Computer Science 320
Parallel Image Generation
The Mandelbrot Set
The Mandelbrot Set
A set of points defined as follows:
given a point (x, y), compute a sequence of other points
(ai, bi), i = 0, 1, 2, …
a0 = 0
b0 = 0
ai+1 = ai2 – bi2 + x
bi+1 = 2 * ai * bi + y
If each point in (ai, bi) stays finite, (x, y) is in; otherwise,
not in
The Mandelbrot Set
If each point in (ai, bi) stays finite, (x, y) is in; otherwise,
not in
Can’t do an infinite # of points, so
if (ai, bi) ever exceeds a distance of 2 from the origin, it’s
not in
or: if (ai2 + bi2 )1/2 > 2 for some i
The Mandelbrot Set
or: if (ai2 + bi2 )1/2 > 2 for some i
If we do 1,000 points, if i reaches this limit before the
distance exceeds 2, we’ll call the point in, even if further
tests might show it to be out
Program Resources
• Will use the Parallel Java Graphics (PJG)
format, which is lossless and larger, but
uses faster compression, than PNG format
• Will generate color values for each point
and save these in a PJG file
Program Inputs
• Image width and height
• Coordinates of the image center point
• Image resolution in pixels per unit
• Maximum number of iterations to test for membership
• Exponent in formula to calculate pixel hues
• Output file name
Pixel Matrix, Image File, Hue Table
// Create image matrix to store results.
matrix = new int [height] [width];
image = new PJGColorImage (height, width, matrix);
// Create table of hues for different iteration counts.
huetable = new int [maxiter+1];
for (int i = 0; i < maxiter; ++ i){
huetable[i] = HSB.pack(/*hue*/ (float) Math.pow(((double)i) /
((double)maxiter),
gamma),
/*sat*/ 1.0f,
/*bri*/ 1.0f);
}
huetable[maxiter] = HSB.pack (1.0f, 1.0f, 0.0f);
Optimizing Matrix Access
// Compute all rows and columns.
for (int r = 0; r < height; ++ r){
int[] matrix_r = matrix[r];
double y = ycenter + (yoffset - r) / resolution;
for (int c = 0; c < width; ++ c){
double x = xcenter + (xoffset + c) / resolution;
Allows access
to a cell with a
single index
operation
Iterate Until Convergence
// Iterate until convergence.
int i = 0;
double aold = 0.0;
double bold = 0.0;
double a = 0.0;
double b = 0.0;
double zmagsqr = 0.0;
while (i < maxiter && zmagsqr <= 4.0){
++ i;
a = aold * aold – bold * bold + x;
b = 2.0 * aold * bold + y;
zmagsqr = a * a + b*b;
aold = a;
bold = b;
}
// Record number of iterations for pixel.
matrix_r[c] = huetable[i];
Parallelize the Program
• Each pixel value can be computed independently,
so divide the matrix rows among the threads
• All inputs are shared
• No per-thread or local variables need
synchronization
• No padding needed
The Parallel for Loop
// Compute all rows and columns.
new ParallelTeam().execute(new ParallelRegion(){
public void run() throws Exception{
execute (0, height-1, new IntegerForLoop(){
public void run (int first, int last){
for (int r = first; r <= last; ++ r){
int[] matrix_r = matrix[r];
double y = ycenter + (yoffset - r) / resolution;
Behavior of Parallel Program