Rectification

The pure rotation case is a special case: a camera rotating about its optical center acquires images of a 3D scene as if the scene were represented on a plane infinitely far from the pin-hole.

When $\mathbf{t} = 0$, that is, when the coordinates of the two pin-holes in the two views coincide $\mathbf{t}_1 = \mathbf{t}_2$, transformation (9.40) reduces in size, yielding an equation compatible with a homography and therefore valid for every image point, regardless of whether a dominant plane is present. Therefore, when the two views share the same pin-hole (that is, in the case of pure rotation or a change in intrinsic parameters), it is possible to perform a transformation that is exact for all image points. This process of projecting points from one camera to another while changing the intrinsic parameters and rotation is called rectification.

To rectify an image, that is, to generate a dense image 1 from the points of image 2, it is necessary to use the homography matrix

$$\mathbf{H}_{1,2}=\mathbf{K}_2 \mathbf{R}_2\mathbf{R}_1^{-1} \mathbf{K}^{-1}_1$$ (9.46)

which makes it possible to obtain every point in image 1 from the points in image 2; that is, for each pixel $(u_1, v_1)$ of the image to be generated, homographic transformation $\mathbf{H}$ is applied to obtain point $(u_2,v_2)$ in the source image, from which the pixel value is copied.

Through transformation (9.46), it is possible to transform an image acquired by a camera with parameters $(\mathbf{K}_2,\mathbf{R}_2)$ into an image from a virtual camera with parameters $(\mathbf{K}_1, \mathbf{R}_1)$.

As applies to all homographies, a method for obtaining matrix $\mathbf{H}$ without knowledge of the intrinsic and extrinsic parameters, using only correspondences between the views of the two cameras, is presented in Section 9.5.1. The homography can then be factorized to recover the parameters that generated it.