Tracking Lines in Higher Order Tensor Fields

While tensors occur in many areas of science and engineering, little has been done to visualize tensors with order higher than two. Tensors of higher orders can be used for example to describe complex diffusion patterns in magnetic resonance imaging (MRI). Recently, we presented a method for tracking lines in higher order tensor fields that is a generalization of methods known from first order tensor fields (vector fields) and symmetric second order tensor fields. Here, this method is applied to magnetic resonance imaging where tensor fields are used to describe diffusion patterns for example of hydrogen in the human brain. These patterns align to the internal structure and can be used to analyze interconnections between different areas of the brain, the so called tractography problem. The advantage of using higher order tensor lines is the ability to detect crossings locally, which is not possible in second order tensor fields. In this paper, the theoretical details will be extended and tangible results will be given on MRI data sets

HOT–Lines: Tracking Lines in Higher Order Tensor Fields

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application

Convolution and Fourier Transform of Second Order Tensor Fields

The goal of this paper is to transfer convolution, correlation and Fourier transform to second order tensor fields. Convolution of two tensor fields is defined using matrix multiplication. Convolution of a tensor field with a scalar mask can thus be described by multiplying the scalars with the real unit matrix. The Fourier transform of tensor fields defined in this paper corresponds to Fourier transform of each of the tensor components in the field. It is shown that for this convolution and Fourier transform, the well known convolution theorem holds and optimization in speed can be achieved by using Fast Fourier transform algorithms. Furthermore, pattern matching on tensor fields based on this convolution is described

Tracking Lines in Higher Order Tensor Fields: Tracking Lines in Higher Order Tensor Fields

While tensors occur in many areas of science and engineering, little has been done to visualize tensors with order higher than two. Tensors of higher orders can be used for example to describe complex diffusion patterns in magnetic resonance imaging (MRI). Recently, we presented a method for tracking lines in higher order tensor fields that is a generalization of methods known from first order tensor fields (vector fields) and symmetric second order tensor fields. Here, this method is applied to magnetic resonance imaging where tensor fields are used to describe diffusion patterns for example of hydrogen in the human brain. These patterns align to the internal structure and can be used to analyze interconnections between different areas of the brain, the so called tractography problem. The advantage of using higher order tensor lines is the ability to detect crossings locally, which is not possible in second order tensor fields. In this paper, the theoretical details will be extended and tangible results will be given on MRI data sets

Interactive Glyph Placement for Tensor Fields: Tracking Lines in Higher Order Tensor Fields

Visualization of glyphs has a long history in medical imaging but gains much more power when the glyphs are properly placed to fill the screen. Glyph packing is often performed via an iterative approach to improve the location of glyphs. We present an alternative implementation of glyph packing based on a Delaunay triangulation to speed up the clustering process and reduce costs for neighborhood searches. Our approach does not require a re–computation of acceleration structures when a plane is moved through a volume, which can be done interactively. We provide two methods for initial placement of glyphs to improve the convergence of our algorithm for glyphs larger and glyphs smaller than the data set’s voxel size. The main contribution of this paper is a novel approach to glyph packing that supports simpler parameterization and can be used easily for highly efficient interactive data exploration, in contrast to previous methods

An open environment CT-US fusion for tissue segmentation during interventional guidance.

Therapeutic ultrasound (US) can be noninvasively focused to activate drugs, ablate tumors and deliver drugs beyond the blood brain barrier. However, well-controlled guidance of US therapy requires fusion with a navigational modality, such as magnetic resonance imaging (MRI) or X-ray computed tomography (CT). Here, we developed and validated tissue characterization using a fusion between US and CT. The performance of the CT/US fusion was quantified by the calibration error, target registration error and fiducial registration error. Met-1 tumors in the fat pads of 12 female FVB mice provided a model of developing breast cancer with which to evaluate CT-based tissue segmentation. Hounsfield units (HU) within the tumor and surrounding fat pad were quantified, validated with histology and segmented for parametric analysis (fat: -300 to 0 HU, protein-rich: 1 to 300 HU, and bone: HU>300). Our open source CT/US fusion system differentiated soft tissue, bone and fat with a spatial accuracy of ∼1 mm. Region of interest (ROI) analysis of the tumor and surrounding fat pad using a 1 mm(2) ROI resulted in mean HU of 68±44 within the tumor and -97±52 within the fat pad adjacent to the tumor (p<0.005). The tumor area measured by CT and histology was correlated (r(2) = 0.92), while the area designated as fat decreased with increasing tumor size (r(2) = 0.51). Analysis of CT and histology images of the tumor and surrounding fat pad revealed an average percentage of fat of 65.3% vs. 75.2%, 36.5% vs. 48.4%, and 31.6% vs. 38.5% for tumors <75 mm(3), 75-150 mm(3) and >150 mm(3), respectively. Further, CT mapped bone-soft tissue interfaces near the acoustic beam during real-time imaging. Combined CT/US is a feasible method for guiding interventions by tracking the acoustic focus within a pre-acquired CT image volume and characterizing tissues proximal to and surrounding the acoustic focus

Tensor Lines in Tensor Fields of Arbitrary Order: Tracking Lines in Higher Order Tensor Fields

This paper presents a method to reduce time complexity of the computation of higher–order tensor lines. The method can be applied to higher–order tensors and the spherical harmonics representation, both widely used in medical imaging. It is based on a gradient descend technique and integrates well into fiber tracking algorithms. Furthermore, the method improves the angular resolution in contrast to discrete sampling methods which is especially important to tractography, since there, small errors accumulate fast and make the result unusable. Our implementation does not interpolate derived directions but works directly on the interpolated tensor information. The specific contribution of this paper is a fast algorithm for tracking lines tensor fields of arbitrary order that increases angular resolution compared to previous approaches

Fabric-like Visualization of Tensor Field Data on Arbitrary Surfaces in Image Space

Tensors are of great interest to many applications in engineering and in medical imaging, but a proper analysis and visualization remains challenging. It already has been shown that, by employing the metaphor of a fabric structure, tensor data can be visualized precisely on surfaces where the two eigendirections in the plane are illustrated as thread-like structures. This leads to a continuous visualization of most salient features of the tensor data set. We introduce a novel approach to compute such a visualization from tensor field data that is motivated by image-space line integral convolution (LIC). Although our approach can be applied to arbitrary, non-selfintersecting surfaces, the main focus lies on special surfaces following important features, such as surfaces aligned to the neural pathways in the human brain. By adding a postprocessing step, we are able to enhance the visual quality of the of the results, which improves perception of the major patterns