A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance

For strongly directed anisotropic processes such as coherence-enhancing diffusion filtering it is crucial to use numerical schemes with highly accurate directional behavior. We show that this is not possible in a satisfactory way when discretizations are limited to 3 x 3 stencils. As a consequence, we investigate a novel algorithm based on 5 x 5 stencils. It utilizes recently discovered differentiation filters with optimized rotation invariance. By juxtaposing it with several common algorithms we demonstrate its superior behavior with respect to the following properties: rotation invariance, avoidance of blurring artifacts (dissipativity), and accuracy. The latter one is evaluated by deriving an analytical solution for coherence-enhancing diffusion filtering of images with circular symmetry. Furthermore, we show that the new scheme is 3 to 4 times more efficient than explicit schemes on 3 x 3 stencils. It does not require to solve linear systems of equations, and it can be easily implemented in any dimension

Optimized Anisotropic Rotational Invariant Diffusion Scheme on Cone-Beam CT

Cone-beam computed tomography (CBCT) is an important image modality for dental surgery planning, with high resolution images at a relative low radiation dose. In these scans the mandibular canal is hardly visible, this is a problem for implant surgery planning. We use anisotropic diffusion filtering to remove noise and enhance the mandibular canal in CBCT scans. For the diffusion tensor we use hybrid diffusion with a continuous switch (HDCS), suitable for filtering both tubular as planar image structures. We focus in this paper on the diffusion discretization schemes. The standard scheme shows good isotropic filtering behavior but is not rotational invariant, the diffusion scheme of Weickert is rotational invariant but suffers from checkerboard artifacts. We introduce a new scheme, in which we numerically optimize the image derivatives. This scheme is rotational invariant and shows good isotropic filtering properties on both synthetic as real CBCT data

Coherence Filtering to Enhance the Mandibular Canal in Cone-Beam CT data

Segmenting the mandibular canal from cone beam CT data, is difficult due to low edge contrast and high image noise. We introduce 3D coherence filtering as a method to close the interrupted edges and denoise the structure of the mandibular canal. Coherence Filtering is an anisotropic non-linear tensor based diffusion algorithm for edge enhancing image filtering. We test different numerical schemes of the tensor diffusion equation, non-negative, standard discretization and also a rotation invariant scheme of Weickert [1]. Only the\ud scheme of Weickert did not blur the high spherical images frequencies on the image diagonals of our test volume. Thus this scheme is chosen to enhance the small curved mandibular canal structure. The best choice of the diffusion equation parameters c1 and c2, depends on the image noise. Coherence filtering on the CBCT-scan works well, the noise in the mandibular canal is gone and the edges are connected. Because the algorithm is tensor based it cannot deal with edge joints or splits, thus is less fit for more complex image structures

Fast space-variant elliptical filtering using box splines

The efficient realization of linear space-variant (non-convolution) filters is a challenging computational problem in image processing. In this paper, we demonstrate that it is possible to filter an image with a Gaussian-like elliptic window of varying size, elongation and orientation using a fixed number of computations per pixel. The associated algorithm, which is based on a family of smooth compactly supported piecewise polynomials, the radially-uniform box splines, is realized using pre-integration and local finite-differences. The radially-uniform box splines are constructed through the repeated convolution of a fixed number of box distributions, which have been suitably scaled and distributed radially in an uniform fashion. The attractive features of these box splines are their asymptotic behavior, their simple covariance structure, and their quasi-separability. They converge to Gaussians with the increase of their order, and are used to approximate anisotropic Gaussians of varying covariance simply by controlling the scales of the constituent box distributions. Based on the second feature, we develop a technique for continuously controlling the size, elongation and orientation of these Gaussian-like functions. Finally, the quasi-separable structure, along with a certain scaling property of box distributions, is used to efficiently realize the associated space-variant elliptical filtering, which requires O(1) computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201

Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes

In diffusion MRI (dMRI), a good sampling scheme is important for efficient acquisition and robust reconstruction. Diffusion weighted signal is normally acquired on single or multiple shells in q-space. Signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue, or the laboratory coordinate frame. The Electrostatic Energy Minimization (EEM) method, originally proposed for single shell sampling scheme in dMRI, was recently generalized to multi-shell schemes, called Generalized EEM (GEEM). GEEM has been successfully used in the Human Connectome Project (HCP). However, EEM does not directly address the goal of optimal sampling, i.e., achieving large angular separation between sampling points. In this paper, we propose a more natural formulation, called Spherical Code (SC), to directly maximize the minimal angle between different samples in single or multiple shells. We consider not only continuous problems to design single or multiple shell sampling schemes, but also discrete problems to uniformly extract sub-sampled schemes from an existing single or multiple shell scheme, and to order samples in an existing scheme. We propose five algorithms to solve the above problems, including an incremental SC (ISC), a sophisticated greedy algorithm called Iterative Maximum Overlap Construction (IMOC), an 1-Opt greedy method, a Mixed Integer Linear Programming (MILP) method, and a Constrained Non-Linear Optimization (CNLO) method. To our knowledge, this is the first work to use the SC formulation for single or multiple shell sampling schemes in dMRI. Experimental results indicate that SC methods obtain larger angular separation and better rotational invariance than the state-of-the-art EEM and GEEM. The related codes and a tutorial have been released in DMRITool.Comment: Accepted by IEEE transactions on Medical Imaging. Codes have been released in dmritool https://diffusionmritool.github.io/tutorial_qspacesampling.htm

Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)

In this work we study the formulation of convection/diffusion equations on the 3D motion group SE(3) in terms of the irreducible representations of SO(3). Therefore, the left-invariant vector-fields on SE(3) are expressed as linear operators, that are differential forms in the translation coordinate and algebraic in the rotation. In the context of 3D image processing this approach avoids the explicit discretization of SO(3) or $S_2$, respectively. This is particular important for SO(3), where a direct discretization is infeasible due to the enormous memory consumption. We show two applications of the framework: one in the context of diffusion-weighted magnetic resonance imaging and one in the context of object detection

Improved 3D Heart Segmentation Using Surface Parameterization for Volumetric Heart Data

Imaging modalities such as CT, MRI, and SPECT have had a tremendous impact on diagnosis and treatment planning. These imaging techniques have given doctors the capability to visualize 3D anatomy structures of human body and soft tissues while being non-invasive. Unfortunately, the 3D images produced by these modalities often have boundaries between the organs and soft tissues that are difficult to delineate due to low signal to noise ratios and other factors. Image segmentation is employed as a method for differentiating Regions of Interest in these images by creating artificial contours or boundaries in the images. There are many different techniques for performing segmentation and automating these methods is an active area of research, but currently there are no generalized methods for automatic segmentation due to the complexity of the problem. Therefore hand-segmentation is still widely used in the medical community and is the €œGold standard€� by which all other segmentation methods are measured. However, existing manual segmentation techniques have several drawbacks such as being time consuming, introduce slice interpolation errors when segmenting slice-by-slice, and are generally not reproducible. In this thesis, we present a novel semi-automated method for 3D hand-segmentation that uses mesh extraction and surface parameterization to project several 3D meshes to 2D plane . We hypothesize that allowing the user to better view the relationships between neighboring voxels will aid in delineating Regions of Interest resulting in reduced segmentation time, alleviating slice interpolation artifacts, and be more reproducible

Tensor-valued diffusion MRI in under 3 minutes: An initial survey of microscopic anisotropy and tissue heterogeneity in intracranial tumors

Purpose: To evaluate the feasibility of a 3-minute b-tensor encoding protocol for diffusion MRI-based assessment of the microscopic anisotropy and tissue heterogeneity in a wide range of intracranial tumors. Methods: B-tensor encoding was performed in 42 patients with intracranial tumors (gliomas, meningiomas, adenomas, metastases). Microscopic anisotropy and tissue heterogeneity were evaluated by estimating the anisotropic kurtosis ($MK_A$) and isotropic kurtosis ($MK_I$), respectively. An extensive imaging protocol was compared with a faster 3-minute protocol. Results: The fast imaging protocol yielded parameters with characteristics in terms of bias and precision similar to the full protocol. Glioblastomas had lower microscopic anisotropy than meningiomas $(MK_A = 0.29 \pm 0.06$ versus $0.45\pm0.08, p = 0.003)$. Metastases had higher tissue heterogeneity $(MK_I = 0.57\pm0.07)$ than both the glioblastomas $(0.44\pm0.06, p < 0.001)$ and meningiomas $(0.46\pm0.06, p = 0.03)$. Conclusion: Evaluation of the microscopic anisotropy and tissue heterogeneity in intracranial tumor patients is feasible in clinically relevant times frames.Comment: Submitted to Magnetic Resonance in Medicin