Measurement of Transformation Distortion

It is important to track both the minimum and maximum values of the Jacobian during the estimation procedure. The Jacobian measures the differential volume change of a point being mapped through the transformation. At the start of the estimation, the transformation is the identity mapping and therefore has a Jacobian of one. If the minimum Jacobian goes negative, the transformation is no longer a one-to-one mapping and as a result folds the domain inside out [30]. Conversely, the reciprocal of the maximum value of the Jacobian corresponds to the minimum value of the Jacobian of the inverse mapping. Thus, as the maximum value of the Jacobian goes to infinity, the minimum value of the Jacobian of the inverse mapping goes to zero. In the present approach, the inverse transformation consistency constraint was used to penalize transformations that deviated from their inverse transformation. A limitation of this approach is that cost function in Eq. 3. is an average metric and can not enforce the pointwise constraints that $$\min_x \{J(h)\} = 1/\max_x \{J(g)\}$$ and $$\min_x \{J(g)\} = 1/\max_x \{J(h)\}$$. This point is illustrated by Tables 1 and 2 by the fact that the minimum values of $J(h)$ and $J(g)$ differ from the reciprocal of the maximum values of $J(g)$ and $J(h)$, respectively, However, these extremal Jacobian values do give an upper bound on the worst case distortions produced by the transformations demonstrating the consistency between the forward and reverse transformations.

Copyright © 2002 • The University of Iowa. All rights reserved. Iowa City, Iowa 52242
Questions or Comments: gary-christensen@uiowa.edu