PicoGA: Evolving a 3D Projection Matrix

In my previous blog post PicoGA  – a tiny GA for C# I talk about PicoGA and I demonstrate a few problems that can be solved using it. One of the examples is evolving a 3D Projection Matrix from a few pixel to world coordinate mappings taken from a given photograph.

The example didn’t use “height” in the picture because it contained no heights that I could be sure of. In this example, I’m using the heights of the goal posts in the picture below to show that it  can correctly handle World (x,y,z) to Screen(x,y) mapping.

From this picture I’ve identified 9 points that I’ve determined the World(x,y,z) coordinates (the football field is 105×68 meters, the goal is 2.44 meters high and 7.32 meters wide). The coordinates are specified below the picture.

new-pitch-with numbers

I simply use MS Paint (yay!) to determine the pixel coordinate, and I use the given dimensions of the football field (105×68) to determine the world coordinates;

    new WorldToScreenCase(688, 349, 0,0 ,0), // 1
    new WorldToScreenCase(454, 155, 52.5,0,0), // 2
    new WorldToScreenCase(353, 70,  105,0,0), // 3
    new WorldToScreenCase(127, 169, 52.5, 34,0), // 4
    new WorldToScreenCase(155, 411, 0, 34,0), // 5
    new WorldToScreenCase(220, 364, 0, 34 - 7.32*0.5, 2.44), // 6
    new WorldToScreenCase(80,  378, 0, 34+7.32*0.5,2.44), // 7
    new WorldToScreenCase(92,  57,  105, 34 + 7.32*0.5, 2.44), // 8
    new WorldToScreenCase(145, 56,  105, 34-7.32*0.5,2.44), // 9

It takes a while, here’s a video of evolution in action;

but the program is successful in evolving an appropriate matrix;


As you can see, the evolved matrix (the red dots) hits the measured points (blue dots) with quite high accuracy. This matrix can now be used for animating something moving across the football field!

As I stated in my previous post, there are analytical methods for doing this, but I’ve been unable to locate any, and this is a cool use for GA…

This is the code that I use to map from world coordinates to screen coordinates, using my evolved matrix;

private static Point3D ProjectCase(Point3D point, List<double> l)
	Point3D p = new Point3D(
		point.X - l[12] * 100,
		point.Y - l[13] * 100,
		point.Z - l[14] * 100);

	Point3D result = new Point3D(
		l[0] * p.X + l[1] * p.Y + l[2] * p.Z + l[3],
		l[4] * p.X + l[5] * p.Y + l[6] * p.Z + l[7],
		l[8] * p.X + l[9] * p.Y + l[10] * p.Z + l[11]);

	if (result.Z != 0)
		result.X *= l[15] / result.Z;
		result.Y *= l[15] / result.Z;

	return result;


You can find the source code among the PicoGA demos.


About mfagerlund
Writes code in my sleep - and sometimes it even compiles!

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