Geodesic regression for image time-series

Abstract

Registration of image-time series has so far been accomplished (i) by concatenating registrations between image pairs, (ii) by solving a joint estimation problem resulting in piecewise geodesic paths between image pairs, (iii) by kernel based local averaging or (iv) by augmenting the joint estimation with additional temporal irregularity penalties. Here, we propose a generative model extending least squares linear regression to the space of images by using a second-order dynamic formulation for image registration. Unlike previous approaches, the formulation allows for a compact representation of an approximation to the full spatio-temporal trajectory through its initial values. The method also opens up possibilities to design image-based approximation algorithms. The resulting optimization problem is solved using an adjoint method.

Fig. 1

Principle of geodesic regression on the space of images. The interpolation path is determined by the blue and the red images. This geodesic path is maximally close to the dashed images.

Fig. 2

Translation experiment and results for geodesic regression with initial image and momentum free. The translation is well captured.

Fig. 3

Geodesic regression result with initial image fixed. The resulting trajectory is a compromise between the shapes. A perfect match of the square and the circle would require a local contraction (with respect to the diamond shape) followed by an expansion, which cannot be expressed by the approximative model.

Fig. 4

Coronal cross sections through ventricles for piecewise-geodesic approach (top) and for geodesic regression (bottom) with fixed initial image.

Fig. 5

Brain slices of subjects with increasing age (left to right, 38, 52, 58, 73, 81 [years]) and geodesic regression results. Results with initial conditions in the space of the 38 year old (middle row) and the 81 year old (bottom row) subject.

Fig. 6

Brain slices for longitudinal subject data (left to right, age: 67, 68, 71, 73 [years]). Top: original images overlaid with image at age 67. Bottom: geodesic regression results overlaid with original images. Yellow indicates agreement. Zoom in to view.