Theorem 1:

The Fourier transform and the collineation commute with the above representation. That is, if points are transformed between views $0$ and $l$ using Equation 1, the same homography will transform corresponding frequency terms in the Fourier domain also. In other words,

$${\bf X}^l[k] = {\bf M}_l{\bf X}^0[k] \mbox{ , } 0 \le k < N.$$ (2)

Proof: Let ${\bf M}_l = m_{lij}, 1\le i,j \le 3$. Expanding Equation 1 for the $u$ term,

$$u^l[i] = m_{l11} u^0[i] + m_{l12} v^0[i] + m_{l13} w^0[i]$$

Taking the Fourier transform of the above equation and using the linearity property of Fourier transforms, we get

$$U^l[k] = m_{l11} U^0[k] + m_{l12} V^0[k] + m_{l13} W^0[k]$$

Similarly for $V^l[k]$ and $W^l[k]$. It is now easy to see that

$${\bf X}^l[k] = {\bf M}_l{\bf X}^0[k]$$

giving us the desired result. $\Box$

Given a set of $L$ views, the recognition problem can be formulated as the identification of a view-independent function $f(\cdot)$ such that $f({\bf x}^0, {\bf x}^1, \ldots, {\bf x}^L) = 0$. This recognition constraint can be linear or nonlinear in image coordinates. The algebraic relation given by $f(\cdot)$ can then be used to settle the question whether the $L$ observed views were of the same object.