By Budinsky F.

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The three transformations are: {1,4,3}→{9,10,12}, {1,4,3}→{5,4,7}, {1,4,3}→{6,7,3}. The Sierpinski triangle provides a good example of the two hallmarks of a fractal: selfsimilarity and non-integer dimension. The self-similarity is evident from the construction. To see why it makes sense to assign a non-integer dimension to this object, consider the following. 1 Fractal dimension What do we mean by “dimension”? Intuitively, we know that a line has dimension 1, a solid square has dimension 2, and a solid cube has dimension 3.

Similarly, the constant oi raises or lowers the grayscale values, and so controls brightness. The ~ is called the spatial part of w . transformation w i i Partitioned Iterated Function Systems (PIFS) 45 ~ w i Di ~ −1 w i Ri Fig. 2 Spatial affine transformation and its inverse. 2) are the basic affine transformations on grayscale images that we will use in fractal image encoding. 2 Contraction mappings on grayscale images When is wi: F → F a contraction? 1). Using the formula for change of variables in a multiple integral, we have I d2 ( wi ( f ), wi ( g)) = 2 2 wi ( f )( x, y) − wi ( g)( x, y) dxdy wi ( Di ) 2 = si det A i I 2 f ( x, y) − g( x, y) dxdy Di 2 ≤ si det A i d2 ( f , g) , 2 where Ai is the matrix part of wi, det Ai is the determinant of Ai, and si is the contrast factor.

This is the same IFS used to produce the images in Figs. 5. 2 Code sample to compute probabilities associated with IFS for the random algorithm. Each probability is proportional to the determinant of the matrix part of the affine transformation. This determinant, in turn, is proportional to the area controlled by that transformation. You can modify this code to experiment with non-uniform distributions for different parts of the attractor image. 3 Code sample implementing the random IFS algorithm.