Linear Invariant Tensor Interpolation Applied to Cardiac Diffusion Tensor MRI

Abstract

Purpose Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape. Methods Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets. Results EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI. Conclusion GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.

The effects of noise over the complete space of diffusion tensor shape

“…More recently, we have used this technique to devise novel tensor interpolation methods that appear to outperform previous methods including Euclidean and log-Euclidean (Gahm et al, 2012). Moving forward, the mathematical framework could also be used to constrain tensor-field reconstruction; for tensor-field denoising; and for compressed sensing acquisition and reconstruction of tensor field data.…”

Microstructurally Anchored Cardiac Kinematics by Combining In Vivo DENSE MRI and cDTI

“…cDTI acquired diffusion tensors were interpolated using linear invariant interpolation [6] to the location of the FE quadrature points where the deformation invariants were computed (Fig. 1D).…”

Geometries and Interpolations for Symmetric Positive Definite Matrices

“…This presents an attractive opportunity to avoid the fattening effect, and it is also useful for interpretation. An abundance of approaches to decouple shape and rotation have appeared [23,16,35,28,22,18,9], seemingly independent of each other, and we shall review some of the most important ones below, in order of increasing complexity.…”

“…Gahm et al [16] utilize the following combination of R-and K-invariants from [23], which allows an analytical reconstruction of the eigenvalues along the interpolated path 2 :…”

“…2 Note that there are some typos in the definitions of R 2 and R 3 in [16]. The LIT interpolation between the same two thin, ellipsoidal tensors as before.…”