Technologies for generating natural shapes, such as mountains, trees, and clouds, play a very important role in visual simulations, which is one of the main applications of computer graphics. This paper reports research trends in methods of generating natural shapes by using fractal theory.

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A fractal based image coding method using contour lines of segmented regions is discussed in this paper. We classify the regions of an image into two categories, using pyramidal region segmentation. First one is the category of the regions do not have to preserve the waveform of image. And the regions categorised in this class are encoded using only fractal parameters. Another one is the category of the regions have to preserve the waveform of image. We can encode this part of image using any waveform coding method(DCT, VQ, etc.). On the other hand, we classify those blocks containing region contour lines into two categories using the complexity of the shape of contour lines. First one is with simple contours. Blocks in this category are encoded usng the same method for the regions do not have to preserve the waveform of image. Another one is with complex contiurs. Blocks in this category are encoded using both waveform coding method and fractal coding method.

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We propose an IFS (Iterated Function System) estimation method using the wavelet transform to solve the inverse problem of finding the IFS for a given image. An isotropic Laplacian-Gaussian 2D wavelet transform is applied to the analysis of local scaling properties of fractal objects, such as the 2D Cantor set with measures. It is shown that contraction, probability and translation parameters of an affine contraction constituting the IFS are determined efficiently by the wavelet transform, and the minimum number of affine contractions is also determined by zero-crossing representation of the wavelet transform.

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An IFS (Iterated Function System) using contractions can generate complex images by simple computation. For image representation by an IFS, it is necessary to solve an inverse problem, that is, to find functions providing invariant sets of an IFS. However, it is unclear how to find such functions and how natural images can be represented as invariant sets of an IFS. In the biginning of this paper, we propose a local parallel random IFS algorithm for generationg fractal images, which is suitable for parallel computers. Next, we propose a method to determine an IFS, consisting of affine contractions and associated propabilities, for an image block, using minimum square error criteria. Using this method, the IFS estimation and reconstruction characteristics are evaluated from the view point of signal-to-noise ratio. We also propose an adaptive method for local IFS estimation, which satisfies the condition of given permissible error. It is shown that the proposed method improves both representation effiency and signal-to-noise ratio compared to those of the fixed method.

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The Peano scan is a path traversing an array of points and is very convoluted as a randam walk. This scan can be applied to sequential image transformation, instead of raster scan, and does not produce periodic patterns that are often observed in the case of raster scan. Also it is noted that the Peano scan has the same properties as that of quadtree or octree. If the distribution of values in a 2D or 3D array has clusters, then the clusters are preserved on the scan. This is the reason why this scan is applied to run-length compression and color quantization. The Peano scan, however, has restrictions in nature: the scanned array must be a square and the width and height must be a power of 2, because the scan is obtained by recursive division of the array, as quadtree. In the present article, some generalized Peano scans for any rectangular array are introduced, and halftoning is discussed as one of the applications of the Peano scans.

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A progressive build-up coding method for color image data base is proposed. In the proposed method, the image resolution of the reconstructed image is increased in the DCT spatial frequency domain. Furthermore, in searching for the images, only the important image areas are displayed with high resolution. The proposed method reduces the number of DCT calculation and round error, and improves the coding efficiency.

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