Homework #10

Reading: pg. 507-524; 921-953

1. Comment on the quadtree, HV, and triangular partitioning methods for use in fractal image compression. Comment on the artifacts (see pg.915-917 of the text); how do they relate to the underlying affine maps?

2. Construct an IFS for a fractal tree (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. Does this seem like a good randomization strategy?

3. The castle fractal has the IFS code {w1: .5, 0, 0, .5; 0, 0, w2: .5, 0, 0, .5; 2, 0, w3: .4, 0, 0, .4; 0, 1, w4: .5, 0, 0, .5; 2, 1}. Show that the resulting fractal is self-similar by showing that each component map is a similarity transformation, and compute the dimension associated with the IFS. Is this the self-similarity dimension (see pg.271-272 of the text) of the castle fractal, or is there too much overlap of the IFS? Plot the castle fractal.

4. How are fractals related to diffusion limited aggregates (DLAs) and percolation? In what ways is the fractal dimension an interesting or useful quantity for studying DLAs?

5. Use the random midpoint displacement method to create three different fractal curves on [0,1]. (Use the same random number generator in each case.) Repeat with a different random number generator. Comment.

b. Use random midpoint displacement on a polygon to make a fractal island. Experiment with the mean and variance of your random number generator until you have achieved a reasonable result.

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