Quantifying Tensor Field Similarity with Global Distributions and Optimal Transport

Strain tensor fields quantify tissue deformation and are important for functional analysis of moving organs such as the heart and the tongue. Strain data can be readily obtained using medical imaging. However, quantification of similarity between different data sets is difficult. Strain patterns vary in space and time, and are inherently multidimensional. Also, the same type of mechanical deformation can be applied to different shapes; hence, automatic quantification of similarity should be unaffected by the geometry of the objects being deformed. This work introduces the application of global distributions used to classify shapes and vector fields in the pattern recognition literature, in the context of tensorial strain data. In particular, the distribution of mechanical properties of a field are approximated using a 3D histogram, and the Wasserstein distance from optimal transport theory is used to measure the similarity between histograms. To measure the method's consistency in matching deformations across different objects, the proposed approach was evaluated by sorting strain fields according to their similarity. Performance was compared to sorting via maximum shear distribution (a 1D histogram) and tensor residual magnitude (in perfectly registered objects). The technique was also applied to correlate muscle activation to muscular contraction observed via tagged MRI. The results show that the proposed approach accurately matches deformation regardless of the shape of the object being deformed. Sorting accuracy surpassed 1D shear distribution and was on par with residual magnitude, but without the need for registration between objects.