at various scales or sizes. Using this data structure interpolation can be effectively carried out starting from the top level of the pyramid and then expanding downwards, until we reach the base of the pyramid where we can fill up the missing piece in the original data. Several iterations of the reduction and expansion process might be needed. The chief reason for this is to improve the pixel amplitude in the hole at the top level of the pyramid, from where we start the interpolation process. The pyramid forming process is extremely inexpensive and thus several iterations can be easily afforded. There are however some boundary issues with this method. Artifacts can be generated at the boundary primarily because a boundary node never gets a contribution from the whole 5 point window either during reduction or expansion. Also amplitude balancing for the pixels in the missing part of the dataset at the top of the pyramid, from where the interpolation starts, is an important issue. Developing sophisticated and systematic method for this amplitude balancing has not been attempted and remains a topic for future research. The blurring effect seen in most of the results is due to this amplitude balancing issue.

Another important issue with this method, with regards to seismic data, is that it may not be possible to construct too many levels of the pyramid. In such a case however we can easily construct a small prediction filter to fill in the data at the coarsest pyramid that we can have and start filling up from there. Moreover since the hole is bound to shrink during the pyramid formation process the filter estimation may actually be more convenient. The method outlined in this paper looks at the problem of missing data interpolation at many different spatial scales, rather than trying to estimate the missing data directly from the original image. It is however different in principle from many multi-scale approaches proposed earlier. The main issue that most multi-scale methods try and address is the problem of estimating a PEF effectively. The pyramid scheme on the other hand tries to stay away from the process of PEF estimation and instead tries to fill in data at multiple spatial resolutions.