Question : Translating coordinates from one coordinate system to another

On a sheet of paper:
The X coordinate increases from left to right
The Y coordinate increases from top to bottom
(0,0) is at the top left corner.

I have two points on this sheet: (x1, y1) and (x2,y1) (i.e., they're both on the same horizontal line)

I now photocopy this sheet. During photocopying, the original paper shifted slightly, and also rotate slightly. As a result, my two original points now became (on this new sheet of paper):
(x3,y3) and (x4,y4).

Now, given any point (a1,b1) on the original sheet of paper, what would its coordinates be on the new sheet of paper?

I thought I had derived the correct formulat for this, but I always seem to be a few pixels off.

Thanks.

Answer : Translating coordinates from one coordinate system to another

You have probably already derived the correct equations, but let's go through it anyway.

You want a rigid transformation that translates (x1, y1) to (x3, y3), and (x2, y1) to (x4, y4).

We can think of the rigid transformation as a translation, followed by a rotation.  We'll start with a translation that moves (x1, y1) to (x3, y3).  This is easy:

(a, b) ---> (a + x3 - x1, b + y3 - y1)

will do the trick.

Now, we want a transformation that rotates around the point (x3, y3) so that the point originally at (x2, y1), which has been mapped over to (x2 + x3 - x1, y3), rotates over to (x4, y4).

Let's work out the angle of this rotation.  Recall that the direction of a vector (x, y) can be determined by the atan2(y, x) function (most math libraries have a atan2 which determines the quadrant of the vector as well as its arctan - don't use the simple atan( y/x ) function for this).

So the total rotation is just the difference between the two angular directions, that is

theta = atan2( y4 - y3, x4 - x3 ) - atan2( x2 - x1, 0 )

(remember that we need vectors that start at the point (x3, y3), so this needs to be subtracted from both of the points).

Okay, now, we simply apply a rotation operation.

(c, d) ----->  ( c * cos(theta) - d * sin(theta), c * sin(theta) + d * cos(theta) )

where (c, d) = (a + x3 - x1, b + y3 - y1), from before.


If you are finding that you might be off by a few pixels near the edge of your image, you're suffering from a resolution problem: the true locations of your two points aren't really (x3, y3) and (x4, y4), but those are just the closest co-ordinates.  This does not affect the translation, since the same (sub-pixel) error just gets propagated throughout the image.  But for the rotation, a small error in the rotation near (x4, y4) could turn into a larger error farther away.  Basically, the error will be something around the resolution of your image divided by abs(x2 - x1).  If x2 and x1 are close together, you can see why you might be off by a few pixels.

There is really no good way to correct for this, other than getting more accurate co-ordinates for (x3, y3) and (x4, y4), or getting mappings for more than two points, and doing a best fit on the translation and rotation you want.

Hope this helps!