What are Fractal Images? sets of points generated by iterating - PDF document

Matthias Book: Parallel Fractal Image Generation May 3, 2001 What are Fractal Images? sets of points generated by iterating Parallel Fractal Image Generation relatively simple calculations Matthias Book contain infinite detail show self-similarity CS 580 - Parallel Processing have a fractal dimension The University of Montana May 3, 2001 What are Fractal Images? What ar e Fr act al I m ages? The complex plane The it erat ion formula yi Imaginary axis yi n R e Z n I m Z n 1 - 1.00000 0.50000 2 - 0.25000 -0.50000 C x C = - 1.0 + 0.5i 0.5 0.5 3 - 1.18750 0.75000 4 - 0.15234 -1.28125 | C| 5 - 2.61839 0.89038 yi 6 5.06320 -4.16273 7 7.30768 - 41.65354 8 - 1682.61511 -608.28115 -1.5 -1.0 -0.5 0.5 x - 1.5 - 1.0 - 0.5 0.5 x 9 2461186.65194 2047006.620082 Real axis 10 1867203633030.90000 10076130739563.00000 -0.5 - 0.5 = Z 0 0 2 = + = − = + C x yi Z Z C i : 1 n + 1 n What ar e Fr act al I m ages? What ar e Fr act al I m ages? I t erat ion approaching infinit y I t erat ion converging n | Z n | yi 1 0.35355339059327 yi 2 0.45069390943300 n | Z n | 3 0.47005027989035 1 1.11803 4 0.40997608405816 C = - 1.0 + 0.5i 2 0.55902 5 0.33506252161220 0.5 0.5 3 1.40451 6 0.34958508021450 4 1.29028 7 0.39034473986338 5 2.76564 8 0.40606669062459 6 6.55472 9 0.39395730588684 7 42.28971 10 0.37488377936362 8 1789.18964 ... ... - 1.5 - 1.0 - 0.5 0.5 x - 1.5 - 1.0 - 0.5 0.5 x 9 3201199.12507 110 0.38268343236512 10 10247675838353.00000 111 0.38268343236508 112 0.38268343236507 C = 0.25 - 0.25i 113 0.38268343236508 114 0.38268343236510 - 0.5 - 0.5 115 0.38268343236510 116 0.38268343236509 = = Z 0 Z 0 117 0.38268343236509 0 0 118 0.38268343236509 2 2 = + = + 119 0.38268343236509 Z + Z C Z + Z C n 1 n n 1 n 120 0.38268343236509 CS 580 - Parallel Processing 1

Mat t hias Book: Par allel Fr act al I m age Gener at ion May 3, 2001 What ar e Fr act al I m ages? What ar e Fr act al I m ages? I t erat ion looping The Mandelbrot set n | Z n | yi 1 0.70710678118655 yi 2 0.50000000000000 3 0.55901699437495 4 0.73154374441998 5 0.18023862492280 Z → ∞ ∞ 0.5 0.5 6 0.69951506698028 7 0.45627830615575 8 0.56915055236267 9 0.68760238111038 10 0.24992959329357 ... ... - 1.5 - 1.0 - 0.5 0.5 - 1.5 - 1.0 - 0.5 0.5 x x 140 0.48608739226858 141 0.48154429128127 Z → fix point 142 0.52267815373996 143 0.45621548679314 144 0.52219422862102 Z looping - 0.5 - 0.5 145 0.48025876780351 C = - 0.5 - 0.5i = = Z 0 Z 0 146 0.48822527911022 0 0 147 0.51699488717534 = 2 + = 2 + 148 0.45968041794762 Z Z C Z Z C + + n 1 n n 1 n 149 0.52254879060228 150 0.47565161324317 What ar e Fr act al I m ages? What ar e Fr act al I m ages? I mage charact erist ics I mage charact erist ics I t er at ion m axim um : I t er at ion m axim um : 150 1, 500, 000 self- similarit y m ore it erat ions � higher accur acy infit e det ail Parallel Algorit hm Design Parallel Algorit hm Design Part it ioning yi sm allest unit of com put at ion: Mot ivat ion: # # # # # # # # # # # # 1 pixel # # # # # # deep zoom s r equir e high it er at ion count # # # # # # � # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # � m ax. it er at ion count alw ays r eached # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 0.5 on t he inside of t he set # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # � high im age r esolut ions desir ed # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # � e.g. 8x11" f r act al at 600 dpi: 31,680,000 pixels, # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # each w it h < = 2, 000, 000 it er at ions - 1.5 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # - 1.0 - 0.5 0.5 x # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # � long com put at ion t im es w it h ser ial algor it hm s # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # - 0.5 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # par allel com put ing can decr ease com put at ion t im e # # # # # # # # # # # # # # # ASCI I out put : # # # # # # # # # # # # # # # # # ' # ' = Z appr oaches inf init y # # # # # # # # # # # # ' ' = Z does not appr oach inf init y CS 580 - Parallel Processing 2