IEEE Transactions on Visualization and Computer Graphics (Proceedings of VisWeek 2010),, Nov 2010
Symmetric second-order tensor fields play a central role in scientific and biomedical studies as ... more Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.
Research scientists and medical professionals use imaging technology, such as computed tomography... more Research scientists and medical professionals use imaging technology, such as computed tomography (CT) and magnetic resonance imaging (MRI) to measure a wide variety of biological and physical objects. The increasing sophistication of imaging technology creates demand for equally sophisticated computational techniques to analyze and visualize the image data. Analysis and visualization codes are often crafted for a specific experiment or set of images, thus imaging scientists need support for quickly developing codes that are reliable, robust, and efficient.
In this paper, we present the design and implementation of Diderot, which is a parallel domain-specific language for biomedical image analysis and visualization. Diderot supports a high-level model of computation that is based on continuous tensor fields. These tensor fields are reconstructed from discrete image data using separable convolution kernels, but may also be defined by applying higher-order operations, such as differentiation (∇). Early experiments demonstrate that Diderot provides both a high-level concise notation for image analysis and visualization algorithms, as well as high sequential and parallel performance.
PURPOSE: Various methods exist for interpolating diffusion tensor fields, but none of them linear... more PURPOSE: Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape.
METHODS: Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets.
RESULTS: EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI.
CONCLUSIONS: GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.
Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstr... more Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstructural organization of biological tissues. Multiple images are necessary to reconstruct the tensor data and each acquisition is subject to complex thermal noise. As such, measures of tensor invariants, which characterize components of tensor shape, derived from the tensor data will be biased from their true values. Previous work has examined this bias, but over a narrow range of tensor shape. Herein, we define the mathematics for constructing a tensor from tensor invariants, which permits an intuitive and principled means for building tensors with a complete range of tensor shape and salient microstructural properties. Thereafter, we use this development to evaluate by simulation the effects of noise on characterizing tensor shape over the complete space of tensor shape for three encoding schemes with different SNR and gradient directions. We also define a new framework for determining the distribution of the true values of tensor invariants given their measures, which provides guidance about the confidence the observer should have in the measures. Finally, we present the statistics of tensor invariant estimates over the complete space of tensor shape to demonstrate how the noise sensitivity of tensor invariants varies across the space of tensor shape as well as how the imaging protocol impacts measures of tensor invariants.
A common goal of multivariate visualization is to enable data inspection at discrete points, whil... more A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.
In algorithms for processing diffusion tensor images, two common ingredients are interpolating te... more In algorithms for processing diffusion tensor images, two common ingredients are interpolating tensors, and measuring the distance between them. We propose a new class of interpolation paths for tensors, termed geodesic-loxodromes, which explicitly preserve clinically important tensor attributes, such as mean diffusivity or fractional anisotropy, while using basic differential geometry to interpolate tensor orientation. This contrasts with previous Riemannian and Log-Euclidean methods that preserve the determinant. Path integrals of tangents of geodesic-loxodromes generate novel measures of over-all difference between two tensors, and of difference in shape and in orientation.
International Journal of High Performance Computing Applications, Jun 1996
The Collaboratory for Microscopic Digital Anatomy (CMDA) will allow biologists at remote location... more The Collaboratory for Microscopic Digital Anatomy (CMDA) will allow biologists at remote locations to acquire over the network three-dimensional data on biological structure using a sophisticated high-voltage electron microscope (HVEM) located in San Diego. The CMDA will transparently distribute to high-performance machines on the network computationally intensive tasks for the derivation of three-dimensional datasets from HVEM images using tomography and for volume visualization. Implementation of the CMDA includes development of automated microscope functions for specimen positioning, image fo- cus, registration, and acquisition; an asynchronous communications system for transporting messages and large data sets between processes; a resource manager for access control and routing of tasks to computing or acquisition resources; and systems for tomographic reconstruction and three-dimensional volume visualization. The CMDA is designed to be extensible to other unique centralized instruments. It will increase access to and use of these instruments and high-performance computers by biologists and facilitate remote collaboration.
A reoccurring theme in the diffusion tensor imaging literature is the per-voxel estimation of a s... more A reoccurring theme in the diffusion tensor imaging literature is the per-voxel estimation of a symmetric 3 ×3 tensor describing the measured diffusion. In this work we attempt to generalize this approach by calculating 2 or 3 or up to k diffusion tensors for each voxel. We show that our procedure can more accurately describe the diffusion particularly when crossing fibers or fiber-bundles are present in the datasets.
Current methods for extracting models of white matter architecture from diffusion tensor MRI are ... more Current methods for extracting models of white matter architecture from diffusion tensor MRI are generally based on fiber tractography. For some purposes a compelling alternative may be found in analyzing the first and second derivatives of diffusion anisotropy. Anisotropy creases are ridges and valleys of locally extremal anisotropy, where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. We describe a crease extraction algorithm that generates high-quality polygonal models of crease surfaces, then demonstrate the method on a measured diffusion tensor dataset, and visualize the result in combination with tractography to confirm its anatomic relevance.
This paper outlines the mathematical development and application of two analytically orthogonal t... more 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.
Geometric models of white matter architecture play an increasing role in neuroscientific applicat... more Geometric models of white matter architecture play an increasing role in neuroscientific applications of diffusion tensor imaging, and the most popular method for building them is fiber tractography. For some analysis tasks, however, a compelling alternative may be found in the first and second derivatives of diffusion anisotropy. We extend to tensor fields the notion from classical computer vision of ridges and valleys, and define anisotropy creases as features of locally extremal tensor anisotropy. Mathematically, these are the loci where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of the major white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. The crease extraction algorithm we present generates high-quality polygonal models of crease surfaces, which are further simplified by connected-component analysis. We demonstrate anisotropy creases on measured diffusion MRI data, and visualize them in combination with tractography to confirm their anatomic relevance.
Guided by empirically established connections between clinically important tissue properties and ... more Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
IEEE Transactions on Visualization and Computer Graphics (Proceedings of VisWeek 2010),, Nov 2010
Symmetric second-order tensor fields play a central role in scientific and biomedical studies as ... more Symmetric second-order tensor fields play a central role in scientific and biomedical studies as well as in image analysis and feature-extraction methods. The utility of displaying tensor field samples has driven the development of visualization techniques that encode the tensor shape and orientation into the geometry of a tensor glyph. With some exceptions, these methods work only for positive-definite tensors (i.e. having positive eigenvalues, such as diffusion tensors). We expand the scope of tensor glyphs to all symmetric second-order tensors in two and three dimensions, gracefully and unambiguously depicting any combination of positive and negative eigenvalues. We generalize a previous method of superquadric glyphs for positive-definite tensors by drawing upon a larger portion of the superquadric shape space, supplemented with a coloring that indicates the quadratic form (including eigenvalue sign). We show that encoding arbitrary eigenvalue magnitudes requires design choices that differ fundamentally from those in previous work on traceless tensors that arise in the study of liquid crystals. Our method starts with a design of 2-D tensor glyphs guided by principles of scale-preservation and symmetry, and creates 3-D glyphs that include the 2-D glyphs in their axis-aligned cross-sections. A key ingredient of our method is a novel way of mapping from the shape space of three-dimensional symmetric second-order tensors to the unit square. We apply our new glyphs to stress tensors from mechanics, geometry tensors and Hessians from image analysis, and rate-of-deformation tensors in computational fluid dynamics.
Research scientists and medical professionals use imaging technology, such as computed tomography... more Research scientists and medical professionals use imaging technology, such as computed tomography (CT) and magnetic resonance imaging (MRI) to measure a wide variety of biological and physical objects. The increasing sophistication of imaging technology creates demand for equally sophisticated computational techniques to analyze and visualize the image data. Analysis and visualization codes are often crafted for a specific experiment or set of images, thus imaging scientists need support for quickly developing codes that are reliable, robust, and efficient.
In this paper, we present the design and implementation of Diderot, which is a parallel domain-specific language for biomedical image analysis and visualization. Diderot supports a high-level model of computation that is based on continuous tensor fields. These tensor fields are reconstructed from discrete image data using separable convolution kernels, but may also be defined by applying higher-order operations, such as differentiation (∇). Early experiments demonstrate that Diderot provides both a high-level concise notation for image analysis and visualization algorithms, as well as high sequential and parallel performance.
PURPOSE: Various methods exist for interpolating diffusion tensor fields, but none of them linear... more PURPOSE: Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape.
METHODS: Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets.
RESULTS: EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI.
CONCLUSIONS: GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.
Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstr... more Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstructural organization of biological tissues. Multiple images are necessary to reconstruct the tensor data and each acquisition is subject to complex thermal noise. As such, measures of tensor invariants, which characterize components of tensor shape, derived from the tensor data will be biased from their true values. Previous work has examined this bias, but over a narrow range of tensor shape. Herein, we define the mathematics for constructing a tensor from tensor invariants, which permits an intuitive and principled means for building tensors with a complete range of tensor shape and salient microstructural properties. Thereafter, we use this development to evaluate by simulation the effects of noise on characterizing tensor shape over the complete space of tensor shape for three encoding schemes with different SNR and gradient directions. We also define a new framework for determining the distribution of the true values of tensor invariants given their measures, which provides guidance about the confidence the observer should have in the measures. Finally, we present the statistics of tensor invariant estimates over the complete space of tensor shape to demonstrate how the noise sensitivity of tensor invariants varies across the space of tensor shape as well as how the imaging protocol impacts measures of tensor invariants.
A common goal of multivariate visualization is to enable data inspection at discrete points, whil... more A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.
In algorithms for processing diffusion tensor images, two common ingredients are interpolating te... more In algorithms for processing diffusion tensor images, two common ingredients are interpolating tensors, and measuring the distance between them. We propose a new class of interpolation paths for tensors, termed geodesic-loxodromes, which explicitly preserve clinically important tensor attributes, such as mean diffusivity or fractional anisotropy, while using basic differential geometry to interpolate tensor orientation. This contrasts with previous Riemannian and Log-Euclidean methods that preserve the determinant. Path integrals of tangents of geodesic-loxodromes generate novel measures of over-all difference between two tensors, and of difference in shape and in orientation.
International Journal of High Performance Computing Applications, Jun 1996
The Collaboratory for Microscopic Digital Anatomy (CMDA) will allow biologists at remote location... more The Collaboratory for Microscopic Digital Anatomy (CMDA) will allow biologists at remote locations to acquire over the network three-dimensional data on biological structure using a sophisticated high-voltage electron microscope (HVEM) located in San Diego. The CMDA will transparently distribute to high-performance machines on the network computationally intensive tasks for the derivation of three-dimensional datasets from HVEM images using tomography and for volume visualization. Implementation of the CMDA includes development of automated microscope functions for specimen positioning, image fo- cus, registration, and acquisition; an asynchronous communications system for transporting messages and large data sets between processes; a resource manager for access control and routing of tasks to computing or acquisition resources; and systems for tomographic reconstruction and three-dimensional volume visualization. The CMDA is designed to be extensible to other unique centralized instruments. It will increase access to and use of these instruments and high-performance computers by biologists and facilitate remote collaboration.
A reoccurring theme in the diffusion tensor imaging literature is the per-voxel estimation of a s... more A reoccurring theme in the diffusion tensor imaging literature is the per-voxel estimation of a symmetric 3 ×3 tensor describing the measured diffusion. In this work we attempt to generalize this approach by calculating 2 or 3 or up to k diffusion tensors for each voxel. We show that our procedure can more accurately describe the diffusion particularly when crossing fibers or fiber-bundles are present in the datasets.
Current methods for extracting models of white matter architecture from diffusion tensor MRI are ... more Current methods for extracting models of white matter architecture from diffusion tensor MRI are generally based on fiber tractography. For some purposes a compelling alternative may be found in analyzing the first and second derivatives of diffusion anisotropy. Anisotropy creases are ridges and valleys of locally extremal anisotropy, where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. We describe a crease extraction algorithm that generates high-quality polygonal models of crease surfaces, then demonstrate the method on a measured diffusion tensor dataset, and visualize the result in combination with tractography to confirm its anatomic relevance.
This paper outlines the mathematical development and application of two analytically orthogonal t... more 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.
Geometric models of white matter architecture play an increasing role in neuroscientific applicat... more Geometric models of white matter architecture play an increasing role in neuroscientific applications of diffusion tensor imaging, and the most popular method for building them is fiber tractography. For some analysis tasks, however, a compelling alternative may be found in the first and second derivatives of diffusion anisotropy. We extend to tensor fields the notion from classical computer vision of ridges and valleys, and define anisotropy creases as features of locally extremal tensor anisotropy. Mathematically, these are the loci where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of the major white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. The crease extraction algorithm we present generates high-quality polygonal models of crease surfaces, which are further simplified by connected-component analysis. We demonstrate anisotropy creases on measured diffusion MRI data, and visualize them in combination with tractography to confirm their anatomic relevance.
Guided by empirically established connections between clinically important tissue properties and ... more Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
Uploads
In this paper, we present the design and implementation of Diderot, which is a parallel domain-specific language for biomedical image analysis and visualization. Diderot supports a high-level model of computation that is based on continuous tensor fields. These tensor fields are reconstructed from discrete image data using separable convolution kernels, but may also be defined by applying higher-order operations, such as differentiation (∇). Early experiments demonstrate that Diderot provides both a high-level concise notation for image analysis and visualization algorithms, as well as high sequential and parallel performance.
METHODS: Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets.
RESULTS: EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI.
CONCLUSIONS: GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.
In this paper, we present the design and implementation of Diderot, which is a parallel domain-specific language for biomedical image analysis and visualization. Diderot supports a high-level model of computation that is based on continuous tensor fields. These tensor fields are reconstructed from discrete image data using separable convolution kernels, but may also be defined by applying higher-order operations, such as differentiation (∇). Early experiments demonstrate that Diderot provides both a high-level concise notation for image analysis and visualization algorithms, as well as high sequential and parallel performance.
METHODS: Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets.
RESULTS: EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI.
CONCLUSIONS: GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost.