Computing Fenchel-Nielsen Coordinates in Teichmüller Shape Space

We develop Fenchel–Nielsen coordinates for representations of surface groups into $${{\mathrm{Sp}}}(2n,{\mathbb {R}})$$ with maximal Toledo invariant. Analogous to classical Fenchel–Nielsen coordinates on the Teichmuller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigation of the gluing. As applications we obtain results for non-closed surfaces, which have been known only for closed surfaces before: we count the number of connected components of their representation space and prove continuity for the limit curve for a certain type of representations.

J.J. Moreau in [1–3] discussed the problem of dynamics of unilateral systems of finite degree of freedom, mainly in the framework of Euclidean space. He introduced the concept of soft contact and arrived at the general measure differential inclusion of a nonsmooth motion.

Chapter 12 - Mapping Class Groups

An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S2 satisfying f (x) = f(-x) for all x E S2 there exists a conformal metric on S2 whose Gauss curvature is equal to f. A natural conformal invariant in dimension four is 1 2 Q (AR R2+ 3 IRicI2), where R denotes the scalar curvature and Ric the Ricci tensor. This formula can also be written in the form

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.

Statistical shape modeling (SSM) is a valuable and powerful tool to generate a detailed representation of complex anatomy that enables quantitative analysis and the comparison of shapes and their variations. SSM applies mathematics, statistics, and computing to parse the shape into a quantitative representation (such as correspondence points or landmarks) that will help answer various questions about the anatomical variations across the population. Complex anatomical structures have many diverse parts with varying interactions or intricate architecture. For example, the heart is a four-chambered anatomy with several shared boundaries between chambers. Coordinated and efficient contraction of the chambers of the heart is necessary to adequately perfuse end organs throughout the body. Subtle shape changes within these shared boundaries of the heart can indicate potential pathological changes that lead to uncoordinated contraction and poor end-organ perfusion. Early detection and robust quantification could provide insight into ideal treatment techniques and intervention timing. However, existing SSM approaches fall short of explicitly modeling the statistics of shared boundaries. In this paper, we present a general and flexible data-driven approach for building statistical shape models of multi-organ anatomies with shared boundaries that captures morphological and alignment changes of individual anatomies and their shared boundary surfaces throughout the population. We demonstrate the effectiveness of the proposed methods using a biventricular heart dataset by developing shape models that consistently parameterize the cardiac biventricular structure and the interventricular septum (shared boundary surface) across the population data.

<p>Patterns in biological and medical data exhibit highly structured geometries. Leveraging geometrical comparisons through quantitative descriptors or via contractive difference measures can facilitate the development of interpretable, unsupervised, and efficient models for clinical diagnosis and biological discovery.</p> <p>In medical image analysis, abnormality-related patterns often exhibit as quantifiable geometrical deformations. Guided by this principle, the following clinical challenges are addressed: a three-dimensional quantitative modeling of bone erosion and proliferation, and the development of robust detection algorithms for real-world applications. First, building upon radiological consensus, a bone erosion is modeled as significant concave regions on the cortical layer and a model-based 3D automatic erosion detection framework is proposed. This approach involves: (a) the cortical surface reconstruction and (b) the curvature-driven erosion identification, implemented via computational differential geometry and level set optimization. Second, the bone proliferation is mathematically formalized by integrating pathological insights with statistical shape analysis. The resulting Deformation-Controllable Elastic Shape (DCES) model provides an unsupervised solution for 3D proliferation quantification.</p> <p>Beyond medical images, the rapid growth of protein structure data demands advanced geometrical comparison methods. While existing approaches primarily rely on sequential geometrical comparisons, non-sequential comparison methods remain underdeveloped. To bridge this gap, the non-sequential comparison is framed as the Entropy-regularized Partial Linear Sum Assignment Problem (epLSAP) and solved by epLSAP-Align, a solver based on Sinkhorn algorithms that enables efficient topology-agnostic comparisons with tunable fidelity-coverage tradeoffs. In summary, these contributions help analyze biomedical structures from geometrical perspectives, enabling practical clinical/biology applications with geometrical insights.</p>

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex structures on S, which we call the Fenchel-Nielsen Teichmüller space associated to the pair (P,X). This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers [1], [2] and [3], and we compared it to the classical Teichmüller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.

This paper provides developments in statistical shape analysis of shape graphs, and demonstrates them using such complex objects as Retinal Blood Vessel (RBV) networks and neurons. The shape graphs are represented by sets of nodes and edges (articulated curves) connecting some nodes. The goals are to utilize nodes (locations, connectivity) and edges (edge weights and shapes) to: (1) characterize shapes, (2) quantify shape differences, and (3) model statistical variability. We develop a mathematical representation, elastic Riemannian metrics, and associated tools for shape graphs. Specifically, we derive tools for shape graph registration, geodesics, statistical summaries, shape modeling, and shape synthesis. Geodesics are convenient for visualizing optimal deformations, and PCA helps in dimension reduction and statistical modeling. One key challenge lies in comparing shape graphs with vastly different complexities (in number of nodes and edges). This paper introduces a novel multi-scale representation to handle this challenge. Using the notions of (1) "effective resistance" to cluster nodes and (2) elastic shape averaging of edge curves, it reduces graph complexity while retaining overall structures. This allows shape comparisons by bringing graphs to similar complexities. We demonstrate these ideas on 2D RBV networks from the STARE and DRIVE databases and 3D neurons from the NeuroMorpho database.

The study of relations among Euclidean, spherical and hyperbolic geometries dates back to the beginning of last century. The attempt to prove Euclid’s fifth postulate led C. F. Gauss to discover hyperbolic geometry in the 1820’s. Only a few years passed before this geometry was rediscovered independently by N. Lobachevski (1829) and J. Bolyai (1832). The strongest evidence given by the founders for its consistency is the duality between hyperbolic and spherical trigonometries. This duality was first demonstrated by Lambert in his 1770 memoir [139] Some theorems, for example the law of sines, can be stated in a form that is valid in spherical, Euclidean, and hyperbolic geometries [25].

This paper presents a comparison of three-dimensional (3-D) segmentations of the prostate, based on two-dimensional (2-D) manually segmented contours, obtained using ultrasound (US) and magnetic resonance (MR) imaging data collected from 40 patients diagnosed with localized prostate cancer and scheduled to receive brachytherapy treatment. The approach we propose here for 3-D prostate segmentation first uses radial basis function interpolation to construct a 3-D point distribution model for each prostate. Next, a modified principal axis transformation is utilized for rigid registration of the US and MR images of the same prostate in preparation for the following shape comparison. Then, statistical shape models are used to capture the segmented 3-D prostate geometries for the subsequent cross-modality comparison. Our study includes not only cross-modality geometric comparisons in terms of prostate volumes and dimensions, but also an investigation of interchangeability of the two imaging modalities in terms of automatic contour segmentation at the pre-implant planning stage of prostate brachytherapy treatment. By developing a new scheme to compare the two imaging modalities in terms of the segmented 3-D shapes, we have taken a first step necessary for building coupled US-MR segmentation strategies for prostate brachytherapy pre-implant planning, which at present is predominantly informed by US images only.

The Marnoso Arenacea Formation provides the most extensive correlation of individual flow deposits (beds) yet documented in an ancient turbidite system. These correlations provide unusually detailed constraints on bed shape, which is used to deduce flow evolution and assess the validity of numerical and laboratory models. Bed volumes have an approximately log‐normal frequency distribution; a small number of flows dominated sediment supply to this non‐channelized basin plain. Turbidite sandstone within small‐volume (<0·7 km3) beds thins downflow in an approximately exponential fashion. This shape is a property of spatially depletive flows, and has been reproduced by previous mathematical models and laboratory experiments. Sandstone intervals in larger‐volume (0·7–7 km3) beds have a broad thickness maximum in their proximal part. Grain‐size trends within this broad thickness maximum indicate spatially near‐uniform flow for distances of ∼30 km, although the flow was temporally unsteady. Previous mathematical models and laboratory experiments have not reproduced this type of deposit shape. This may be because models fail to simulate the way in which near bed sediment concentration tends towards a constant value (saturates) in powerful flows. Alternatively, the discrepancy may be the result of relatively high ratios of flow thickness and sediment settling velocity in submarine flows, together with very gradual changes in sea‐floor gradient. Intra‐bed erosion, temporally varying discharge, and reworking of suspension fallout as bedload could also help to explain the discrepancy in deposit shape. Most large‐volume beds contain an internal erosion surface underlain by inversely graded sandstone, recording waxing and waning flow. It has been inferred previously that these characteristics are diagnostic of turbidites generated by hyperpycnal flood discharge. These turbidites are too voluminous to have been formed by hyperpycnal flows, unless such flows are capable of eroding cubic kilometres of sea‐floor sediment. It is more likely that these flows originated from submarine slope failure. Two beds comprise multiple sandstone intervals separated only by turbidite mudstone. These features suggest that the submarine slope failures occurred as either a waxing and waning event, or in a number of stages.

SynopsisThe solutions of stochastic differential equations are used to construct Markov processes on the Banach manifold C(S, M) of continuous maps from a compact metric space S into a smooth complete finite dimensional Riemannian manifold M. In the special case where S is a single point the construction gives a large class of diffusion processes on the manifold M, including (under certain curvature conditions) the Brownian motion process.

The form and shape of a hypothesis space imposes natural objective constraints to any inferential process. This contribution summarizes what is currently known and the mathematics that are thought to be needed for new developments in this area. For example, it is well known that the quality of best possible estimators deteriorates with increasing volume, dimension and curvature of the hypothesis space. It is also known that regular statistical parametric models are finite dimensional Riemannian manifolds admitting a family of dual affine connections. Fisher information is the metric induced on the hypothesis space by the Hellinger distance. Nonparametric models are infinite dimensional manifolds. Global negative curvature implies asymptotic inadmissibility of uniform priors. When there is uncertainty about the model and the prior, entropic methods are more robust than standard Bayesian inference. The presence of some types of singularities allow the existence of faster than normal estimators …, etc. The large number of fundamental statistical concepts with geometric and topological content suggest to try to look at Riemannian Geometry, Algebraic Geometry, K‐theory, Algebraic Topology, Knot‐theory and other branches of current mathematics, not as empty esoteric abstractions but as allies for statistical inference.