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

SLIDE 1

Matthias Book: Parallel Fractal Image Generation May 3, 2001 CS 580 - Parallel Processing 1 Parallel Fractal Image Generation

Matthias Book

CS 580 - Parallel Processing The University of Montana May 3, 2001

What are Fractal Images?

sets of points generated by iterating relatively simple calculations contain infinite detail show self-similarity have a fractal dimension

What are Fractal Images?

The complex plane

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

Imaginary axis Real axis

1 : − = i

C x yi | C|

yi x C + = What ar e Fr act al I m ages?

The it erat ion formula

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

0.50000

• 0.50000

0.75000

• 1.28125

0.89038

• 4.16273
• 41.65354
• 608.28115

2047006.620082 10076130739563.00000

• 1.00000
• 0.25000
• 1.18750
• 0.15234
• 2.61839

5.06320 7.30768

• 1682.61511

2461186.65194 1867203633030.90000 1 2 3 4 5 6 7 8 9 10 I m Z n R e Z n n

C = - 1.0 + 0.5i

C Z Z Z

n n

+ = =

+ 2 1

What ar e Fr act al I m ages?

I t erat ion approaching infinit y

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

C = - 1.0 + 0.5i

1.11803 0.55902 1.40451 1.29028 2.76564 6.55472 42.28971 1789.18964 3201199.12507 10247675838353.00000 1 2 3 4 5 6 7 8 9 10 | Z n| n

C Z Z Z

n n

+ = =

+ 2 1

What ar e Fr act al I m ages?

I t erat ion converging

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

0.35355339059327 0.45069390943300 0.47005027989035 0.40997608405816 0.33506252161220 0.34958508021450 0.39034473986338 0.40606669062459 0.39395730588684 0.37488377936362 ... 0.38268343236512 0.38268343236508 0.38268343236507 0.38268343236508 0.38268343236510 0.38268343236510 0.38268343236509 0.38268343236509 0.38268343236509 0.38268343236509 0.38268343236509 1 2 3 4 5 6 7 8 9 10 ... 110 111 112 113 114 115 116 117 118 119 120 | Z n| n

C = 0.25 - 0.25i

C Z Z Z

n n

+ = =

+ 2 1

SLIDE 2

Mat t hias Book: Par allel Fr act al I m age Gener at ion May 3, 2001 CS 580 - Parallel Processing 2

What ar e Fr act al I m ages?

I t erat ion looping

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

C = - 0.5 - 0.5i

0.70710678118655 0.50000000000000 0.55901699437495 0.73154374441998 0.18023862492280 0.69951506698028 0.45627830615575 0.56915055236267 0.68760238111038 0.24992959329357 ... 0.48608739226858 0.48154429128127 0.52267815373996 0.45621548679314 0.52219422862102 0.48025876780351 0.48822527911022 0.51699488717534 0.45968041794762 0.52254879060228 0.47565161324317 1 2 3 4 5 6 7 8 9 10 ... 140 141 142 143 144 145 146 147 148 149 150 | Z n| n

C Z Z Z

n n

+ = =

+ 2 1

What ar e Fr act al I m ages?

The Mandelbrot set

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

C Z Z Z

n n

+ = =

+ 2 1 Z → ∞ ∞ Z → fix point Z looping

What ar e Fr act al I m ages?

I mage charact erist ics

self- similarit y infit e det ail I t er at ion m axim um : 150

What ar e Fr act al I m ages?

I mage charact erist ics

m ore it erat ions

higher accur acy

I t er at ion m axim um : 1, 500, 000

Parallel Algorit hm Design

Mot ivat ion:

• deep zoom s r equir e high it er at ion count
• m ax. it er at ion count alw ays r eached
• n 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

long com put at ion t im es w it h ser ial algor it hm s

par allel com put ing can decr ease com put at ion t im e

Parallel Algorit hm Design

Part it ioning

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

ASCI I out put : '#' = Z appr oaches inf init y ' ' = Z does not appr oach inf init y

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

sm allest unit of com put at ion: 1 pixel

SLIDE 3

Mat t hias Book: Par allel Fr act al I m age Gener at ion May 3, 2001 CS 580 - Parallel Processing 3

Parallel Algorit hm Design

Blockw ise Agglom erat ion

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

Pr oble m : load not evenly balanced

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

Parallel Algorit hm Design

Linew ise Agglom erat ion

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

Solut ion: load now r elat ively evenly balanced

0.5

• 0.5

0.5

• 0.5
• 1.0
• 1.5

yi x

Parallel Algorit hm Design

Com m unicat ion

t asks don't have consecut ive lines of im age

synchr onizat ion f or sequent ial out put r equir ed

1 2 3 4

• ut put line
• ut put line
• ut put line
• ut put line

send not if icat ion send not if icat ion send not if icat ion send not if icat ion

Pr oble m : only 1 t ask act ive at any t im e

Parallel Algorit hm Design

Non- blocking com m unicat ion

Solut ion: use non- blocking com m unicat ion

• w hile lines t o com put e

non- blocking r eceive f r om pr evious node com put e lines and st or e in out put buf f er if not if icat ion received if at least one line in buf f er send f ir st line and delet e f r om buf f er else cont inue com put at ion t ill one line com plet e send line and delet e f r om buf f er non- blocking send not if icat ion t o next node Parallel Algorit hm Design

Proj ect ed perf orm ance

Processors Speedup

ideal proj ect ed

close t o ideal speedup, since all nodes can do com put at ions all t he t im e

Parallel Fract al I mage Generat or

Tha nk you!