Problem Formulation

When a planar object is imaged from multiple view points or when a scene is imaged by cameras having the same optical centre, the images are related by a unique homography [4]. A homography or a collineation is a mapping from one plane to another such that the collinearity of a set of points is preserved. In other words, a homography, more precisely a projective homography, is an invertible mapping $h$ from $\kern 0.05em \hbox{\vrule width 0.05em height 1.55ex} \kern -0.05em {\rm P} ^2$ to itself such that three points $x_1$, $x_2$ and $x_3$ lie on the same line if and only if $h(x_1)$, $h(x_2)$ and $h(x_3)$ are also collinear.

Plane-to-plane homographies can be categorised into isometry, similarity, affine and projective [4]. The later classes subsume the earlier ones, i.e., isometry $\subset$ similarity $\subset$ affine $\subset$ projective. Projective homography is mathematically most general. In this paper, we derive the rank constraints for affine homographies, and later show that the constraints are valid in most practical situations of imaging a scene from multiple points, when the image to image homography is projective. Let the image-to-image transformation of points from view 0 to view $l$ be given by a $3\times 3$ matrix ${\bf M}_l$.

$${\bf x}^l[i] = {\bf M}_l {\bf x}^0[i]$$ (1)

Let O be a set of $N$ points on the boundary of a planar object and let ${\bf P}_l$ be its images in views ${\bf\cal V}_l$ where $l$ is the view index. Let $(u^l[i], v^l[i], w^l[i])$ be the homogeneous coordinates of points on the closed boundary in view ${\bf\cal V}_l$. This shape is represented by a sequence of vectors of complex numbers as shown below.

$${\bf x}^l[i] = \left[ \begin{array}{c} u^l[i] + j 0 v^l[i] + j 0 w^l[i] + j 0 \end{array} \right]$$

($w^l[i]$ need not be 1). Let the Fourier domain representation of the sequence ${\bf x}^l[i], 0 \leq i < N$ be ${\bf X}^l[k], 0 \leq k < N$ such that

$${\bf X}^l[k] = \left[ \begin{array}{c} U^l[k] V^l[k] W^l[k] \end{array} \right]$$

where $U^l[k] = U^l_R[k] + j U_I^l[k]$, $V^l[k] = V^l_R[k] + j V^l_I[k]$, $W^l[k] = W^l_R[k] + jW_I^l[k]$ are respectively the Fourier transforms of the individual sequences $(u^l[i] + j0), (v^l[i] + j0), (w^l[i] + j0)$. The subscripts $R$ and $I$ denote the real and imaginary components of the corresponding complex number. Note that the sequences ${\bf X}^l[k]$ are periodic and conjugate symmetric.