Trying to derive a 2D transformation matrix using only the images?

Not sure if you want manual or automatic Manual If you specify the transformed coordinates of the four corners of your rectangle, then you can derive the transformation equations: (From Pierre Wellner's Interacting with Paper on the DigitalDesk and more details in his Thesis ) Now you just have to solve for the coefficients of the equation With four point pairs, the two sets of four simultaneous linear equations can be quickly solved by Gaussian Elimination to find the values of c1-8 Lastly, you can turn those equations into the 3x3 matrices you want. The above equations are powerful enough to do non-linear transformations and you can simplify it into the 3x3 affine shear matrix But I would just stick with the nonliner equations (above) since they can handle perspective distortion Automatic Same method, but you can use an edge-detector comboined with a line detection algorithm to find a set of 4-ish lines that makeup a rectangle If your image rectangles really stand out (whiteish images on a dark background), then you can use corner detection available from libraries like OpenCV's Feature Detection (see cvCornerHarris ) You can intersect those lines to find the four corners and use the transformation equation.

Not sure if you want manual or automatic... Manual If you specify the transformed coordinates of the four corners of your rectangle, then you can derive the transformation equations: (From Pierre Wellner's Interacting with Paper on the DigitalDesk and more details in his Thesis) Now you just have to solve for the coefficients of the equation. With four point pairs, the two sets of four simultaneous linear equations can be quickly solved by Gaussian Elimination to find the values of c1-8. Lastly, you can turn those equations into the 3x3 matrices you want.

The above equations are powerful enough to do non-linear transformations and you can simplify it into the 3x3 affine shear matrix. But I would just stick with the nonliner equations (above) since they can handle perspective distortion. Automatic Same method, but you can use an edge-detector comboined with a line detection algorithm to find a set of 4-ish lines that makeup a rectangle.

If your image rectangles really stand out (whiteish images on a dark background), then you can use corner detection available from libraries like OpenCV's Feature Detection (see cvCornerHarris). You can intersect those lines to find the four corners and use the transformation equation.

I should let Mike speak for himself, but I for one feel hopelessly inadequate to understand your explanation. I think many of us are only passingly familiar with mathematics. – Carl Smotricz Dec 6 '09 at 19:15 Oh.

Well Gaussian Elimination is well documented in books such as "Numerical Algorithms". But I would like to stress that this is far superior to 3x3 affine transformations as it can handle perspective distortion (a non-linear non-affine transformation). If there is interest, I can expand on my answer.

– Frank Krueger Dec 6 '09 at 19:19 Comment Span. I would like to add that I have used this exact method in a production project to excellent effect. My apologies if I didn't do it justice, but don't let my glib description steer you away from this powerful technique.

Please read the linked-to materials. – Frank Krueger Dec 6 '09 at 19:22 This is PERFECT, Frank. I don't think the explanation was glib, and it gives me a map for what I need to do.

I'm certain there are perspective transformations going on in there. I can derive the c1-c8 constants from 4 non-linear points... they needn't be the corners, right? I just remembered that two of the corners are clipped in the second image, but it shouldn't matter in this case... there are a number of very distinct reference points I can use.

I'll plug those in to convert the rest. I'm not married to the idea of using 3x3 matrices... I just needed a method to translate points. – Mike Dec 6 '09 at 19:43 I think your image link is broken and I have a feeling the lack of that image is why I seem to be missing the part about how to derive a set of non-linear equations from the two separate set of corresponding points.

If you could expound on how to derive the transformation equations I think that would help my out a lot. – jpierson Dec 6 '097 at 17:19.

You only need 3 points to define a 3x3 transformation matrix. If you have the points (0,0), (0,1) and (1,0) and transform them by the matrix a be c d e f 0 0 1, you'll get (c,f), (b,e) and (a,d).

Oh... we have 6 unknowns, but there are 2 equations for each pair of dots... so 3 pairs will give us 6 equations. Clever! – Carl Smotricz Dec 6 '09 at 19:13.

I think you should start by providing a list of, say 6 3 points (for 6 unknowns) with X/Y coordinates before and after transformation. Then somebody more clever than I should pop that into a set of linear equations and then feed it to (say) Wolfram Alpha for solving. The top of Java's documentation for AffineTransform shows how the matrix needs to be set up: x' m00 m01 m02 x m00x + m01y + m02 y' = m10 m11 m12 y = m10x + m11y + m12 1 0 0 1 1 1 Removing most of the fluff leaves: x' m00x + m01y + m02 y' = m10x + m11y + m12 Then you just set up a set of 6 x 2 equations like this: m00x + m01y + m02 - x' = 0 m10x + m11y + m12 - y' = 0 (repeat for 2 other x/y before/after pairs) and throw them at an equation solver.

This is a simplification of my answer where you simply drop the c3 and c7 terms. But, good work spelling the actual linear algebra involved. Your "equation solver" is my "Gaussian Elimination".

– Frank Krueger Dec 6 '09 at 19:25 Thank you! I'm happy to see I came close. I think I did my last GE 30 years ago; since then I've been happy to let a program do it.

Glad to see Mike was taken care of by a pro. – Carl Smotricz Dec 6 '09 at 19:51.

I cant really gove you an answer,but what I can give you is a way to a solution, that is you have to find the anglde that you relate to or peaks your interest. A good paper is one that people get drawn into because it reaches them ln some way.As for me WW11 to me, I think of the holocaust and the effect it had on the survivors, their families and those who stood by and did nothing until it was too late.