Rotational flows for interpolation between sampled surfaces

Abstract

We introduce a locally defined shape-maintaining method for interpolating between corresponding oriented samples (vertices) from a pair of surfaces. We have applied this method to interpolate synthetic data sets in two and three dimensions and to interpolate medially represented shape models of anatomical objects in three dimensions. In the plane, each oriented vertex follows a circular arc as if it was rotating to its destination. In three dimensions, each oriented vertex moves along a helical path that combines in-plane rotation with translation along the axis of rotation.We show that our planar method provides shapemaintaining interpolations when the reference and target objects are similar. Moreover, the interpolations are size maintaining when the reference and target objects are congruent. In three dimensions, similar objects are interpolated by an affine transformation. We use measurements of the fractional anisotropy of such global affine transformations to demonstrate that our method is generally more shape preserving than the alternative of interpolating vertices along linear paths irrespective of changes in orientation. In both two and three dimensions we have experimental evidence that when non-shape-preserving deformations are applied to template shapes, the interpolation tends to be visually satisfying with each intermediate object appearing to belong to the same class of objects as the end points.

Image Estimation from Marker Locations for Dose Calculation in Prostate Radiation Therapy

“…This procedure is similar to organ segmentation except that the seed locations, rather than image intensities, are what drive deformation within the trained shape space describing day-to-day shape variations of the prostate [12]. The diffeomorphism implied by the two m-reps is then computed with a shape interpolation method called rotational flows [13]. The deformation is used to transform the planning image to generate an estimated treatment image for dose calculation.…”

“…In 3D cases, each oriented vertex moves along a helical path along the axis of rotation. It can be shown that the interpolation is shape-maintaining if the two objects are similar and is size-maintaining if the two models are congruent [13]. …”