Codomain scale space and regularization for high angular resolution diffusion imaging

Regularization is an important aspect in high angular resolution diffusion imaging (HARDI), since, unlike with classical diffusion tensor imaging (DTI), there is no a priori regularity of raw data in the co-domain, i.e. considered as a multispectral signal for fixed spatial position. HARDI preprocessing is therefore a crucial step prior to any subsequent analysis, and some insight in regularization paradigms and their interrelations is compulsory. In this paper we posit a codomain scale space regularization paradigm that has hitherto not been applied in the context of HARDI. Unlike previous (first and second order) schemes it is based on infinite order regularization, yet can be fully operationalized. We furthermore establish a closed-form relation with first order Tikhonov regularization via the Laplace transform

Coarse-to-fine partitioning of signals

An empirically acquired signal can be analyzed in a multi-scale framework. Its multi-scale structure induces a hierarchical partitioning of the signal domain into topologically meaningful segments. A method is proposed to operationalize this using elementary results from singularity theory for certain generic solutions of the one-dimensional heat equatio

Two canonical representations for regularized high angular resolution diffusion imaging

Two canonical representations for regularization of unit spherefunctions encountered in the context of high angular resolution diffusionimaging (HARDI) are discussed. One of these is based on spherical harmonicdecomposition, and its one-parameter extension via Tikhonov regularization.This case is well-established, and is mainly reviewed for thesake of completeness. The second one is new, and is based on a higherorder diffusion tensor decomposition. A homogeneous representation ofthis type has been proposed in the literature, but we show that thisis inconvenient for the purpose of regularization. We instead construct aheterogeneous representation that can be regarded as "canonical", to theextent that its behaviour under regularization mimics that of sphericalharmonics

Decomposition of higher-order homogeneous tensors and applications to HARDI

High Angular Resolution Diffusion Imaging (HARDI) holds the promise to provide insight in connectivity of the human brain in vivo. Based on this technique a number of different approaches has been proposed to estimate the ¿ber orientation distribution, which is crucial for ¿ber tracking. A spherical harmonic representation is convenient for regularization and the construction of orientation distribution functions (ODFs), whereas maxima detection and ¿ber tracking techniques are most naturally formulated using a tensor representation. We give an analytical formulation to bridge the gap between the two representations, which admits regularization and ODF construction directly in the tensor basis

Riemann-Finsler geometry and its applications to diffusion magnetic resonance imaging

Riemannian geometry has become a popular mathematical framework for the analysis of diffusion tensor images (DTI) in diffusion weighted magnetic resonance imaging (DWMRI). If one declines from the a priori constraint to model local anisotropic diffusion in terms of a 6-degrees-of-freedom rank-2 DTI tensor, then Riemann-Finsler geometry appears to be the natural extension. As such it provides an interesting alternative to the Riemannian rationale in the context of the various high angular resolution diffusion imaging (HARDI) schemes proposed in the literature. The main advantages of the proposed Riemann-Finsler paradigm are its manifest incorporation of the DTI model as a limiting case via a "correspondence principle" (operationalized in terms of a vanishing Cartan tensor), and its direct connection to the physics of DWMRI expressed by the (appropriately generalized) Stejskal-Tanner equation and Bloch-Torrey equations furnished with a diffusion term

Using Top-Points as Interest Points for Image Matching

We consider the use of so-called top-points for object retrieval. These points are based on scale-space and catastrophe theory, and are invariant under gray value scaling and offset as well as scale-Euclidean transformations. The differential properties and noise characteristics of these points are mathematically well understood. It is possible to retrieve the exact location of a top-point from any coarse estimation through a closed-form vector equation which only depends on local derivatives in the estimated point. All these properties make top-points highly suitable as anchor points for invariant matching schemes. In a set of examples we show the excellent performance of top-points in an object retrievaltask

Cardiac motion estimation using multi-scale feature points

Heart illnesses influence the functioning of the cardiac muscle and are the major causes of death inthe world. Optic flow methods are essential tools to assess and quantify the contraction of the cardiacwalls, but are hampered by the aperture problem. Harmonic phase (HARP) techniques measure thephase in magnetic resonance (MR) tagged images. Due to the regular geometry, patterns generated bya combination of HARPs and sine HARPs represent a suitable framework to extract landmark features.In this paper we introduce a new aperture-problem free method to study the cardiac motion by trackingmulti-scale features such as maxima, minima, saddles and corners, on HARP and sine HARP taggedimages

Cardiac motion estimation using covariant derivatives and Helmholtz decomposition

The investigation and quantification of cardiac movement is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider new aperture problem-free methods to track cardiac motion from 2-dimensional MR tagged images and corresponding sine-phase images. Tracking is achieved by following the movement of scale-space maxima, yielding a sparse set of linear features of the unknown optic flow vector field. Interpolation/reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev-norm expressed in covariant derivatives (rather than standard derivatives). These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. They are defined on a fiber bundle where sections coincide with vector fields. Furthermore, the optic flow vector field is decomposed in a divergence free and a rotation free part, using our multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in a single non-singular analytic kernel operator. Finally, we combine this multi-scale Helmholtz decomposition with vector field reconstruction (based on covariant derivatives) in a single algorithm and present some experiments of cardiac motion estimation. Further experiments on phantom data with ground truth show that both the inclusion of covariant derivatives and the inclusion of the multi-scale Helmholtz decomposition improves the optic flow reconstruction