Inferring transforms [computer graphics

Abstract

In simple two-dimensional texture mapping you take a 2D image and render it on the screen after some transformation or distortion. To accomplish this you will need to take each [X, Y] location on the screen and calculate a [U, V] texture coordinate to place there. A particularly common transformation is: U=(aX+bY+c)/(gX+hY+j), V=(dX+eY+f)/(gX+hY+j). By picking the proper values for the coefficients a...j, we can fly the 2D texture around to an arbitrary position, orientation, and perspective projection on the screen. You can, in fact, generate the coefficients by a concatenation of 3D rotation, translation, scale, and perspective matrices. However, the author discusses a more direct approach to finding a...j. It turns out that the 2D-to-2D mapping is completely specified if you give four arbitrary points in screen space and the four arbitrary points in texture space they must map to. The only restriction is that no three of the input or output points may be collinear. This method of transformation specification proves useful, for example, in taking flat objects digitized in perspective and processing them into orthographic views.