Motion-Compensated Interpolation in Echocardiography: A Lie Advection-Based Approach

To better understand cardiac structures and dynamics via echocardiography, it is essential to have cardiac image sequences with sufficient spatio-temporal resolution. However, in echocardiography, there is an inherent tradeoff between temporal and spatial resolution, which limits the ability to acquire images with both high temporal and spatial resolution simultaneously. Motion-compensated interpolation, a post-acquisition technique, enhances the temporal resolution without compromising the spatial resolution. This paper introduces a novel motion-compensated interpolation algorithm based on the advection equation in fluid mechanics. Considering the incompressibility of cardiac tissue, we derive a solution in terms of Lie series for the advection problem. Subsequently, we construct a bidirectional advection energy model to estimate the optimal velocity fields that can simultaneously advect two cardiac images towards each other. The process continues until they converge at a midpoint where the image similarity peaks. To preserve the topology of the cardiac structures and ensure that image deformations are diffeomorphic, the advection process is carried out gradually with a smooth velocity field. To reduce the contribution of the blood signal in optimizing for the best tissue advection velocity, a nonlocal regularization pre-processing is applied to echocardiography data. Our algorithm, tested on 2D and 3D echocardiography, outperforms existing motion-compensated interpolation algorithms in estimating cardiac motions. It preserves cardiac topology during image deformations and reduces interpolation artifacts, especially in low frame rate recordings. By training a neural network on the data generated by our algorithm, we achieved over 75 times faster computation without compromising image quality.