Projective Mappings for Image Warping

The homogeneous representation for points provides a consistent notation for affine and projective mappings. Homogeneous notation was used in projective geometry [Maxwell46], [Coxeter78] long before its introduction to computer graphics [Roberts66]. The homogeneous notation is often misunderstood so we will take a moment to clarify its use and properties. In familiar Euclidean geometry we represent points of the real planeR by vectors of the form (x;y) . The superscript T denotes matrix transpose; that is, a column vector. We will use column vectors for points and vectors. Projective geometry deals with the projective plane, a superset of the real plane, whose homogeneous coordinates are (x; y;w) . In projective geometry the 2-D real point (x;y) is represented by the homogeneous vector p = (x; y;w) = (xw;yw;w) , where w is an arbitrary nonzero number. Vectors of the form (xw;yw;w) for w 6= 0 form the equivalence class of homogeneous representations for the real point (x;y) . To recover the actual coordinates from a homogeneous vector, we simply divide by the homogeneous component; e.g., the homogeneous vector p = (xw;yw;w) = (x; y;w) represents the actual point (x;y) = (x=w;y=w) . This division, a projection onto the w = 1 plane, cancels the effect of scalar multiplication by w. When representing real points with homogeneous notation we could use any nonzero w, but it is usually most convenient to choosew = 1 so that the real coordinates can be recovered without division. Projective space also includes the points at infinity: vectors of the form (x; y;0) , excluding (0;0;0) . The points at infinity lie on the line at infinity. We will see later how augmentation of the real plane by these points simplifies projective geometry. In homogeneous notation, 2-D points are represented by 3-vectors and 3-D points are represented by 4-vectors. For affine and projective mappings, we denote points in source space by ps = (u; v; q) and points in the destination space by pd = (x; y;w) .