# Image recovery and segmentation using the fractal dimension

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## Abstract

It has been observed that real world objects are inherently composed of complex, rough and jumbled surfaces while current representational schemes use generalized cylinders or splines to describe natural surfaces. Therefore, there is a need to have a model for describing all naturally occurring surfaces.

Alex Pentland [8] has shown that the fractal dimension is a representation, capable of succinctly describing the surfaces of natural objects, such as mountains, trees, clouds etc. His paper describes a method of computing the fractal dimension.

In this work, Alex Pentland's [8] algorithm for evaluating the fractal dimension was applied to a set of images. The results of this algorithm when studied, reveal a distinct distribution of smooth edges such as the texture of the shirt in Figure 2, the facial variations in the same image, the surface variations of the road in the Figure 3 and the varied distribution of regions with people and houses in Figure 4. In addition to obtaining Sun Raster images for the fractal dimension using Pentland's [8] algorithm, this work also included evaluating the hard edges present in the images using the Sobel operator edge-detection scheme. On applying the Sobel operator method to Figure 3, the resultant image distinctly indicated the hard-edges in the original image by clearing bringing out the outlines of different people and the outlines of other objects present in the image. The Sobel edge-detected image had all the sharp edges represented by sudden variations of homogeneous regions in the original image (Figure 3).

Finally, the results obtained using the fractal dimension and those obtained using the Sobel operator were logically OR-ed to capture both hard and soft edges. The test images used for this logical OR operation were the results of the fractal dimension algorithm and the Sobel operator edge-detection technique on Figure 3. As can be seen from the final result (Figure 49), both the soft edge distribution of the fractal dimension image (Figure 47) and the hard edges found in the Sobel operator image (Figure 48) were successfully captured in the logically OR-ed image.

The final results of this work were thus able to represent up to a degree, a previously non-existent model which attempted to integrate the results of Alex Pentland's [8] work using the fractal dimension for evaluating soft edges and results obtained from the Sobel operator for hard edge detection. This work indicates mixed results in coming up with the new model suggested and definitely has potential for improvement.