Transform Image to Cartesian from Polar Coordinates

[Mirkules]

Hi All,

 

I am trying to create an image transform that will plot all images in the cartesian coordinate system from a polar system.  So far, I haven't had luck. Images are coming out distorted.  My input is a picture of a perfect circle, and the expected output is a straight line.  The circle actually gets pretty distorted but it is nowhere near the straight line that I'm expecting.  I will be converting other pictures to cartesian coordinates later, so I'm only using the circle as baseline.  I am not sure if it is a problem with math or if it is a bug in the program.

 

All the images are Black and White ONLY (this is part of a bigger project where the image is converted to only B&W prior to hitting this portion of the code), so colors are not a concern (which is why, as you will see in the code, only the red channel is tested).

 

The flow of the program is:

for every point (i, j) in the original image $im:

 

if the pixel is black, find the radius and angle.

convert radius and angle to cartesian coordinates using formula x = r cos (angle) and y = r sin (angle)

plot a pixel in the new image at x,y  (if x or y is less than 0, "wrap" around)

 

I've attached the code.  If anyone has patience to look through it, and if you have any idea what the problem might be, I would be greatful.

 

Thanks

mk

 

function translate_polar_to_cartesian($im)
{
$PI =  3.14159265;

$w = 1000;
$h = 1000;
$cart_im = imagecreate($w,$h);

$im_w = imagesx($im);
$im_h = imagesy($im);

$back = imagecolorallocate($cart_im, 255, 255, 255);
$front = imagecolorallocate($cart_im, 0, 0, 0);
for ($i = 0; $i < $im_w; $i++)
{
	for ($j = 0; $j < $im_h; $j++)
	{
		//echo "i: $i, new x: $new_x<br/>";

		$im_rgb = imagecolorat($im, $i, $j);
		$c = $back;
		$red = ($im_rgb >> 16) & 0xFF;
		if ($red < 255)
		{
			$c = $front;
			$r = sqrt($i*$i + $j*$j);

			if ($i != 0)
				$t = tan($j/$i);
			else
				$t = $PI/2;

			$x = $r * cos($t);
			$y = $r * sin($t);
			if ($x < 0)
				$x = imagesx($im) - $x;

			if ($y < 0)
				$y = imagesy($im) - $y;

			imagesetpixel( $cart_im, $x, $y, $c);
			//echo "r: $r, t: $t<br/>";
		}	
	}	
}
return $cart_im;	
}

[btherl]

What value does tan($j/$i) represent?

[Mirkules]

$j is the y coordinate of the image's pixel, while $i is the x coordinate. $t = tan($j/$i) is the angle of the line segment specified by (0,0), ($i,$j) with respect to the x-axis (polar axis).

 

 

[btherl]

I don't quite get that.  Let's say $j = pi/2 and $i = 1.  That point doesn't exist in the image but let's use it anyway.

 

tan($j/$i)

= tan((pi/2)/1)

= tan(pi/2)

= ???

[Mirkules]

You are right, it makes no sense because it should be arctan.

 

Still, I can't make it work. I get the input image back now, only lightly distorted. Still not the desired output.

 

It could be the way I am displaying the image at the end. I found this: http://www.codeguru.com/forum/archive/index.php/t-278733.html

 

I rewrote the code to look more like that, but still no luck. I think it might be the way I am displaying the coordinates at the very end when I obtain radius and angle. What do you think?

[Barand]

If you plot

 

x = r cos(t), y = r sin(t)

 

then aren't you just redoing the polar plot?

 

x = t, y = rt should give a straight line in the form y = Mx

[Mirkules]

Thank you -- you are right, I was replotting the image in polar space.  This brings me a step closer.

 

Now it's an issue of displaying floating point numbers. Here's a sample of the points that are being plotted:

 

x: 0.1624891092172, y: 61.814237842102

x: 0.14648417839356, y: 61.660360037872

x: 0.13040330788913, y: 61.522353661088

x: 0, y: 61

x: -0.016391974308005, y: 61.008196170679

x: -0.032775144404076, y: 61.032778078669

x: -0.099668652491162, y: 60.299253726725

 

As you can see, I am in floating point hell.  The issue isn't just multiplying by 100 (or 10), rounding off and setting the pixel.  The problem is that when I do that, I get a lot of empty space between the dots, resulting in a non-continuous image.  I attached my input and output images so you can see what I mean. I'm not sure if this is just a function of converting from polar to rectangular space, or if it's something wrong with the way I'm displaying the images (I think it's the latter). I am trying to mimic this: http://microship.com/resources/rotation-scaling.html

 

I'm not sure if that's possible at all.

 

Anyway, thank you very much for your help, I appreciate it!!

 

[attachment deleted by admin]

[Barand]

Perhaps store results in arrays. Get min and max values and scale the axes accordingly.

[Mirkules]

I'm not exactly sure what you mean by "scale the axes"? How could I do that in an image?

[Barand]

[pre]

            | Ymax

            |

            |

-------------+-------------                                         

Xmin        |        Xmax

            |

            | Ymin

[/pre]

 

The values in polar mode are different from those required in cartesian mode

[Mirkules]

Yeah, I understand that I need to convert the X coordinate and Y coordinate so that they are between the xMin/xMax and yMin/yMax bounds. What I don't think I understand is what is the function to get a point like x=1.5674243 between the xMin and xMax?

[btherl]

Let's say your points range from -50 to 100, and your image is 300 pixels wide.

 

$imageWidth = 300;
$xMin = -50;
$xMax = 100;
$xRange = $xMax - $xMin; # That's 150
$xShifted = $xOriginal - $xMin; # Shift so it has a 0 offset
$xScaled = $xShifted * $imageWidth / $xRange; # And scale it to the image width, 300

 

Is that what you were after?

[Mirkules]

btherl, thank you for the scaling. Even though it's not directly what I was asking for in this post, I would have had to tackle that problem eventually.

 

The problem I'm having is that I cannot connect the dots. If you look at the images I attached, you will see the original smiley face, the version of the smiley translated into polar with my script and a version done with Photoshop's Filter->Distort->Polar Coordinates.

 

What I'm really trying to achieve is continuity, just like PS. Any deas how PS may be doing that?

 

[attachment deleted by admin]

[Barand]

Your result of a horizontal straight line is the result of y = constant rather then y = mx, which should give a sloping line.

 

BTW, Is your test circle centred on the origin, the polar coords calculation assumes that it is.

[btherl]

Since your original image is not continuous, the only way to make a continuous output would be to extrapolate a bit.  Maybe you could assume that for all input pixels i1 and i2 which are adjacent in the input, you should draw a line in the output from f(i1) to f(i2).