On the non-local geometry of turbulence

A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component three-dimensional fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. The characterization stage is based on the joint probability density function (p.d.f.), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the joint p.d.f. of each structure, that specify its location as a point in a multi-dimensional ‘feature space’. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This then allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry. Results are presented of a first application of this technique to a passive scalar field obtained from 5123 direct numerical simulation of scalar mixing by forced, isotropic turbulence (Reλ = 265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behaviour of passive scalar stirring and mixing by turbulence are discussed

A Parallel Feature-preserving Mesh Variable Offsetting Method with Dynamic Programming

Mesh offsetting plays an important role in discrete geometric processing. In this paper, we propose a parallel feature-preserving mesh offsetting framework with variable distance. Different from the traditional method based on distance and normal vector, a new calculation of offset position is proposed by using dynamic programming and quadratic programming, and the sharp feature can be preserved after offsetting. Instead of distance implicit field, a spatial coverage region represented by polyhedral for computing offsets is proposed. Our method can generate an offsetting model with smaller mesh size, and also can achieve high quality without gaps, holes, and self-intersections. Moreover, several acceleration techniques are proposed for the efficient mesh offsetting, such as the parallel computing with grid, AABB tree and rays computing. In order to show the efficiency and robustness of the proposed framework, we have tested our method on the quadmesh dataset, which is available at [https://www.quadmesh.cloud]. The source code of the proposed algorithm is available on GitHub at [https://github.com/iGame-Lab/PFPOffset]

Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence

We report the multi-scale geometric analysis of Lagrangian structures in forced isotropic turbulence and also with a frozen turbulent field. A particle backward-tracking method, which is stable and topology preserving, was applied to obtain the Lagrangian scalar field φ governed by the pure advection equation in the Eulerian form ∂_tφ + u · ∇φ = 0. The temporal evolution of Lagrangian structures was first obtained by extracting iso-surfaces of φ with resolution 1024^3 at different times, from t = 0 to t = T_e, where T_e is the eddy turnover time. The surface area growth rate of the Lagrangian structure was quantified and the formation of stretched and rolled-up structures was observed in straining regions and stretched vortex tubes, respectively. The multi-scale geometric analysis of Bermejo-Moreno & Pullin (J. Fluid Mech., vol. 603, 2008, p. 101) has been applied to the evolution of φ to extract structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. In this multi-scale sense, we observe, for the evolving turbulent velocity field, an evolutionary breakdown of initially large-scale Lagrangian structures that first distort and then either themselves are broken down or stretched laterally into sheets. Moreover, after a finite time, this progression appears to be insensible to the form of the initially smooth Lagrangian field. In comparison with the statistical geometry of instantaneous passive scalar and enstrophy fields in turbulence obtained by Bermejo-Moreno & Pullin (2008) and Bermejo-Moreno et al. (J. Fluid Mech., vol. 620, 2009, p. 121), Lagrangian structures tend to exhibit more prevalent sheet-like shapes at intermediate and small scales. For the frozen flow, the Lagrangian field appears to be attracted onto a stream-surface field and it develops less complex multi-scale geometry than found for the turbulent velocity field. In the latter case, there appears to be a tendency for the Lagrangian field to move towards a vortex-surface field of the evolving turbulent flow but this is mitigated by cumulative viscous effects

Doctor of Philosophy

dissertationIn this dissertation, we advance the theory and practice of verifying visualization algorithms. We present techniques to assess visualization correctness through testing of important mathematical properties. Where applicable, these techniques allow us to distinguish whether anomalies in visualization features can be attributed to the underlying physical process or to artifacts from the implementation under verification. Such scientific scrutiny is at the heart of verifiable visualization - subjecting visualization algorithms to the same verification process that is used in other components of the scientific pipeline. The contributions of this dissertation are manifold. We derive the mathematical framework for the expected behavior of several visualization algorithms, and compare them to experimentally observed results in the selected codes. In the Computational Science & Engineering community CS&E, this technique is know as the Method of Manufactured Solution (MMS). We apply MMS to the verification of geometrical and topological properties of isosurface extraction algorithms, and direct volume rendering. We derive the convergence of geometrical properties of isosurface extraction techniques, such as function value and normals. For the verification of topological properties, we use stratified Morse theory and digital topology to design algorithms that verify topological invariants. In the case of volume rendering algorithms, we provide the expected discretization errors for three different error sources. The results of applying the MMS is another important contribution of this dissertation. We report unexpected behavior for almost all implementations tested. In some cases, we were able to find and fix bugs that prevented the correctness of the visualization algorithm. In particular, we address an almost 2 0 -year-old bug with the core disambiguation procedure of Marching Cubes 33, one of the first algorithms intended to preserve the topology of the trilinear interpolant. Finally, an important by-product of this work is a range of responses practitioners can expect to encounter with the visualization technique under verification

3D fusion of histology to multi-parametric MRI for prostate cancer imaging evaluation and lesion-targeted treatment planning

Multi-parametric magnetic resonance imaging (mpMRI) of localized prostate cancer has the potential to support detection, staging and localization of tumors, as well as selection, delivery and monitoring of treatments. Delineating prostate cancer tumors on imaging could potentially further support the clinical workflow by enabling precise monitoring of tumor burden in active-surveillance patients, optimized targeting of image-guided biopsies, and targeted delivery of treatments to decrease morbidity and improve outcomes. Evaluating the performance of mpMRI for prostate cancer imaging and delineation ideally includes comparison to an accurately registered reference standard, such as prostatectomy histology, for the locations of tumor boundaries on mpMRI. There are key gaps in knowledge regarding how to accurately register histological reference standards to imaging, and consequently further gaps in knowledge regarding the suitability of mpMRI for tasks, such as tumor delineation, that require such reference standards for evaluation. To obtain an understanding of the magnitude of the mpMRI-histology registration problem, we quantified the position, orientation and deformation of whole-mount histology sections relative to the formalin-fixed tissue slices from which they were cut. We found that (1) modeling isotropic scaling accounted for the majority of the deformation with a further small but statistically significant improvement from modeling affine transformation, and (2) due to the depth (mean±standard deviation (SD) 1.1±0.4 mm) and orientation (mean±SD 1.5±0.9°) of the sectioning, the assumption that histology sections are cut from the front faces of tissue slices, common in previous approaches, introduced a mean error of 0.7 mm. To determine the potential consequences of seemingly small registration errors such as described above, we investigated the impact of registration accuracy on the statistical power of imaging validation studies using a co-registered spatial reference standard (e.g. histology images) by deriving novel statistical power formulae that incorporate registration error. We illustrated, through a case study modeled on a prostate cancer imaging trial at our centre, that submillimeter differences in registration error can have a substantial impact on the required sample sizes (and therefore also the study cost) for studies aiming to detect mpMRI signal differences due to 0.5 – 2.0 cm3 prostate tumors. With the aim of achieving highly accurate mpMRI-histology registrations without disrupting the clinical pathology workflow, we developed a three-stage method for accurately registering 2D whole-mount histology images to pre-prostatectomy mpMRI that allowed flexible placement of cuts during slicing for pathology and avoided the assumption that histology sections are cut from the front faces of tissue slices. The method comprised a 3D reconstruction of histology images, followed by 3D–3D ex vivo–in vivo and in vivo–in vivo image transformations. The 3D reconstruction method minimized fiducial registration error between cross-sections of non-disruptive histology- and ex-vivo-MRI-visible strand-shaped fiducials to reconstruct histology images into the coordinate system of an ex vivo MR image. We quantified the mean±standard deviation target registration error of the reconstruction to be 0.7±0.4 mm, based on the post-reconstruction misalignment of intrinsic landmark pairs. We also compared our fiducial-based reconstruction to an alternative reconstruction based on mutual-information-based registration, an established method for multi-modality registration. We found that the mean target registration error for the fiducial-based method (0.7 mm) was lower than that for the mutual-information-based method (1.2 mm), and that the mutual-information-based method was less robust to initialization error due to multiple sources of error, including the optimizer and the mutual information similarity metric. The second stage of the histology–mpMRI registration used interactively defined 3D–3D deformable thin-plate-spline transformations to align ex vivo to in vivo MR images to compensate for deformation due to endorectal MR coil positioning, surgical resection and formalin fixation. The third stage used interactively defined 3D–3D rigid or thin-plate-spline transformations to co-register in vivo mpMRI images to compensate for patient motion and image distortion. The combined mean registration error of the histology–mpMRI registration was quantified to be 2 mm using manually identified intrinsic landmark pairs. Our data set, comprising mpMRI, target volumes contoured by four observers and co-registered contoured and graded histology images, was used to quantify the positive predictive values and variability of observer scoring of lesions following the Prostate Imaging Reporting and Data System (PI-RADS) guidelines, the variability of target volume contouring, and appropriate expansion margins from target volumes to achieve coverage of histologically defined cancer. The analysis of lesion scoring showed that a PI-RADS overall cancer likelihood of 5, denoting “highly likely cancer”, had a positive predictive value of 85% for Gleason 7 cancer (and 93% for lesions with volumes \u3e0.5 cm3 measured on mpMRI) and that PI-RADS scores were positively correlated with histological grade (ρ=0.6). However, the analysis also showed interobserver differences in PI-RADS score of 0.6 to 1.2 (on a 5-point scale) and an agreement kappa value of only 0.30. The analysis of target volume contouring showed that target volume contours with suitable margins can achieve near-complete histological coverage for detected lesions, despite the presence of high interobserver spatial variability in target volumes. Prostate cancer imaging and delineation have the potential to support multiple stages in the management of localized prostate cancer. Targeted biopsy procedures with optimized targeting based on tumor delineation may help distinguish patients who need treatment from those who need active surveillance. Ongoing monitoring of tumor burden based on delineation in patients undergoing active surveillance may help identify those who need to progress to therapy early while the cancer is still curable. Preferentially targeting therapies at delineated target volumes may lower the morbidity associated with aggressive cancer treatment and improve outcomes in low-intermediate-risk patients. Measurements of the accuracy and variability of lesion scoring and target volume contouring on mpMRI will clarify its value in supporting these roles

Surface Shape Perception in Volumetric Stereo Displays

In complex volume visualization applications, understanding the displayed objects and their spatial relationships is challenging for several reasons. One of the most important obstacles is that these objects can be translucent and can overlap spatially, making it difficult to understand their spatial structures. However, in many applications, for example medical visualization, it is crucial to have an accurate understanding of the spatial relationships among objects. The addition of visual cues has the potential to help human perception in these visualization tasks. Descriptive line elements, in particular, have been found to be effective in conveying shape information in surface-based graphics as they sparsely cover a geometrical surface, consistently following the geometry. We present two approaches to apply such line elements to a volume rendering process and to verify their effectiveness in volume-based graphics. This thesis reviews our progress to date in this area and discusses its effects and limitations. Specifically, it examines the volume renderer implementation that formed the foundation of this research, the design of the pilot study conducted to investigate the effectiveness of this technique, the results obtained. It further discusses improvements designed to address the issues revealed by the statistical analysis. The improved approach is able to handle visualization targets with general shapes, thus making it more appropriate to real visualization applications involving complex objects

A Novel 3-D Segmentation Algorithm for Anatomic Liver and Tumor Volume Calculations for Liver Cancer Treatment Planning

Three-Dimensional (3-D) imaging is vital in computer-assisted surgical planning including minimal invasive surgery, targeted drug delivery, and tumor resection. Selective Internal Radiation Therapy (SIRT) is a liver directed radiation therapy for the treatment of liver cancer. Accurate calculation of anatomical liver and tumor volumes are essential for the determination of the tumor to normal liver ratio and for the calculation of the dose of Y-90 microspheres that will result in high concentration of the radiation in the tumor region as compared to nearby healthy tissue. Present manual techniques for segmentation of the liver from Computed Tomography (CT) tend to be tedious and greatly dependent on the skill of the technician/doctor performing the task. This dissertation presents the development and implementation of a fully integrated algorithm for 3-D liver and tumor segmentation from tri-phase CT that yield highly accurate estimations of the respective volumes of the liver and tumor(s). The algorithm as designed requires minimal human intervention without compromising the accuracy of the segmentation results. Embedded within this algorithm is an effective method for extracting blood vessels that feed the tumor(s) in order to plan effectively the appropriate treatment. Segmentation of the liver led to an accuracy in excess of 95% in estimating liver volumes in 20 datasets in comparison to the manual gold standard volumes. In a similar comparison, tumor segmentation exhibited an accuracy of 86% in estimating tumor(s) volume(s). Qualitative results of the blood vessel segmentation algorithm demonstrated the effectiveness of the algorithm in extracting and rendering the vasculature structure of the liver. Results of the parallel computing process, using a single workstation, showed a 78% gain. Also, statistical analysis carried out to determine if the manual initialization has any impact on the accuracy showed user initialization independence in the results. The dissertation thus provides a complete 3-D solution towards liver cancer treatment planning with the opportunity to extract, visualize and quantify the needed statistics for liver cancer treatment. Since SIRT requires highly accurate calculation of the liver and tumor volumes, this new method provides an effective and computationally efficient process required of such challenging clinical requirements

A comparison of hole-filling methods in 3D

This paper presents a review of the most relevant current techniques that deal with hole-filling in 3D models. Contrary to earlier reports, which approach mesh repairing in a sparse and global manner, the objective of this review is twofold. First, a specific and comprehensive review of hole-filling techniques (as a relevant part in the field of mesh repairing) is carried out. We present a brief summary of each technique with attention paid to its algorithmic essence, main contributions and limitations. Second, a solid comparison between 34 methods is established. To do this, we define 19 possible meaningful features and properties that can be found in a generic hole-filling process. Then, we use these features to assess the virtues and deficiencies of the method and to build comparative tables. The purpose of this review is to make a comparative hole-filling state-of-the-art available to researchers, showing pros and cons in a common framework.• Ministerio de Economía y Competitividad: Proyecto DPI2013-43344-R (I+D+i) • Gobierno de Castilla-La Mancha: Proyecto PEII-2014-017-PpeerReviewe