Abstract This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.
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Geodesic Regression Characterizes 3D Shape Changes in the Female Brain During Menstruation
Women are at higher risk of Alzheimer’s and other neurological diseases after menopause, and yet research con- necting female brain health to sex hormone fluctuations is limited. We seek to investigate this connection by develop- ing tools that quantify 3D shape changes that occur in the brain during sex hormone fluctuations. Geodesic regression on the space of 3D discrete surfaces offers a principled way to characterize the evolution of a brain’s shape. However, in its current form, this approach is too computationally expensive for practical use. In this paper, we pro- pose approximation schemes that accelerate geodesic regression on shape spaces of 3D discrete surfaces. We also provide rules of thumb for when each approximation can be used. We test our approach on synthetic data to quantify the speed-accuracy trade-off of these approximations and show that practitioners can expect very significant speed-up while only sacrificing little accuracy. Finally, we apply the method to real brain shape data and produce the first characterization of how the female hippocampus changes shape during the menstrual cycle as a function of progesterone: a characterization made (practically) possible by our approximation schemes. Our work paves the way for comprehensive, practical shape analyses in the fields of bio-medicine and computer vision. Our implementation is publicly avail- able on GitHub.
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- Award ID(s):
- 2240158
- PAR ID:
- 10524941
- Publisher / Repository:
- IEEE Xplore
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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It is well-known that morphological features in the brain undergo changes due to traumatic events and associated disorders such as post-traumatic stress disorder (PTSD). However, existing approaches typically offer group-level comparisons, and there are limited predictive approaches for modeling behavioral outcomes based on brain shape features that can account for heterogeneity in PTSD, which is of paramount interest. We propose a comprehensive shape analysis framework representing brain sub-structures, such as the hippocampus, amygdala, and putamen, as parameterized surfaces and quantifying their shape differences using an elastic shape metric. Under this metric, we compute shape summaries (mean, covariance, PCA) of brain sub-structures and represent individual brain shapes by their principal scores under a shape-PCA basis. These representations are rich enough to allow visualizations of full 3D structures and help understand localized changes. In order to validate the elastic shape analysis, we use the principal components (PCs) to reconstruct the brain structures and perform further evaluation by performing a regression analysis to model PTSD and trauma severity using the brain shapes represented
via PCs and in conjunction with auxiliary exposure variables. We apply our method to data from the Grady Trauma Project (GTP), where the goal is to predict clinical measures of PTSD. The framework seamlessly integrates accurate morphological features and other clinical covariates to yield superior predictive performance when modeling PTSD outcomes. Compared to vertex-wise analysis and other widely applied shape analysis methods, the elastic shape analysis approach results in considerably higher reconstruction accuracy for the brain shape and reveals significantly greater predictive power. It also helps identify local deformations in brain shapes associated with PTSD severity. -
null (Ed.)We carry out a structural and algorithmic study of a mobile sensor coverage optimization problem targeting 2D surfaces embedded in a 3D workspace. The investigated settings model multiple important applications including camera net- work deployment for surveillance, geological monitoring/survey of 3D terrains, and UVC-based surface disinfection for the prevention of the spread of disease agents (e.g., SARS-CoV- 2). Under a unified general “sensor coverage” problem, three concrete formulations are examined, focusing on optimizing visibility, single-best coverage quality, and cumulative quality, respectively. After demonstrating the computational intractabil- ity of all these formulations, we describe approximation schemes and mathematical programming models for near-optimally solving them. The effectiveness of our methods is thoroughly evaluated under realistic and practical scenarios.more » « less
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Abstract Protein‐protein interactions are critical to protein function, but three‐dimensional (3D) arrangements of interacting proteins have proven hard to predict, even given the identities and 3D structures of the interacting partners. Specifically, identifying the relevant pairwise interaction surfaces remains difficult, often relying on shape complementarity with molecular docking while accounting for molecular motions to optimize rigid 3D translations and rotations. However, such approaches can be computationally expensive, and faster, less accurate approximations may prove useful for large‐scale prediction and assembly of 3D structures of multi‐protein complexes. We asked if a reduced representation of protein geometry retains enough information about molecular properties to predict pairwise protein interaction interfaces that are tolerant of limited structural rearrangements. Here, we describe a reduced representation of 3D protein accessible surfaces on which molecular properties such as charge, hydrophobicity, and evolutionary rate can be easily mapped, implemented in the MorphProt package. Pairs of surfaces are compared to rapidly assess partner‐specific potential surface complementarity. On two available benchmarks of 185 overall known protein complexes, we observe predictions comparable to other structure‐based tools at correctly identifying protein interaction surfaces. Furthermore, we examined the effect of molecular motion through normal mode simulation on a benchmark receptor‐ligand pair and observed no marked loss of predictive accuracy for distortions of up to 6 Å Cα‐RMSD. Thus, a shape reduction of protein surfaces retains considerable information about surface complementarity, offers enhanced speed of comparison relative to more complex geometric representations, and exhibits tolerance to conformational changes.
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Accepted Manuscript1.0
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