The Effect of Metric Selection on Averaging Diffusion Tensors – When and Why do Tensors Swell?

Introduction In many diffusion tensor imaging (DTI) analysis methods, including registration, realignment and re-slicing, averaging or interpolating tensors is required. Defining how distance is measured between tensors, through a metric, determines the interpolation results. It was demonstrated that when using a conventional Euclidean metric, the resulted tensor might have a larger volume (determinant) than the original tensors [1]. This effect was termed the “swelling effect” [2] and a family of geometric metrics that include the Affine-invariant metric [3] and the Log-Euclidean metric [1] were suggested in order to minimize it. However, recently it was shown that using the geometric metrics introduces bias in the estimation of the diffusion quantities, which renders these metrics inappropriate for diffusion tensor analysis [4]. In this work we seek to find the sources of the swelling effect by performing tensor averaging using a Euclidean metric and a Log-Euclidean metric, and observing the bias in estimating FA, ADC and volume. We show that (i) using the Log-Euclidean metric reduces swelling, but introduces other types of biases in the estimation of ADC and FA; (ii) depending on the type of noise, a swollen tensor may be a preferred estimate. We argue that unwanted swelling effect is limited to certain scenarios, yet neither metrics help in avoiding it then.