Zhizhou Wang

A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L/sup p/ norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm.

Fiber tract mapping from diffusion tensor MRI

To understand evolving pathology in the central nervous system (CNS) and develop effective treatments, it is essential to correlate the nerve fiber connectivity with the visualization of function. Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. We present a novel algorithm for automatic fiber tract mapping in the CNS specifically, the spinal cord. The automatic fiber tract mapping problem is solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing is achieved via a new weighted total variation (TV)-norm minimization (for vector-valued data) which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Fiber tracts are then determined as the smooth integral curves of this vector field in a variational framework.

Tensor Field Segmentation Using Region Based Active Contour Model

Tensor fields (matrix valued data sets) have recently attracted increased attention in the fields of image processing, computer vision, visualization and medical imaging. Tensor field segmentation is an important problem in tensor field analysis and has not been addressed adequately in the past. In this paper, we present an effective region-based active contour model for tensor field segmentation and show its application to diffusion tensor magnetic resonance images (MRI) as well as for the texture segmentation problem in computer vision. Specifically, we present a variational principle for an active contour using the Euclidean difference of tensors as a discriminant. The variational formulation is valid for piecewise smooth regions, however, for the sake of simplicity of exposition, we present the piecewise constant region model in detail. This variational principle is a generalization of the region-based active contour to matrix valued functions. It naturally leads to a curve evolution equation for tensor field segmentation, which is subsequently expressed in a level set framework and solved numerically. Synthetic and real data experiments involving the segmentation of diffusion tensor MRI as well as structure tensors obtained from real texture data are shown to depict the performance of the proposed model.

A Constrained Variational Principle for Direct Estimation and Smoothing of the Diffusion Tensor Field from DWI

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from diffusion weighted imaging (DWI). The constrained variational principle involves the minimization of a regularization term in an LP norm, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The original nonlinear form leads to a more accurate (when compared to the linearized form) estimated tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a possibly known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of cholesky factors and estimated. variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Both synthetic and real data experiments are shown to depict the performance of the tensor field estimation algorithm. Fiber tracts in a rat brain are then mapped using a particle system based visualization technique.

Registration Assisted Image Smoothing and Segmentation

Image segmentation is a fundamental problem in Image Processing, Computer Vision and Medical Imaging with numerous applications. In this paper, we address the atlas-based image segmentation problem which involves registration of the atlas to the subject or target image in order to achieve the segmentation of the target image. Thus, the target image is segmented with the assistance of a registration process. We present a novel variational formulation of this registration assisted image segmentation problem which leads to solving a coupled set of nonlinear PDEs that are solved using efficient numerical schemes. Our work is a departure from earlier methods in that we have a unified variational principle wherein registration and segmentation are simultaneously achieved. We present several 2D examples on syntheticand real data sets along with quantitative accuracy estimates of the registration.

Simultaneous smoothing and estimation of the tensor field from diffusion tensor MRI

Diffusion tensor magnetic resonance imaging (DT-MRI) is a relatively new imaging modality in the field of medical imaging. This modality of imaging allows one to capture the structural connectivity if any between functionally meaningful regions for example, in the brain. The data however can be noisy and requires restoration. In this paper, we present a unified model for simultaneous smoothing and estimation of diffusion tensor field from DT-MRI. The diffusion tensor field is estimated directly from the raw data with L/sup P/ smoothness and positive definiteness constraints. The data term we employ is from the original Stejskal-Tanner equation instead of the linearized version as usually done in literature. In addition, we use Cholesky decomposition to ensure positive definiteness of the diffusion tensor. The unified model is discretized and solved numerically using limited memory quasi-Newton method. Both synthetic and real data experiments are shown to depict the algorithm performance.

Line Integral Convolution for Visualization of Fiber Tract Maps from DTI

Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. However, robust and accurate acquisition and processing algorithms are needed to accurately map the nerve connectivity. In this paper, we present a novel algorithm for extracting and visualizing the fiber tracts in the CNS specifically, the spinal cord. The automatic fiber tract mapping problem will be solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former, smoothing is achieved via a weighted TV-norm minimization which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Visualization of the fiber tracts is achieved by adapting a known Computer Graphics technique called the line integral convolution, which has the advantage of being able to cope with singularities in the vector field and is a resolution independent way of visualizing the 3D vector field corresponding to the dominant eigen vectors of the diffusion tensor field. Examples are presented to depict the performance of the visualization scheme on three DT-MR data sets, one from a normal and another from an injured rat spinal cord and a third from a rat brain.

Deformable Pedal Curves and Surfaces: Hybrid Geometric Active Models for Shape Recovery

In this paper, we propose significant extensions to the “snake pedal” model, a powerful geometric shape modeling scheme introduced in (Vemuri and Guo, 1998). The extension allows the model to automatically cope with topological changes and for the first time, introduces the concept of a compact global shape into geometric active models. The ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition. In this new modeling scheme, object shapes are represented using a parameterized function—called the generator—which accounts for the global shape of an object and the pedal curve (surface) of this global shape with respect to a geometric snake to represent any local detail. Traditionally, pedal curves (surfaces) are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point. Local shape control is achieved by introducing a set of pedal points—lying on a snake—for each point on the generator. The model dubbed as a “snake pedal” allows for interactive manipulation via forces applied to the snake. In this work, we replace the snake by a geometric snake and derive all the necessary mathematics for evolving the geometric snake when the snake pedal is assumed to evolve as a function of its curvature. Automatic topological changes of the model may be achieved by implementing the geometric snake in a level-set framework. We demonstrate the applicability of this modeling scheme via examples of shape recovery from a variety of 2D and 3D image data.

Automatic fiber tractography from DTI and its validation

To understand evolving pathology in the central nervous system (CNS) and develop effective treatments, it is essential to correlate the nerve fiber connectivity with the visualization of function. Such information is fundamental in CNS processes since anatomical connections determine where information is passed and processed Diffusion tensor imaging (DTI) can provide the fundamental information required for viewing structural connectivity. However robust and accurate acquisition and processing algorithms are needed to accurately map the nerve connectivity. In this paper we present a novel, algorithm for automatic fiber tract mapping in the CNS specifically, a rat spinal cord as well as validate the mapped fibers using ex-vivo fluoro images of the excised rat. The novelty of our work lies in the fiber tract mapping as well as the validation experiment. The automatic fiber tract mapping problem will be solved in two phases, namely a data smoothing phase and a fiber tract mapping phase. In the former smoothing is achieved via a new weighted TV-norm minimization which strives to smooth while retaining all relevant detail. For the fiber tract mapping, a smooth 3D vector field indicating the dominant anisotropic direction at each spatial location is computed from the smoothed data. Fiber tracts are then determined as the smooth integral curves of this vector field in a variational framework Examples are presented for DTI data sets from a normal and injured rat spinal cords respectively.

Diffusion Tensor MR Image Restoration

Diffusion tensor magnetic resonance imaging (DT-MRI) can provide the fundamental information required to visualize structural connectivity. However, this high-dimensional data can be rather noisy and requires restoration. In this paper, we present a novel unified formulation involving a variational principle for simultaneous smoothing and estimation of the diffusion tensor field from DT-MRI. This tensor field is estimated directly from the measurements using a combination of L p smoothness and positive definiteness constraints respectively. The data term we employ is the Stejskal-Tanner equation instead of its linearized version as usually employed in the published literature. In addition, we impose the positive definite constraint via the Cholesky decomposition of the tensors in the field. Our unified variational principle is discretized and solved numerically using the limited memory quasi-Newton method. Algorithm performance is depicted via both synthetic and real data experiments.