ECE 448: Spring 2015 Lab 5 Mandelbrot Set Fractal 1.

ECE 448: Spring 2015Lab 5

Mandelbrot Set Fractal

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Introduction to Lab5

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Benoit Mandelbrot

• Mandelbrot was born in Poland in 1924.

• Father of fractal geometry.

• He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

What is a Fractal?Fractals are mathematical structures defined by two

properties– Iterative– Self-similar

• Zooming in or out on the image reveals deep repetition of patterns

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The Mandelbrot Set

• The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal.

• Mathematically, the Mandelbrot set can be generated using a very simple iterative formula, called the quadratic recurrence equation, applied to points in complex plane

 zn+1 = zn2 + c 

 That is, a complex number c, is in the Mandelbrot set if the absolute value of zn never exceeds a certain number.

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The Mandelbrot Set

Pseudocode

for cy = -1 to 1, step 2/400 do

for cx = -2 to 1, step 3/600 do

{

zx = 0

zy = 0

iteration = 0

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In the pseudocode below, c=cx+i·cy, corresponds to one pixel of the display region. The plotted region should have the following limits

-2 ≤ cx=Re[c] ≤ 1

-1 ≤ cy=Im[c] ≤ 1

Pseudocode Cont…

// z = z2 + c =

// =(zx2 – zy2 + cx) + i · (2 · zx · zy + cy)

while (zx2 + zy2 < 4 && iteration < MAX_ITER ){

zxtemp = zx2 – zy2 + cx

zytemp = 2 · zx · zy + cy

zx = zxtemp

zy = zytemp

iteration++

}

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x = x_conv(cx) // conversion to the x-coordinate of a pixel

y = y_conv(cy) // conversion to the y-coordinate of a pixel

if zx2 + zy2 < 4

color(x,y) = fractal_color

else

color(x,y) = background_color

}

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Pseudocode Cont…

x_conv() and y_convThe functions x_conv() and y_conv() are used to convert the coordinates of the complex number c into x and y coordinates of the pixel.

 x = x_conv(cx) = 20 + (cx-(-2))*(600/3) = 20 + 200*(cx+2)

y = y_conv(cy) = 440 – (cy-(-1))*(400/2) = 440 - 200*(cy+1)

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The magnification of the last image relative to the first one is about 10,000,000,000 to 1. 

Fixed Point Representation• You can use Q4.28 representation

– 4 integer bits– 28 fractional bits

• Addition/Subtraction performed as usual

• Multiplication of two Q4.28 numbers results in a Q8.56 number, which should be converted back to Q4.28

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VGA Display Area

• Use a rectangular area of the size (600 x 400) to display the Mandelbrot set fractal.

• Left and Right border = 20 pixels each

• Top and Bottom border = 40 pixels each

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Cont..

• Top border should display “The Mandelbrot Set” in the center.

• Bottom border should display– Percentage of the display area, increasing

every 0.5%.– Progress bar – Total execution time with the step 0.1 s.

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Input/Output Scheme

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• Use BTNS as the Start/Pause button to start/pause the computations.

• The color of the fractal and background should also change, depending on the positions of switches

Final VGA Display Outlook

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THE MANDELBROT SET

95.5% 33.1 s

Bonus Tasks

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• Increase the speed of calculations, by evaluating 4 values of c in parallel.

• Determine the maximum speed-up possible by evaluating N values of c in parallel, where N is limited only by the available FPGA resources.

• Add colors by assigning a different color to each value of c, based on the number of iterations required for the decision based on the following formula:

color = iteration mod 8