Oguz C. Durumeric

Thickness of knots

Abstract Classical knot theory studies one-dimensional filaments; in this paper we model knots as more physically “real”, e.g., made of some “rope” with nonzero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot? For a smooth knot K , the “injectivity radius” R ( K ) is the supremum of radii of embedded tubular neighborhoods. The “thickness” of K , a new measure of knot complexity, is the ratio of R ( K ) to arc-length. We relate thickness to curvature, self-distance, distortion, and (for knot types) edge-number.

Growth of fundamental groups and isoembolic volume and diameter

Some properties of fundamental groups of Riemannian manifolds M will be studied without a lower bound assumption on Ricci curvature. The main method is to relate the local packing to global packing instead of using the Bishop-Gromov relative volume comparison. This method allows us to control the volume growth of the universal cover M and yields bounds on the number of generators of π 1 (M) in terms of some isoembolic geometric invariants of M.

Sensitivity analysis of Jacobian determinant used in treatment planning for lung cancer

Four-dimensional computed tomography (4DCT) is regularly used to visualize tumor motion in radiation therapy for lung cancer. These 4DCT images can be analyzed to estimate local ventilation by finding a dense correspondence map between the end inhalation and the end exhalation CT image volumes using deformable image registration. Lung regions with ventilation values above a threshold are labeled as regions of high pulmonary function and are avoided when possible in the radiation plan. This paper investigates a sensitivity analysis of the relative Jacobian error to small registration errors. We present a linear approximation of the relative Jacobian error. Next, we give a formula for the sensitivity of the relative Jacobian error with respect to the Jacobian of perturbation displacement field. Preliminary sensitivity analysis results are presented using 4DCT scans from 10 individuals. For each subject, we generated 6400 random smooth biologically plausible perturbation vector fields using a cubic B-spline model. We showed that the correlation between the Jacobian determinant and the Frobenius norm of the sensitivity matrix is close to -1, which implies that the relative Jacobian error in high-functional regions is less sensitive to noise. We also showed that small displacement errors on the average of 0.53 mm may lead to a 10% relative change in Jacobian determinant. We finally showed that the average relative Jacobian error and the sensitivity of the system for all subjects are positively correlated (close to +1), i.e. regions with high sensitivity has more error in Jacobian determinant on average.

Current-and Varifold-Based Registration of Lung Vessel and Airway Trees

Registering lung CT images is an important problem for many applications including tracking lung motion over the breathing cycle, tracking anatomical and function changes over time, and detecting abnormal mechanical properties of the lung. This paper compares and contrasts current-and varifold-based diffeomorphic image registration approaches for registering tree-like structures of the lung. In these approaches, curve-like structures in the lung—for example, the skeletons of vessels and airways segmentation—are represented by currents or varifolds in the dual space of a Reproducing Kernel Hilbert Space (RKHS). Current and varifold representations are discretized and are parameterized via of a collection of momenta. A momenta corresponds to a line segment via the coordinates of the center of the line segment and the tangent direction of the line segment at the center. A varifold-based registration approach is similar to currents except that two varifold representations are aligned independent of the tangent vector orientation. An advantage of varifolds over currents is that the orientation of the tangent vectors can be difficult to determine especially when the vessel and airway trees are not connected. In this paper, we examine the image registration sensitivity and accuracy of current-and varifold-based registration as a function of the number and location of momentum used to represent tree like-structures in the lung. The registrations presented in this paper were generated using the Deformetrica software package ([Durrleman et al. 2014]).

Detecting Out-of-Phase Ventilation Using 4DCT to Improve Radiation Therapy for Lung Cancer

Functional avoidance radiation therapy (RT) uses lung function images to identify and minimize irradiation of high-function lung tissue. Lung function can be estimated by local expansion ratio (LER) of the lung, which we define in this paper as the ratio of the maximum to the minimum local lung volume in a breathing cycle. LER is computed using deformable image registration. The end exhale (0EX) and the end inhale (100IN) phases of four-dimensional computed tomography (4DCT) are often used to estimate LER, which we refer to as LER3D. However, the lung may have out-of-phase ventilation, i.e., local lung volume change is out of phase with respect to global lung expansion and contraction. We propose the LER4D measure which estimates the LER measure using all phases of 4DCT. The purpose of this paper is to quantify the amount of out-of-phase ventilation of the lung. Out-of-phase ventilation is defined to occur when the LER4D measure is \(5\%\) or more than the LER3D measure. 4DCT scans of 14 human subjects were used in this study. Low-function (high-function) regions are defined as regions that have less (greater) than \(10\%\) expansion. Our results show that on average \(19.3\%\) of the lung had out-of-phase ventilation; \(3.8\%\) of the lung had out-of-phase ventilation and is labeled as low-function by both LER3D and LER4D; \(9.6\%\) of the lung is labeled as low-function by LER3D while high-function by LER4D; and \(5.9\%\) of the lung had out-of-phase ventilation and is labeled as high-function by both LER3D and LER4D. We conclude that out-of-phase ventilation is common in all 14 human subjects we have investigated.

Population Shape Collapse in Large Deformation Registration of MR Brain Images

This paper examines the shape collapse problem that occurs when registering a pair of images or a population of images of the brain to a reference (target) image coordinate system using diffeomorphic image registration. Shape collapse occurs when a foreground or background structure in an image with non-zero volume is transformed into a set of zero or near zero volume as measured on a discrete voxel lattice in the target image coordinate system. Shape collapse may occur during image registration when the moving image has a structure that is either missing or does not sufficiently overlap the corresponding structure in the target image[4]. Such a problem is common in image registration algorithms with large degrees of freedom such as many diffeomorphic image registration algorithms. Shape collapse is a concern when mapping functional data. For example, loss of signal may occur when mapping functional data such as fMRI, PET, SPECT using a transformation with a shape collapse if the functional signal occurs at the collapse region. This paper proposes an novel shape collapse measurement algorithm to detect the regions of shape collapse after image registration in pairwise registration. We further compute the shape collapse for a population of pairwise transformations such as occurs when registering many images to a common atlas coordinate system. Experiments are presented using the SyN diffeomorphic image registration algorithm. We demonstrate how changing the input parameters to the SyN registration algorithm can mitigate some of the collapse image registration artifacts.

Geometric finiteness in large families in dimension 3

Abstract We prove a diffeomorphism type finiteness theorem in dimension 3 for families C covered by finitely many distance-like functions with bounded twist. C contains families of bounded isoembolic volume and twist. However, the subfamilies of C with fixed volume are not Hausdorff precompact. Our methods do not involve Hausdorff limits and they yield explicit estimates on upper bounds for the number of diffeomorphism types.