Estimation of vector fields in unconstrained and inequality constrained variational problems for segmentation and registration

Vector fields arise in many problems of computer vision, particularly in non-rigid registration. In this paper, we develop coupled partial differential equations (PDEs) to estimate vector fields that define the deformation between objects, and the contour or surface that defines the segmentation of the objects as well.We also explore the utility of inequality constraints applied to variational problems in vision such as estimation of deformation fields in non-rigid registration and tracking. To solve inequality constrained vector field estimation problems, we apply tools from the Kuhn-Tucker theorem in optimization theory. Our technique differs from recently popular joint segmentation and registration algorithms, particularly in its coupled set of PDEs derived from the same set of energy terms for registration and segmentation. We present both the theory and results that demonstrate our approach

Quantification of cortical folding using MR image data

The cerebral cortex is a thin layer of tissue lining the brain where neural circuits perform important high level functions including sensory perception, motor control and language processing. In the third trimester the fetal cortex folds rapidly from a smooth sheet into a highly convoluted arrangement of gyri and sulci. Premature birth is a high risk factor for poor neurodevelopmental outcome and has been associated with abnormal cortical development, however the nature of the disruption to developmental processes is not fully understood. Recent developments in magnetic resonance imaging have allowed the acquisition of high quality brain images of preterms and also fetuses in-utero. The aim of this thesis is to develop techniques which quantify folding from these images in order to better understand cortical development in these two populations. A framework is presented that quantifies global and regional folding using curvature-based measures. This methodology was applied to fetuses over a wide gestational age range (21.7 to 38.9 weeks) for a large number of subjects (N = 80) extending our understanding of how the cortex folds through this critical developmental period. The changing relationship between the folding measures and gestational age was modelled with a Gompertz function which allowed an accurate prediction of physiological age. A spectral-based method is outlined for constructing a spatio-temporal surface atlas (a sequence of mean cortical surface meshes for weekly intervals). A key advantage of this method is the ability to do group-wise atlasing without bias to the anatomy of an initial reference subject. Mean surface templates were constructed for both fetuses and preterms allowing a preliminary comparison of mean cortical shape over the postmenstrual age range 28-36 weeks. Displacement patterns were revealed which intensified with increasing prematurity, however more work is needed to evaluate the reliability of these findings.Open Acces

Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams

This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive

Tighter Connections Between Formula-SAT and Shaving Logs

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the $O(n^2)$ dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to $O(n^2/\log^2 n)$ in several ways and using a variety of ingenious tricks. This line of research, also known as "the art of shaving log factors", lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time $O(n^2/\log^3 n)$? Perhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on Boolean formulas (Formula-SAT) faster than exhaustive search. They show that an $O(n^2/\log^{1000} n)$ algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear. In this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fr\'echet distance problem from Computational Geometry, we show that an $O(n^2/\log^{7+\varepsilon} n)$ runtime would imply new Formula-SAT algorithms. Our main result is a reduction from SAT on formulas of size $s$ over $n$ variables to LCS on sequences of length $N=2^{n/2} \cdot s^{1+o(1)}$. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with $N=2^{n/2} \cdot s^c$, for some $c \geq 100$

A fast implementation of near neighbors queries for Fr\'echet distance (GIS Cup)

This paper describes an implementation of fast near-neighbours queries (also known as range searching) with respect to the Fr\'echet distance. The algorithm is designed to be efficient on practical data such as GPS trajectories. Our approach is to use a quadtree data structure to enumerate all curves in the database that have similar start and endpoints as the query curve. On these curves we run positive and negative filters to narrow the set of potential results. Only for those trajectories where these heuristics fail, we compute the Fr\'echet distance exactly, by running a novel recursive variant of the classic free-space diagram algorithm. Our implementation won the ACM SIGSPATIAL GIS Cup 2017.Comment: ACM SIGSPATIAL'17 invited paper. 9 page

Fast assessment of uncertainty in buoyant fluid displacement using a connectivity-based proxy

It is crucial to estimate the uncertainty in flow characteristics of injected fluid. However, because a large suite of geological models is probable given sparse static data, it is impractical to conduct full physics flow simulations on the entire suite of models in order to quantify the uncertainty in fluid displacements. Thus a fast alternative to a full physics simulator is necessary to quickly predict the fluid displacements. Most of the proxies proposed thus far are inappropriate to approximate the buoyant flow of injected fluid for 3D heterogeneous rock during the injection period. In this dissertation, a new proxy will be proposed to quickly predict the buoyant flow of injected fluid during CO2 sequestration. The geological models are ranked based on the extent of the approximated CO2 plumes. By selecting a representative group of models among the ranked models, the uncertainty in the spatial and temporal characteristics of the CO2 plume migrations can be quickly quantified. About 90% of the computational cost of quantifying the uncertainty in the extent of CO2 plumes was saved using the proposed connectivity based proxy. In a geological carbon storage project, the spatial and temporal characteristics of CO2 plume migrations can be monitored by 4D seismic surveys. The images of CO2 plumes obtained from 4D seismic surveys are used as observed data to find subsurface models honoring the spatial and temporal characteristics of the observed CO2 plumes. However, because manually comparing an observed CO2 plume and prior CO2 plumes in a large suite of subsurface models is inefficient, an automatic measure to calculate the dissimilarity between the CO2 plumes is necessary. The most intuitive way to calculate the dissimilarity is the Euclidean distance between vectors representing CO2 plumes. However, this is inappropriate to measure the dissimilarity between CO2 plumes because it does not consider spatial relation between the elements of the vectors. The shape dissimilarity between the CO2 plumes that reflects the spatial relation can be calculated using the Hausdorff distance. The computational cost of calculating the shape dissimilarity between CO2 plumes is significantly reduced by calculating the Hausdorff distance between the representations of the CO2 plumes such as perimeter, surface, and skeleton instead of the original CO2 plumes. An appropriate representation should be chosen according to the spatial characteristics of CO2 plumes.Petroleum and Geosystems Engineerin

Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results