Low-dimensional representation of cardiac motion using Barycentric Subspaces: A new group-wise paradigm for estimation, analysis, and reconstruction

One major challenge when trying to build low-dimensional representation of the cardiac motion is its natural circular pattern during a cycle, therefore making the mean image a poor descriptor of the whole sequence. Therefore, traditional approaches for the analysis of the cardiac deformation use one specific frame of the sequence - the end-diastolic (ED) frame - as a reference to study the whole motion. Consequently, this methodology is biased by this empirical choice. Moreover, the ED image might be a poor reference when looking at large deformation for example at the end-systolic (ES) frame. In this paper, we propose a novel approach to study cardiac motion in 4D image sequences using low-dimensional subspace analysis. Instead of building subspaces relying on a mean value we use a novel type of subspaces called Barycentric Subspaces which are implicitly defined as the weighted Karcher means of k+1 reference images instead of being defined with respect to one reference image. In the first part of this article, we introduce the methodological framework and the algorithms used to manipulate images within these new subspaces: how to compute the projection of a given image on the Barycentric Subspace with its coordinates, and the opposite operation of computing an image from a set of references and coordinates. Then we show how this framework can be applied to cardiac motion problems and lead to significant improvements over the single reference method. Firstly, by computing the low-dimensional representation of two populations we show that the parameters extracted correspond to relevant cardiac motion features leading to an efficient representation and discrimination of both groups. Secondly, in motion estimation, we use the projection on this low-dimensional subspace as an additional prior on the regularization in cardiac motion tracking, efficiently reducing the error of the registration between the ED and ES by almost 30%. We also derive a symmetric and transitive formulation of the registration that can be used both for frame-to-frame and frame-to-reference registration. Finally, we look at the reconstruction of the images using our proposed low-dimensional representation and show that this multi-references method using Barycentric Subspaces performs better than traditional approaches based on a single reference.