Next: Synthetic Examples Up: Witten: Quaternion-based Signal Processing Previous: Quaternionic Gabor Filters

# Quaternionic Disparity Estimation

Given two images, f1 and f2, it is possible to find a vector field, , that relates the local displacement between f1 and f2  (i.e.   ). Therefore, if the QFT of f1 is,
 (17)
then by the shift theorem,
 (18)
Knowing that f1 and f2 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, uref and vref. 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+dx,y+dy), 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. kee=(hee  ). 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