Quaternionic Disparity Estimation

Given two images, f₁ and f₂, it is possible to find a vector field, $\textbf{d}(d_x,d_y)$, that relates the local displacement between f₁ and f₂  (i.e.  $f_1(\textbf{x})=f_2(\textbf{x}+\textbf{d}(\textbf{x})$ ). Therefore, if the QFT of f₁ is,

$$f_1(x,y) \Longrightarrow F^q(u,v),$$ (17)

then by the shift theorem,

$$f_2(x,y)=f_1(x+d_x,y+d_y) \Longrightarrow e^{i 2 \pi u d_x}F^q(u,v)e^{j 2 \pi u d_y}.$$ (18)

Knowing that f₁ and f₂ have local quaternionic phases ($\phi_1$,$\theta_1$,$\psi_1$) and ($\phi_2$,$\theta_2$,$\psi_2$) and assuming that $\phi$ varies only in x and $\theta$ varies only in y, then the displacement $\textbf{d}(\textbf{x})$ is given by

$$\begin{eqnarray} d_x(\textbf{x})=\frac{\phi_2(\textbf{x})-\phi_1(\textbf{x})}{u_... ...tbf{x})=\frac{\theta_2(\textbf{x})-\theta_1(\textbf{x})}{v_{ref}}.\end{eqnarray}$$ (19) (20)

The accuracy of the displacement depends strongly on the choice of the reference frequencies, u_ref and v_ref. The local model approach for quaternions outlined by Bülow (1999) will be used. This model assumes that the local phase at corresponding points of the two images will not differ, $\textbf{$\Omega_1$}$(x,y)=$\textbf{$\Omega_2$}$(x+d_x,y+d_y), where $\textbf{$\Omega$}$=($\phi$,$\theta$). An estimate for $\textbf{d}$ is obtained by a first-order Taylor expansion of $\Omega$ about x

$$\begin{eqnarray} \Omega_2 ( \textbf{x} + \textbf{d} ) \approx \Omega_2({\textbf{x}})+(\textbf{d} \cdot \nabla) \Omega_2(\textbf{x}).\end{eqnarray}$$ (21)

Solving for $\textbf{d}$ in equation 21 gives the disparity estimate for the local model. The disparity is estimated using equation 19 and the reference frequencies given by,

$$u_{ref}=\frac{\partial \phi_1}{\partial x}(\textbf{x}), ~~~~ v_{ref}=\frac{\partial \theta_1}{\partial y}(\textbf{y}).$$ (22)

The local quaternionic phase components for anywhere in an image are given by

$$\begin{eqnarray} \phi(\textbf{x})=\frac{\text{atan2}(n_{\phi}(\textbf{x}),d_{\ph... ...c{\text{atan2}(n_{\theta}(\textbf{x}),d_{\theta}(\textbf{x}))}{2},\end{eqnarray}$$ (23) (24)

where n and d are related to the rotation matrix of a quaternion and are,

$$\begin{eqnarray} n_{\phi}=-2(k^q_{eo}(\textbf{x})k^q_{oo}(\textbf{x}) +k^q_{ee}... ...textbf{x}))^2- (k^q_{eo}(\textbf{x}))^2-(k^q_{oo}(\textbf{x}))^2.\end{eqnarray}$$ (25) (26) (27) (28)

The k-functions are the responses of a symmetric component of the quaternionic Gabor filter to the image (e.g. k_ee=(h_ee $\ast$ $f)(\textbf{x})$). From equations 23 and 24 the derivatives of the local phase components are computed

$$\begin{eqnarray} \frac{\partial}{\partial x}\phi(\textbf{x})~=~\frac{d_{\phi}(\t... ...extbf{x})}{n^2_{\theta}(\textbf{x}) +d^2_{\theta}(\textbf{x})}~~.\end{eqnarray}$$ (29) (30)

Therefore, the disparity depends on the rate of change of the local phase as approximated by the quaternionic Gabor filters.