Affine Invariant

The study of invariants has been pursued actively for many years. Invariants provide us with the ability to come up with representations of the features in a scene that do not depend on the view, and can prove to be extremely handy for purposes of recognising objects from multiple views. In this Subsection we explore the possibility of deriving an affine invariant for a contour.

Let us define a measure called the cross-conjugate product (CCP) on the Fourier representations of two views as

$$\psi(0, l)[k] = ({\bf\bar{X}}^0[k])^{*T}{\bf\bar{X}}^l[k], \;\;\; 0 < k < N$$

$$= ({\bf\bar{X}}^0[k])^{*T} {\bf A}_l {\bf\bar{X}}^0[k] \exp({\frac{j 2 \pi \lambda_l k}{N} }).$$ (5)

The matrix ${\bf A}_l$ can be expressed as a sum of a symmetric matrix and a skew symmetric matrix as ${\bf A}_l = {\bf A}_l^s + {\bf A}_l^{sk}$ where ${\bf A}_l^s = \frac{1}{2} ({\bf A}_l + {\bf A}_l^T)$ and ${\bf A}_l^{sk} = \frac{1}{2} ({\bf A}_l - {\bf A}_l^T)$. The skew symmetric matrix reduces to

$$c \left[ \begin{array}{cr} 0 & 1 -1\;\; & 0 \end{array} \right],$$

where $c = m_{l12} - m_{l21}$ is the difference of the off-diagonal elements of ${\bf A}_l$. We now have

$$\psi(0, l)[k] = {\bf\bar{X}}^0[k]^{*T}\left( {\bf A}_l^s + {\bf A}_l^{sk} \right) {\bf\bar{X}}^0[k]\;\exp({\frac{j 2 \pi \lambda_l k}{N}})$$

The first term of the above equation is purely real and the second term is purely imaginary. We observe that the effect of the transformation matrix $A_l$ on the second term is restricted to a scaling by a factor $c$. We can define a new measure $\kappa$, ignoring scale, for the sequence ${\bf\bar{X}}^l$ in view $l$ as

$$\kappa(l)[k] = {\bf\bar{X}}^l[k]^{*T} \left[ \begin{array}{cr} 0 & 1 \\ -1\;\; & 0 \end{array} \right] {\bf\bar{X}}^l[k].$$ (6)

It can be shown [7] that

$$\kappa (l)[k] = \vert{\bf A}_l\vert\; \kappa(0)[k] ,\;\;\; 0 < k < N$$ (7)

Equation 7 gives a necessary condition for the sequences ${\bf\bar{X}}^l$ and ${\bf\bar{X}}^0$ to be two different views of the same planar shape, or in other words, the values of the measure $\kappa(\cdot)$ in the two views should be scaled versions of each other. This extends to multiple views also. Consider the $M\!\times\!(N-1)$ matrix formed by the coefficients of the $\kappa(\cdot)$ measures for M different views.

$$\Theta = \left[ \begin{array}{ccc} \kappa(0)[1] & \cdots & \... ...kappa(M-1)[1] & \cdots & \kappa(M-1)[N-1] \end{array} \right]$$

The necessary condition for matching of the planar shape in $M$ views then reduces to

$$\mbox{rank}(\Theta) = 1.$$ (8)

It should be noted that this recognition constraint does not require correspondence between views and is valid for any number of views.

Since, the $\kappa$ measures in the various views are only scaled versions of each other, if we normalize the $\kappa$ measure terms in each view with respect to a fixed one then

$$\gamma(l)[k] = \kappa(l)[k] \;\; / \;\;\kappa(l)[p] \;,\;\;\mbox{ p is fixed}$$

$$= (\vert{\bf A}_l\vert \kappa(0)[k] ) \;\;/\;\; (\vert{\bf A}_l\vert \kappa(0)[p] )$$

$$= \kappa(0)[k] \;\; / \;\; \kappa(0)[p]$$

$$\gamma(l)[k] = \gamma(0)[k]$$ (9)

These terms of the normalized $\kappa$ measure - the $\gamma$ measure are independent of the view. Hence, $\gamma$ is an affine view invariant of a contour, whose computation does not need correspondence information across views.