Isomap and Nonparametric Models of Image Deformation

Isomap is an exemplar of a set of data driven non-linear dimensionality reduction techniques that have shown promise for the analysis of images and video. These methods parameterize each image as coordinates on a low-dimensional manifold, but, unlike PCA, the low dimensional parameters do not have an explicit meaning, and are not natural projection operators between the high and low-dimensional spaces. For the important special case of image sets of an unknown object undergoing an unknown deformation, we show that Isomap gives a valuable pre-processing step to find an ordering of the images in terms of their deformation. Using the continuity of deformation implied in the Isomap ordering allows more accurate solutions for a thin-plate spline deformation from a specific image to all others. This defines a mapping between the Isomap coordinates and a specific deformation, which is extensible to give projection functions between the image space and the Isomap space. Applications of this technique are shown for cardiac MRI images undergoing chest cavity deformation due to patient breathing.