Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm

Approximability of the Discrete {Fr\'echet} Distance

<p>The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.</p><p>In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.</p><p>This raises the question of how well we can approximate the Fréchet distance (of two given $d$-dimensional point sequences of length $n$) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be $2^{\Theta(n)}$. Moreover, we design an $\alpha$-approximation algorithm that runs in time $O(n\log n + n^2/\alpha)$, for any $\alpha\in [1, n]$. Hence, an $n^\varepsilon$-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p

Investigating the geochemical alterations in an aquifer due to long-term sequestration of CO2 using time-lapse seismic information

The effects of chemical interaction between injected CO2, brine, and formation rocks are often ignored in sequestration studies because chemical reactions are assumed to be localized to carbonate rocks that make up only a small proportion of the potential reservoirs. It is conjectured in this work that long-term exposure of certain types of clays and cement material to CO2-brine mixtures can induce chemical reactions and subsequent alteration of rock properties that can be subsequently detected in time-lapse seismic surveys. This is demonstrated using a case-study structured after the Cranfield field injection site. Geochemical alterations of the reservoir rock are quantified by performing reactive transport simulations and subsequently using rock physics models to translate the altered petrophysical properties into seismic responses. The study quantifies the long-term geochemical effects of CO2 injection on the seismic response and conversely, presents an approach to invert the reservoir regions contacted by the CO2-saturated brine based on the observed seismic response. Time lapse or passive seismic monitoring is an effective method for mapping the progress of the CO2 plume through the subsurface. But, because of the lack of resolution of the seismic information, it is necessary to use the seismic information together with prior geologic knowledge about the surface in order to identify if there is any migration of CO2 into regions that might be deemed sensitive e.g. overlying aquifers or faults. Because of uncertainties in the prior geologic description of the reservoir, the feasibility of implementing a model selection process is explored in this work. The model selection procedure utilizes the observed well data and reference seismic map to select a subset of models. The flow simulation of CO2 injection and forward seismic modeling were repeated for the newly generated reservoir models, and the seismic responses were compared for the reaction and non-reaction cases. The study showed that the effects of geochemical reactions on petrophysical properties and resultant spatial distribution of fluid saturation were visible in the seismic response. Major differences in seismic responses were detected in regions of the reservoir where significant amount of minerals were dissolved and precipitated. These regions were at the top of the reservoir due to the reactions caused by the buoyant CO2 plume. The presence of carbonate facies, even in small proportion, plays an important role in geochemical reactions and their effect is manifested at the seismic scale. The unique model selection methodology presented in this thesis is efficient at detecting the important features in the seismic and injection response that is induced by the geochemical alterations occurring in the reservoir. The results of this time-lapse study can provide new interpretation of events observed in time-lapse seismic data that might lead to a better assessment of leakage pathways and other risks.Petroleum and Geosystems Engineerin

Registration of histology and magnetic resonance imaging of the brain

Combining histology and non-invasive imaging has been attracting the attention of the medical imaging community for a long time, due to its potential to correlate macroscopic information with the underlying microscopic properties of tissues. Histology is an invasive procedure that disrupts the spatial arrangement of the tissue components but enables visualisation and characterisation at a cellular level. In contrast, macroscopic imaging allows non-invasive acquisition of volumetric information but does not provide any microscopic details. Through the establishment of spatial correspondences obtained via image registration, it is possible to compare micro- and macroscopic information and to recover the original histological arrangement in three dimensions. In this thesis, I present: (i) a survey of the literature relative to methods for histology reconstruction with and without the help of 3D medical imaging; (ii) a graph-theoretic method for histology volume reconstruction from sets of 2D sections, without external information; (iii) a method for multimodal 2D linear registration between histology and MRI based on partial matching of shape-informative boundaries

Computing the Fréchet distance with a retractable leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; then they use this oracle to find the optimum among a finite set of critical values. We present a novel approach that avoids the detour through the decision version. We demonstrate its strength by presenting a quadratic time algorithm for the Fréchet distance between polygonal curves in R^d under polyhedral distance functions, including L_1 and L_infinity. We also get a (1+eps)-approximation of the Fréchet distance under the Euclidean metric. For the exact Euclidean case, our framework currently gives an algorithm with running time O(n^2 log^2 n). However, we conjecture that it may eventually lead to a faster exact algorithm