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Given two images, *f*_{1} and *f*_{2}, it is possible to find a
vector field, , that relates the local
displacement between *f*_{1} and
*f*_{2} (i.e. ).
Therefore, if the QFT of *f*_{1} is,
| |
(17) |

then by the shift theorem,
| |
(18) |

Knowing that *f*_{1} and *f*_{2} have local quaternionic phases
(,,) and (,,) and
assuming that varies only in *x* and varies only in
*y*, then the displacement is given by
| |
(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,
(*x*,*y*)=(*x*+*d*_{x},*y*+*d*_{y}),
where =(,). An estimate for
is obtained by a first-order Taylor expansion of about x
| |
(21) |

Solving for in equation 21 gives the disparity estimate
for the local model. The disparity is estimated using equation
19 and the reference frequencies given by,
| |
(22) |

The local
quaternionic phase components for anywhere in an image are
given by
| |
(23) |

| (24) |

where *n* and *d* are related to the rotation matrix of a quaternion
and are,
| |
(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} ). From equations 23 and
24 the derivatives of the local phase components are computed
| |
(29) |

| (30) |

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

** Next:** Synthetic Examples
** Up:** Witten: Quaternion-based Signal Processing
** Previous:** Quaternionic Gabor Filters
Stanford Exploration Project

4/5/2006