A simplified algorithm for inverting higher order diffusion tensors

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann-Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann-Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices.</p

A Simplified Algorithm for Inverting Higher Order Diffusion Tensors

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices

Stain separation in digital bright field histopathology

Digital pathology employs images that were acquired by imaging thin tissue samples through a microscope. The preparation of a sample from a biopt to the glass slide entering the imaging device is done manually introducing large variability in the samples to be imaged. For visible contrast it is necessary to stain the samples prior to imaging. Different stains attach to different compounds elucidating the different cellular structures. Towards automatic analysis and for visual comparability there is a need to standardize the images to obtain consistent appearances regardless of the potential differences in sample preparation. A standard approach is to unmix the the various stains computationally, normalize each separate stain image and to recombine these. This paper describes a modification to a standard blind method for stain normalization. The performance is quantified in terms of annotated expert data. Theoretical analysis is presented to rationalize the new approach

Multi-scale Riemann-Finsler geometry : applications to diffusion tensor imaging and high angular resolution diffusion imaging

In this brief summary, we reflect the goals and the achievements of this PhD-project by citing three sentences in the original project proposal. "This project investigates the exploitation of the scale degree of freedom in images from the vantage point of differential geometry and tensor calculus in scale space." Indeed, this thesis introduces the necessary tools for differential geometric tensor calculus and derives some theoretical as well as practical results. The scale is considered separately in two different settings. In this thesis, the weight is somewhat shifted from the problem of scale to problems in differential geometry and applications. The applications considered here are multi-directional medical images, namely diffusion weighted magnetic resonance images. The measurements are modeled using symmetric tensors, which is equivalent to a polynomial approximation. This thesis can be roughly separated in to two parts: one that studies second order tensor fields and one that uses higher order tensor fields. The second order tensor fields are studied with tools from Riemann geometry, and the higher order ones respectively with those from Finsler geometry. Each of these can be separated to a theoretical part, that contain the mathematical definitions and derivations and an applied part, where some geometric properties of synthetic, simulated or real data are computed and analyzed. "The goal is to couple geometry to image content based on a specific task." By attaching geometric meaning to the physical properties (of the imaged object) represented by data, we have derived some measures and algorithms to extract information from images. For example a novel method to do fiber tractography, i.e. extract neural connections from diffusion weighted images of brain, is introduced. This method has the special property, that it can propagate through voxels with complex fiber orientations. Techniques to measure relative diffusivity along a curve and to detect inhomogeneities in tensor field are some of the other examples. Since the real data is discrete, the interpolation of tensor fields is also considered."The objective is to foster specific applications in biomedical image analysis, and to extend these to multiple scales." In the Riemannian framework, the concept of scale is introduced in a Gaussian derivative scheme to second order tensor fields. In Finslerian context, a scale parameter was attached to higher order tensors by applying Laplace-Beltrami smoothing, solving the heat equation on the sphere

Sticky vector fields, and other geometric measures on diffusion tensor images

This paper is about geometric measures in diffusion tensor imaging (DTI) analysis, and it is a continuation of our previous work (L. Astola et al., 2007), where we discussed two measures for diffusion tensor (DT) image (fiber tractography) analysis. Its contribution is threefold. First, we show how the so called connectivity measure performs on a real DTI image with three different interpolation methods. Secondly, we introduce a new vector field on DTI images, that points out the locally most coherent direction for fiber tracking, and we illustrate it on bundles of tracked fibers. Thirdly, we introduce an inhomogeneity- (edge-, crossing-) detector for symmetric positive matrix valued images, including DTI images. One possible application is segmentation of diffusion tensor fields

Sticky vector fields, and other geometric measures on diffusion tensor images

This paper is about geometric measures in diffusion tensor imaging (DTI) analysis, and it is a continuation of our previous work (L. Astola et al., 2007), where we discussed two measures for diffusion tensor (DT) image (fiber tractography) analysis. Its contribution is threefold. First, we show how the so called connectivity measure performs on a real DTI image with three different interpolation methods. Secondly, we introduce a new vector field on DTI images, that points out the locally most coherent direction for fiber tracking, and we illustrate it on bundles of tracked fibers. Thirdly, we introduce an inhomogeneity- (edge-, crossing-) detector for symmetric positive matrix valued images, including DTI images. One possible application is segmentation of diffusion tensor fields