Radiological images are increasingly being used in healthcare and medical research. There is, consequently, widespread interest in accurately relating information in the different images for diagnosis, treatment and basic science. This article reviews registration techniques used to solve this problem, and describes the wide variety of applications to which these techniques are applied. Applications of image registration include combining images of the same subject from different modalities, aligning temporal sequences of images to compensate for motion of the subject between scans, image guidance during interventions and aligning images from multiple subjects in cohort studies. Current registration algorithms can, in many cases, automatically register images that are related by a rigid body transformation (i.e. where tissue deformation can be ignored). There has also been substantial progress in non-rigid registration algorithms that can compensate for tissue deformation, or align images from different subjects. Nevertheless many registration problems remain unsolved, and this is likely to continue to be an active field of research in the future.

Motion of an object degrades MR images, as the acquisition is time-dependent, and thus k-space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k-space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstra- Motion of an object can degrade MR images and imposes constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that the acquisition is time-dependent, and the Fourier transform of the image seen during acquisition changes due to the deformation of the object. This causes inconsistencies in k-space and hence ghosts in the image.Standard motion correction methods make assumptions on the type of motions, for example, that it is a translation or a rotation, and use formulas on Fourier transforms to correct the data (1-3). We assume here that these data are acquired in shots. When the data positions at each shot are known, an empirical motion correction method could be used to spatially transform the ghosted image by the transformation corresponding to a shot, pick the k-space lines corresponding to that shot, and repeat this operation for all shots (this is a version of the method used in (1)). We could then rebuild an image by inverse Fourier transform. This method is in general incorrect, as shown by the difference between translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the other hand, correcting rotations requires knowledge of the data at neighboring k-space positions and these are acquired at different times. Before applying the Fourier rotation theorem, we would need to "synchronize" neighboring values. Furthermore, complicated motions such as nonrigid deformations cannot have a simple description in Fourier space. Here, however, we show that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical description of the problems involved; the motion corruption is entirely described by a large matrix acting on the space of images. Thus, inversion of this matrix should correct the motion's effects. This approach is of theoretical interest, but its practical value depends on how easily we can find a solution of the linear system. It turns out ...

In biological tissue, all eigenvalues of the diffusion tensor are assumed to be positive. Calculations in diffusion tensor MRI generally do not take into account this positive definiteness property of the tensor. Here, the space of positive definite tensors is used to construct a framework for diffusion tensor analysis. The method defines a distance function between a pair of tensors and the associated shortest path (geodesic) joining them. From this distance a method for computing tensor means, a new measure of anisotropy, and a method for tensor interpolation are derived. The method is illustrated using simulated and in vivo data. The diffusion of water in biological tissue can be characterized by a positive definite tensor, which means that all eigenvalues are positive (or in the limit nonnegative). This property is in general ignored by standard methods of calculus with tensors because of the lack of a general framework that takes the positivity into account.We present here such a framework, built on a rigorously defined distance function on the curved space of positive definite tensors. Using this distance function, we show how to interpolate between tensors and how to compute quantities such as a tensor mean. The conventional fractional anisotropy (FA) is a measure of the normalized distance of a tensor from its isotropic part and uses the matrix norm for the distance measure. Here we use the distance measured along a geodesic to define a new measure, the geodesic anisotropy (GA). The concept of isotropic part will be defined more clearly below. These ideas are demonstrated on simulated and in vivo diffusion tensor MRI (DT-MRI) data. THEORYDT-MRI produces images that are the matrix components of a diffusion tensor and standard operations such as a distance function (the quantification of a difference), averaging, and interpolation are often required on tensors. To this list we add specific tensor operations such as computations of anisotropies, which quantify the shape of tensor ellipsoids. What we will call the standard method for these operations is to apply them linearly to the matrix components. Then, we have 1. distance function between tensors D 1 and D 2 :for s in [0, 1]. The mean of two tensors D 1 and D 2 is (D 1 ϩ D 2 )/2 (arithmetic mean); the mean of a setThe most common anisotropy index of a generic tensor D, the FA, is proportional to ʈD Ϫ trace(D)/3Iʈ/ʈDʈ (where I is the identity matrix). Under a general linear change of coordinates T, tensors change according to the congruence transformations D ‫ۋ‬ TDT t , where the superscript t denotes transposition. Note that in general the standard distance function is not invariant under congruence transformations; this is the case only when T is orthogonal. The essential characteristic of diffusion tensor matrices is to have all their eigenvalues strictly positive (see also the Discussion). The operations as defined above, however, are not specific to such tensors; tensors with zero, or negative, eigenvalues are handled in the same way as positive def...

The subject of this study is the controversial choice of directions in diffusion tensor MRI (DT-MRI); specifically, the numerical algebra related to this choice. In DT-MRI, apparent diffusivities are sampled in six or more directions and a leastsquares equation is solved to reconstruct the diffusion tensor. Numerical characteristics of the system are considered, in particular the condition number and normal matrix, and are shown to be dependent on the relative orientation of the tensor with respect to the laboratory frame. As a consequence, noise propagation can be anisotropic. However, the class of icosahedral direction schemes is an exception, and icosahedral directions have the same condition number and normal matrix for direction encoding as the ideal scheme with an infinite number of directions. This normal matrix and its condition number are rotationally invariant. Numerical simulations show that for icosahedral schemes with 30 directions the standard deviation of the fractional anisotropy is both low and nearly independent of fiber orientation. Key words: diffusion encoding; icosahedral; rotational invariance; condition numberA recurring question in diffusion tensor MRI (DT-MRI) is the optimal selection of gradient directions for diffusion sensitization. In DT-MRI the acquired images are not directly the components of the tensor, but diffusion sensitized images for at least six directions. To get the tensor itself, a linear least-squares system must be solved. The algebra of this system is specified by the choice of directions and influences the noise propagation to the tensor images.Several alternative and partially contradictory methods have been reported for choosing directions (1-4). In particular, Jones et al. (2) The theory section covers the algebraic reconstruction of the diffusion tensor, in particular for the ideal case in which every direction is sampled. Explicit values are derived for two key quantities associated with the leastsquare system; namely, the condition number and the normal matrix of the system. The rotational dependence of these values for a number of direction schemes is then considered, together with the consequences for the propagation of noise in each case. Due to the mathematical nature of this study, an extensive appendix is also included, which provides additional details of the relevant algebra. The experimental section describes computer simulations used to assess the implications of the theoretical findings and investigates noise propagation in the fractional anisotropy (FA) of the diffusion tensor. Our aim is to provide an algebraic framework for choosing diffusion encoding directions, and thus to identify an optimal set of directions. THEORY Background