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8 July 1994 Hinting about causes of deformation: visual explorations of a novel inverse problem in medical imaging
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In previous papers in this series, we have extended the praxis of image warping by thin-plate splines to accord with information about edge direction and other features of affine derivatives at landmarks. It is interesting to consider the derived warp that is induced solely by the changes of edge induced by a landmark displacement. To the extent that this latter transformation resembles the former we may speak of the landmark displacement as a hypothetical `cause' of the change observed in the edge image, a cause to be estimated by an inverse regression between these energetically oblique subspaces. The apparent `scales' of these specifications are discrepant even when they agree in all large- scale features of deformation. We show how nontrivial regressions arise when edge-information is collinear with interlandmark segments. Other forms of affine constraints on mappings are not interchangeable with landmark displacements, but correspond rather to relationships among diverse subspaces of derivative specifications likewise apparently discrepant in scale and spatial localization. These derivative-constrained splines, which all arise as singular perturbations of the original landmark- driven spline, thereby supply a very flexible and diverse language of image changes to complement various current approaches emphasizing the tracking or `evolution' of features pertaining to single images.
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Fred L. Bookstein and William D. K. Green "Hinting about causes of deformation: visual explorations of a novel inverse problem in medical imaging", Proc. SPIE 2299, Mathematical Methods in Medical Imaging III, (8 July 1994);

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