1. CSE 30331 Lecture 6 – Complex Numbers & Images
Mandelbrot & Julia Sets
Image File Format (.ppm)
C/C++ system() function
g++, make & makefiles
Debuggers

2. Quick Aside Group Project Guidelines Due: Tuesday, September 22nd
… are posted on web page
Due: Tuesday, September 22nd
Initial Group membership
Brief description of project you plan to complete
Initial references you have found …

3. Images (photographic)

4. Images (fractal)

5. Image Representation An image is a rectangular grid of pixels
A pixel is a single picture element
Each pixel has a value representing the color (or intensity) of a single point in the image
Image size is in pixels (640 x 480, etc.)
Image resolution is in pixels / inch

6. Pixel Color Pixel size # of colors possible 1 bit 2 (Black or White)
Binary image
1 byte (8 bit)
256 shades of gray or
256 distinct colors
3 byte (24 bit)
224 colors (true color)
(millions of colors)

7. Color Maps red green blue 1 20 2 100 3 255 … pixel
If each pixel is a single byte, its value is often used as an index into a color map
(a table of actual 3 byte color codes)
pixel
red
green
blue 1 20 2
100 3
255 …
Black
Dark Red
Medium Yellow
Bright Cyan
White

8. Pixel Class class pixel { public:
pixel (unsigned char r = 0, unsigned char g = 0,
unsigned char b = 0)
: red(r), green(g), blue(b)
{ }
setColor (unsigned char r, unsigned char g,
unsigned char b)
{ red = r; green = g; blue = b; }
getColor(unsigned char &r, unsigned char &g,
unsigned char &b)
( r = red; g = green; b = blue; }
private:
unsigned char
red, green, blue; // true color components };

9. Image in memory // matrix template class found in Ford & Topp Ch 5
// is a 2-D grid using a vector of vectors
#include “d_matrix.h”
// declare white image of 500 x 500 pixels
matrix<pixel> image(500,500,pixel(255,255,255));
// set color of pixel3,4
image[3][4].setColor(100,20,255);
// get color (r,g,b) of pixeli,j
unsigned char r,g,b;
image[i][j].getColor(r,g,b);

10. Complex Numbers External format: a + b j a and b are real coefficients
j is sqrt(-1)
Represents a point on a 2D Cartesian plane
Real (horizontal) axis
Imaginary (vertical) axis
Addition x1 + x2 =
(a1 + b1 j) + (a2 + b2 j) = (a1 + a2) + (b1 + b2) j
Subtraction x1 - x2 =
(a1 + b1 j) - (a2 + b2 j) = (a1 - a2) + (b1 - b2) j

11. Complex Numbers Multiplication x1 * x2 =
(a1 + b1 j) * (a2 + b2 j) = (a1a2 - b1b2) + (a1b2 + a2b1) j
Division x1 / x2 =
(Note: multiply top & bottom by complex conjugate)
(a1 + b1 j) / (a2 + b2 j) =
((a1 + b1 j) * (a2 - b2 j)) / ((a2 + b2 j) * (a2 - b2 j)) =
((a1a2+b1b2) / (a2a2+b2b2)) + ((a2b1-a1b2) / (a2a2+b2b2)) j

12. Complex Number Plane

13. Mandelbrot & Julia Sets
Both based on repeated (recursive) application of the following function, where C and Z are both complex numbers
Zn = Zn-1*Zn-1 + C
If the distance of Zn from the origin never exceeds 2.0 then the original point is a member of the set
100 applications of the function is a sufficient test

14. Mandelbrot Set Images

15. Mandelbrot Sets There is ONLY ONE Mandelbrot Set
Initial conditions are ….
Z1 = j and
C = a complex number corresponding to a point on the complex number plane in the range real and imaginary (also corresponding to a pixel in the image being produced)
Zn = Zn-1*Zn-1 + C
For each pixel (complex number C) apply the function and count the number of applications before the magnitude(Zn) > 2.0
If count == 100 then C is in the set, color it Black
If count < 100 then C is not in the set, color it based on the count, indicating the “speed” with which it departed

16. Julia Set Images

17. Julia Sets
There are infinitely many Julia Sets (one for each constant C)
Initial conditions are ….
Z1 = a complex number corresponding to a point on the complex number plane in the range real and imaginary (also corresponding to a pixel in the image)
C is another complex number chosen and held constant during tests of all other points
Zn = Zn-1*Zn-1 + C
For each pixel (complex number Z1) apply the function and count the number of applications before the magnitude(Zn) > 2.0
If count == 100 C is in the set, color it Black
If count < 100 C is not in the set, color it based on the count, indicating the “speed” with which it departed

18. Suggested Ranges Images at least 600 x 600 pixels
Mandelbrot sets (Complex plane)
real
imaginary
Julia Sets (Complex Plane)
real

19. Portable Pix Map (PPM) File format for *.ppm image files
<magic number>
<comment>
<width & height>
<max color value>
<data bytes> P6
# creator: JHS 9/4/2004
255
d0^g%8%#$ <EOF>

20. C++ system() function Requests operating system to run a command
Command may be system command or another program
Examples:
// List all PPM image files in current directory
system(“ls *.ppm”);
// Start Eye of Gnome (eog) to display m1.pp image
system(“eog m1.ppm”);
Requires literal string or C-style char array as argument

21. System() Example of building and executing command with string class
string prog, filename, command;
cout << “Which viewer (gimp, eog)? ”;
cin >> prog;
system(“ls *.ppm”);
cout << “Enter name of file to display: “;
cin >> filename;
command = prog + “ “ + filename + “ &”;
system(command.c_str());

22. G++ g++ is the GNU C++ compiler Command line options Examples:
-c compiles to object file
-o <name> creates named executable
-g compiles to allow debugging
-lm links to math library
Examples:
g++ -g -c ctester.cpp
g++ -g -c complex.cpp
g++ -g -o ctester ctester.o complex.o –lm

23. Make & Makefiles
Make reads instruction in makefile or Makefile and performs indicated actions, by recursive application of rules
Rules based on targets, dependency lists, and time stamps on files
Rule format:
<target> : <list of files target depends on>
<tab> <command to build target>
Rule example:
ctester: ctester.o complex.o
g++ -g -o ctester ctester.o complex.o –lm

24. Using make To make 1st target in makefile or Makefile
To make 1st target in some other file
make –f <makefile_name>
To make specific target
make <target>

25. Phony Targets
Some targets are used to invoke commands BUT NOT actually build a target
They are identified using the term “phony”
Example:
note comments beginning with #
# phony target for use in clearing directory
# of all object files
# use: “make clean”
phony: clean
clean:
rm *.o

26. Makefile for programs # 2
# CSE 331 Program 2 makefile (JHS 9/10/2004 Notre Dame)
all: prog2_1 ctester prog2_2
prog2_1: prog2_1.cpp myVector.h myMatrix.h
g++ -g -o prog2_1 prog2_1.cpp
ctester: ctester.o complex.o
g++ -g -o ctester ctester.o complex.o
complex.o: complex.cpp complex.h
g++ -g -c complex.cpp
ctester.o: ctester.cpp complex.h
g++ -g -c ctester.cpp
prog2_2: prog2_2.o complex.o
g++ -g -o prog2_2 prog2_2.o complex.o
prog2_2.o: prog2_2.cpp complex.h pixel.h myVector.h myMatrix.h
g++ -g -c prog2_2.cpp
phony: clean
clean:
rm *.o

27. Debugging in Linux/Unix
Gdb is the command line debugger
ddd is a GUI version of gdb for Linux
xxgdb is a GUI version of gdb for Unix/X11
All versions support breakpoints, steps into and out of functions, data value examination, etc.
Program must be compiled with –g option to use debuggers

28. Summary Image Pixel ColorTable Complex numbers 2D matrix of pixels
Picture Element
Index into color table or true (RGB) color
ColorTable
Indexed list or true (RGB) color values for pixels
Complex numbers
Real and imaginary components
Represent points on 2D complex plane

29. Summary 2 Mandelbrot & Julia Sets Mandelbrot Set Julia Set
Complex numbers attracted to origin (Mandelbrot Set) or to another point C in the complex plane (Julia Set)
Based on recursive function Zn = Zn-1*Zn-1 + C
Mandelbrot Set
Initially, Z1 is origin and C is point being tested in plane
Julia Set
Initially, Z1 point being tested in plane and C is some point held constant for entire Julia Set
Non-member points are color coded based on value of n when Zn moves more than 2.0 units from origin, escaping the strange attractor

30. Summary 3 Portable Pix Map (simple image file format) P6
#comment (creator and date)
width height
max_color
image data in bytes (rgbrgb....)

31. Summary 4 G++ Make Debugging GNU C++ compiler
Command line options (-g -o –c ....)
Make
Executes rules in makefiles to build targets and perform other tasks
Recursively follows rules back through dependency lists
Phony rules execute commands; do not build targets
Debugging
ddd, xxgdb, gdb