## Abstract

The focus of this research is to explore the structure present in the wavelet decomposition of natural images and use this structure for image compression and representation. We first show empirically that the wavelet coefficients of a natural image, when sorted by magnitude, lie almost exactly on a curve of the form $. This, in turn, helps us to demonstrate that natural images may exhibit self-similarity in the wavelet domain, although no such property may be apparent in the pixel domain. In order to predict which scale-space positions may have large wavelet coefficients, we define a data structure called a "web". We use this together vvith a top-down adaptive search (TAS) algorithm that we propose. Using this, one can predict not only which scale-space positions have large wavelet coefficients, but also the magnitudes of the wavelet coefficients. Further, we are able to find a bound on the error in our prediction. We also show that this prediction scheme has the ability of automatically zooming in onto the edges in the image. Additionally, this method of predicting scale-space positions yields a number (we call it y) which is indicative of the amount of structure present in an image. Finally, we derive a theoretical bound on the value y can attain when the TAS algorithm is applied to an autoregressive process. This theoretical bound further supports the empirical evidence that y is indicative of the amount of structure in the image. We use the prediction scheme to construct an image compressiorl algorithm. The image compression algorithm is comparable in performance to the best available algorithms and also has low computational requirements. Further, the ;algorithm allows the user to control both the mean square error as well as the number of non-zero coefficients in the representation. Finally, we use these properties to formulate a stochastic process to model an image and explore various theoretical properties of the stochastic process.

## Date of this Version

July 1996