Using Homographies

Finding the projection matrix between the lenslet display and the image display would certainly solve the calibration problem. However, a neat insight is that instead of solving for the projection matrix, one can find a homography such that when applied to an image, its projected image in the lenslet array is aligned correctly [And]. Figure 3 illustrates this idea.

Figure 3: The goal is to find homography h.

Given an image A, its image in the display via projection matrix p is A'. We want to find a homography h such that B' = Ahp is a correctly calibrated image. Given a point $p$, projection matrix M and the projected point $p'$ in homogeneous coordinates:

$$\left\vert\begin{array}{ccc} M_{11} & M_{12} & M_{13} \\ ... ...egin{array}{c} x'\\ y'\\ 1'\\ \end{array} \right\vert$$

Let $M_i$ be the $i$'th row of M. Then

$$M_1p = \lambda x'$$ (1)

$$M_2p = \lambda y'$$ (2)

$$M_3p = \lambda$$ (3)

The scale factor $\lambda$ is unknown. However, we can take the ratios of Equations 1 and 2. Similarly for Equations 1 and 3 then we have:

$$y'(M_1p)-x'(M_2p) = 0$$ (4)

$$M_1p-x'(M_3p) = 0$$

For each point $p$ there are such two equations. With 9 unknowns, we need 4 points (each with 2 equations) to create a matrix A of rank 8. The matrix A is composed of only point coordinates, and the vector m comprises the values of the matrix M.

$$\left\vert\begin{array}{ccccc} xy' & yy' & y' & -xx' & \ld... ... M_{13}\\ M_{21}\\ \vdots \end{array}\right\vert = 0$$

The first row of A is shown with respect to equation 4. Since A is rank deficient, its null space exists and in this case is of dimension 1 (the constant scale factor). Vector m can be recovered using SVD techniques as the column of V corresponding to the zero singular value of A [TV98] [HZ00]. In practice, correspondence between points through a homography is known up to a noise factor, so many more points are used, forming an overconstrained system and the column of V corresponding to the smallest singular value of A is used. However, in this case of calibrating the autostereoscopic display, corresponding point are exactly known as explained in section 3.