This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process. The results are applied to the analysis and visualization of diffusion tensor magnetic resonance images of the brain in a healthy volunteer. The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisotropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractional anisotropy structures. Magn Reson Med, 2006.

Cardiac magnetic resonance (CMR) is an essential clinical tool for the assessment of cardiovascular disease. Deep learning (DL) has recently revolutionized the field through image reconstruction techniques that allow unprecedented data undersampling rates. These fast acquisitions have the potential to considerably impact the diagnosis and treatment of cardiovascular disease. Herein, we provide a comprehensive review of DL-based reconstruction methods for CMR. We place special emphasis on state-of-the-art unrolled networks, which are heavily based on a conventional image reconstruction framework. We review the main DL-based methods and connect them to the relevant conventional reconstruction theory. Next, we review several methods developed to tackle specific challenges that arise from the characteristics of CMR data. Then, we focus on DL-based methods developed for specific CMR applications, including flow imaging, late gadolinium enhancement, and quantitative tissue characterization. Finally, we discuss the pitfalls and future outlook of DL-based reconstructions in CMR, focusing on the robustness, interpretability, clinical deployment, and potential for new methods.

Purpose: To establish the correspondence between the two histologically observable and diffusion tensor MRI (DTMRI) measurements of myolaminae orientation for the first time and show that single myolaminar orientations observed in local histology may result from histological artifact.Materials and Methods: DTMRI was performed on six sheep left ventricles (LV), then corresponding direct histological transmural measurements were made within the anterobasal and lateral‐equatorial LV. Secondary and tertiary eigenvectors of the diffusion tensor were compared with each of the two locally observable sheet orientations from histology. Diffusion tensor invariants were calculated to compare differences in microstructural diffusive properties between histological locations with one observable sheet population and two observable sheet populations.Results: Mean difference ± 1SD between DTMRI and histology measured sheet angles was 8° ± 27°. Diffusion tensor invariants showed no significant differences between histological locations with one observable sheet population and locations with two observable sheet populations.Conclusion: DTMRI measurements of myolaminae orientations derived from the secondary and tertiary eigenvectors correspond to each of the two local myolaminae orientations observed in histology. Two local sheet populations may exist throughout LV myocardium, and one local sheet population observed in histology may be a result of preparation artifact. J. Magn. Reson. Imaging 2011.

The spatially varying tensor fields that arise in magnetic resonance imaging are difficult to visualize due to the multivariate nature of the data. To improve the understanding of myocardial structure and function a family of objects called glyphs, derived from superquadric parametric functions, are used to create informative and intuitive visualizations of the tensor fields. The superquadric glyphs are used to visualize both diffusion and strain tensors obtained in canine myocardium. The eigensystem of each tensor defines the glyph shape and orientation. Superquadric functions provide a continuum of shapes across four distinct eigensystems (λ_i, sorted eigenvalues), λ₁ = λ₂ = λ₃ (spherical), λ₁ < λ₂ = λ₃ (oblate), λ₁ > λ₂ = λ₃ (prolate), and λ₁ > λ₂ > λ₃ (cuboid). The superquadric glyphs are especially useful for identifying regions of anisotropic structure and function. Diffusion tensor renderings exhibit fiber angle trends and orthotropy (three distinct eigenvalues). Visualization of strain tensors with superquadric glyphs compactly exhibits radial thickening gradients, circumferential and longitudinal shortening, and torsion combined. The orthotropic nature of many biologic tissues and their DTMRI and strain data require visualization strategies that clearly exhibit the anisotropy of the data if it is to be interpreted properly. Superquadric glyphs improve the ability to distinguish fiber orientation and tissue orthotropy compared to ellipsoids. Magn Reson Med 53:169–176, 2005. Published 2004 Wiley‐Liss, Inc.