Homework #9

Reading: pg. 1-7; 903-919

1. Discuss the advantages and disadvantages of encoding images as IFSs. Does this seem like a good image compression system?

2. Consider a random IFS realizing a ratio list (r1,...,rn) which is not necessarily contracting (that is, some of the r(i) may exceed one) and with each component map affine. If p1,...pn are the associated probabilities, then the random IFS is said to obey the average contractivity condition if it is the case that (r1^p1)*(r2^p2)*...*(rn^pn)<1. Such an IFS has a unique attractor. Construct a random IFS of this sort on H(R2) using at least three component maps, with at least one r(i) exceeding one, and use the random IFS to generate a plot of its attractor.

3. Let W be a Hutchinson map for a contractive IFS on H(X). Prove that if z0 is a fixed point of one of its component maps, and w is another of its component maps, then w(z0) lies in the attractor of W. (Hint: First argue that zo must lie in the attractor of W.)

4. a. Design an IFS for a fern-like image. (Start from the Barnsely fern.) Plot it.

b. Construct an IFS for a leaf (explain the reasoning behind how you chose your collage). Plot it. Randomize it in proportion to the determinants of the component maps, and plot it again.

5. In what ways is an IFS similar to the initiator-generator method used to make classical fractals? In what ways is it different? In what ways is a random IFS similar to the initiator-random generator method used to make random classical fractals (such as the random Koch curve, see pg.400 of the text)? In what ways is it different?

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