Convert quadrangle to rectangle?

Question

I have a scene consisting of one arbitrary quadrilateral. I need to be able to convert this quad to a rectangle. Each square is in 2d coordinates, so they have 4 vertices (x_i, y_i) .

The transformation should have the opposite, because the idea is to return to the original square after manipulating the rectangle.

What would be the easiest way to perform this operation? I heard this is called a perspective conversion, but I found some small tips that make me think it can be pretty easy to do.

Answer 1

Do you know what size of the desired rectangle is? You can map any convex quadrangle to a rectangle with a reversible transform with a perspective transform, if so. All you have to do is get 4 corresponding points (between the quad and the rectangle), say, (X ₁ , Y ₁ ), (X ₂ , Y ₂ ), (X ₃ , Y ₃ ), (X ₄ , Y ₄ ) for a quadrangle and (x ₁ , y ₁ ), (x ₂ , y ₂ ), (x ₃ , y ₃ ), (x ₄ , y ₄ )) for a rectangle, respectively. Then connect it to the final equation in the Borealid link and you will set:

The solution to the above equation (where n = 4) will give you the elements (a, b, c, d, e, ..., h) of the matrix of reversible perspective transformations,

This will allow you to convert points on a rectangle to points on quadrangles. For the inverse transform, just invert the transform matrix.

Also note that as soon as you get the transformed coordinates of the vector [XW YW W] ^T, you need to normalize it so that W = 1. That is, Your final answer will be [XW ​​/ W YW / WW / W] ^T which equals [XY 1] ^{T the} desired answer.

Answer 2

Not all quadrangles are rectangles. For this reason, there is no reversible transformation from square to rectangle; there are more quads than rectangles, so you cannot create a reversible mapping from squares to rectangles.

However, you can create a reversible transformation for a specific quad. As you guessed, this concerns the rotation of perspective, so the quadrangle “appears” as a rectangle in the new coordinate space. See http://alumni.media.mit.edu/~cwren/interpolator/ which contains the Matlab source code for this problem.

Answer 3

This solution uses the Java Advance Image (JAI) API all the magic in the QuadToQuad method. Here is a sample code.

 try { BufferedImage img = UtilImageIO.loadImage(picName); ParameterBlock params = new ParameterBlock(); params.addSource(img); //source is the input image int w = img.getWidth(); //Set to the original width of the image int h = img.getHeight(); //Set to the original height of image Point tl = new Point(x,y); //The new top left corner Point tr = new Point((x1,y1); //The new top right corner Point bl = new Point(x2,y2); //The new bottom left corner Point br = new Point(x3,y3); //The new bottom right corner PerspectiveTransform p = PerspectiveTransform.getQuadToQuad(0,0, 0, h, w, h, w, 0, tl.x, tl.y, bl.x, bl.y, br.x, br.y, tr.x, tr.y).createInverse(); WarpPerspective wa = new WarpPerspective(p); params.add(wa); params.add(Interpolation.getInstance(Interpolation.INTERP_BICUBIC)); //Change the interpolation if you need more speed RenderedOp dest = JAI.create("warp", params); //dest is now the output File outputfile = new File(picName); ImageIO.write(dest, "jpg", outputfile); } catch(Exception e){} 

Hope this helps you. :)