The Mandelbrot Set

In my last post about the mandelbrot set I said:

I would put the Mandelbrot explorer script online however it is very computationally intense. It takes around 5 seconds to generate 1 image.

Now I have created a new explorer which you can use.

I tried to find a way of replicating my original PHP script in JavaScript. That way the processing is all done on the client computer and not my server. For a while I didn’t think it was possible until today I came across pixel manipulation using the Canvas element. Internet Explorer users will be disappointed to know that the Canvas element doesn’t work on their browser and this therefore won’t work.

The code is almost the same as the PHP code except that it uses the
getImageData and putImageData() functions. These functions extract the pixel data from the image and allow it to be manipulated before being saved again, extremely useful.

Why not explore the Mandelbrot Set!

Knowing about how 3D images are produced I knew that Google couldn’t possibly render whole pages in real 3D. Instead it uses a clever rendering effect to create two identical images of the page and convert one into the red component and the other into the blue and green component.

This doesn’t make the page 3D as such but instead when viewed through cyan and red glasses looks like the page comes out of the computer screen. This isn’t really thrilling and can make your eyes and brain hurt after a short while due to retinal rivalry.

For real anaglyphs you need two images one of the scene viewed from the right and another from the left. Then you can use my anaglyph creator to make your own genuine 3D images. And don’t forget to download your 3D glasses to view them!

For those mathematical boffins the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial z_n+1 = z_n² + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z₀=0 and applying the iteration repeatedly, the absolute value of z_n never exceeds a certain number (that number depends on c) however large n gets.

I decided to create a PHP script which could visualise the Mandelbrot set for me. With a bit of research, I found some basic code which I modified to make the script. You are free to use and distribute it as you wish. The script does work but It is isn’t perfect so use at your own risk.

<?php
class Mandelbrot
{
var $im;
function colour($value){
$value=$value*ceil(1200/$this->its);
if($value<256){
$r=255;
$g=$value;
$b=0;
}elseif($value<512){
$r=511-$value;
$g=255;
$b=0;
}elseif($value<768){
$r=0;
$g=255;
$b=$value-512;
}elseif($value<1024){
$r=0;
$g=1023-$value;
$b=255;
}elseif($value<1280){
$r=$value-1024;
$g=0;
$b=255;
}elseif($value<1536){
$r=255;
$g=0;
$b=1535-$value;
}
return imagecolorallocate($this->im, $r, $g, $b);
}
function Mandelbrot()
{
$this->im = @imagecreatetruecolor(200, 200);
$this->y1=is_numeric($_GET['y'])?$_GET['y']:0;
$this->x1=is_numeric($_GET['x'])?$_GET['x']:0;
$zoom=is_numeric($_GET['zoom'])?100*$_GET['zoom']:100;
$this->its=(is_numeric($_GET['its']) and $_GET['its']<=1200 and $_GET['its']>0)?$_GET['its']:100;
$d1 = microtime(1);
for ($y = -100; $y < 100; $y++) {
for ($x = -100; $x < 100; $x++) {
$val=$this->iterate($y/$zoom,$x/$zoom);
imagesetpixel($this->im, $x+100, $y+100, $this->colour($val));
}
}
$d2 = microtime(1);
$diff = $d2 - $d1;
header("Elapsed: $diff");
header('Content-Type: image/png');
imagepng($this->im);
}
function iterate($x,$y)
{
$cr = $y+$this->y1;
$ci = $x+$this->x1;
$zi = 0.0;
$zr = 0.0;
$i = 0;
while ($i < $this->its and $zi2 + $zr2 < 4) {
$i++;
$temp = $zr * $zi;
$zr2 = $zr * $zr;
$zi2 = $zi * $zi;
$zr = $zr2 - $zi2 + $cr;
$zi = $temp + $temp + $ci;
}
return $i;
}
}
$m = new Mandelbrot();
?>

The script works in a simple way. It creates a blank canvas of 200×200 pixels. For each pixel it uses the escape time algorithm to check if a point is within the set or not. A value is then returned for how long it took a point to reach the escape condition. Some points take millions of cycles to reach the escape condition so we set a maximum iteration limit. This defines how much detail we see in the mandelbrot set.

This value is used to define the colour of the point. Instead of using value to define the brightness of the point I use the value to define the hue of the point as there are many more possible hues than possible greyscale values. To convert a value into an RGB value needed to plot the point I use a special algorithm. The image above roughly shows how the RGB values are worked out.

I have made the script take 4 input parameters, the x and y coordinates the amount of zoom and the maximum number of iterations allowed. By messing around with these paarmeters you can explore the mandelbrot set in stunning detail.

The outputted images are simply amazing. The image to the left is found at a magnification of 200,000. Its a miniature replica of the Mandelbrot set.

To show you just how much magnification you can get watch the following animation. It goes from a magnification of 1 to 1 billion. With a little more processing time you could easily go much deeper to many billions of billions of times magnification!

I would put the Mandelbrot explorer script online however it is very computationally intense. It takes around 5 seconds to generate 1 image.

Important Update: You can now explorer the mandelbrot set!

Father’s day was celebrated here in the UK last Sunday. I decided that I’d rise to the occasion and make something special. I can’t say that I’m a whiz with Photoshop because I don’t actually own a copy of it, instead I rely on a rather old version of Paint Shop Pro.

Paint Shop Pro is a nice application. It lacks some of the more advanced canvas and brush features of Photoshop which means that it lacks the creative edge. Also I have yet to find a way to convert Photoshop brushes and documents to Paint Shop Pro documents.

Anyway, I found a great tutorial on psdtuts.com to make a grass effect. My dad loves gardening so I thought it suited him well. I had a quick flick throught the steps to check there was nothing that Paint Shop Pro couldn’t handle before getting to work.

I think the end result was well worth it and it only took me a couple of hours. Sure it isn’t half as good as the tutorial example but my dad was impressed.

You can download the full sized wallpaper from flickr. It’s a bit late for father’s day this year though!

Hopefully, I can find some more time to get to grips with Paint Shop Pro and maybe I can find some money as well so I can buy Photoshop!

Think of cryptography as the lock to a safe. You need a key to open the safe. Steganography is hiding the safe somewhere where it can’t be found easily such as behind the painting on the wall. To find the safe you need directions. So how is steganography useful in the modern world. It can actually be used in several ways. A secret file can be hidden on your computer so it looks like an ordinary file. I’m going to show you today how to hide an MP3 file inside a JPEG or GIF image.

Firstly, we need two files to work with. Create a recording which you wish to hide and save it as an MP3 file then find an innocent GIF or JPEG. Size doesn’t matter but putting a 10MB MP3 file inside a 10×10 GIF image may look slightly wrong. Place your two images in the same directory.

Bring up the command prompt and navigate to the directory in which your files are saved using the cd command. Write the following command

copy /b wood.jpeg + rain.mp3 test.jpeg

Obviously change the file names in the example above. The /b stands for binary so the command prompt will make a binary copy of wood.jpeg and rain.mp3 and combine them in the file test.jpeg.

The file created will operate like a normal JPEG image until you change the extension to mp3 or run the file with a program which only supports MP3 files. This has many uses, you can email inconspicuous images to your friends for them to extract the file hidden within. This technique is not exclusive to MP3 files. You can hide whole zip files inside an image meaning one image can hide an unlimited number of files.

For additional security you could encrypt your files before hiding them inside an image. Steganography and cryptography combined is a powerful weapon against eavesdroppers and code breakers. If they don’t know where the encrypted message is they can’t tamper with it or decrypt it.

Having a little experience with PHP, I decided to create a simple script to create anaglyphs. First I needed to plan out how the script was going to work. The GD library has no native image filters except in PHP5. My plan works like so.

Find the size of the image
For every x coordinate
  For every y coordinate
    Get the red value of the left pixel
    Get the green value of the right pixel
    Get the blue value of the right pixel
    Set the result pixel to red + green + blue
Output image

In PHP this results in:

// The file for the left eye
$left = 'left.jpg';
// The file for the right eye
$right = 'right.jpg';
// Get the dimensions
list($width, $height) = getimagesize($left);
// Define our source images
$src_right = imagecreatefromjpeg($right);
$src_left = imagecreatefromjpeg($left);
// Create the canvas
$bwimage= imagecreatetruecolor($width, $height);
//Reads the origonal colors pixel by pixel
for ($y=0;$y<$height;$y++){
for ($x=0;$x<$width;$x++){
$rgb_left = imagecolorat($src_left,$x,$y);
$r = ($rgb_left >> 16) & 0xFF;
$rgb_right = imagecolorat($src_right,$x,$y);
$g = ($rgb_right >> 8) & 0xFF;
$b = $rgb_right & 0xFF;
//This is where we create the color which is a mix of the red, green and blue channels
$color = imagecolorallocate($bwimage,$r,$g,$b);
imagesetpixel($bwimage,$x,$y,$color);
}
}
// This sets it to a .jpg, but you can change this to png or gif if that is what you are working with
header('Content-type: image/jpeg');
imagejpeg($bwimage);
?>

I didn’t think this would work at first because it seems too simple. My tests revealed the images I was outputing were the same as other programmes so it was working. Having said that since I cant find any red and cyan glasses I can’t really test it properly! Can anyone else verify the images below?

Update: If you want to create your own anaglyphs why not use my free anaglyph creator.

The probe has an instrument called the Surface Stereo Imager. Being a stereo camera means that it has two lenses spaced apart which capture a scene from two very slightly different points. The images sent back can then be turned into 3D images.

Several years ago I picked up a book about stereo images. Using a cross eyed technique you can see two images taken see images taken side by side as one 3D image. Of course for those of us who haven’t learnt how to do the cross eyed technique there are two other methods: those really cool red and cyan glasses or using special polarized glasses.

For your viewing pleasure I have put together some of the images from thePhoenix gallery to form stereo pairs. I have also created an image for people with those red and cyan glasses.

Update: If you want to create your own anaglyphs why not use my free anaglyph creator.

This is my new blog which I have got up and running. The style stills needs a little touching up here and there but it is all working. Watch this space for more interesting stuff!