Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Discrete Curvature and Torsion from Cross-Ratios

Motivated by a M\"obius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular M\"obius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a M\"obius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature

OReX: Object Reconstruction from Planner Cross-sections Using Neural Fields

Reconstructing 3D shapes from planar cross-sections is a challenge inspired by downstream applications like medical imaging and geographic informatics. The input is an in/out indicator function fully defined on a sparse collection of planes in space, and the output is an interpolation of the indicator function to the entire volume. Previous works addressing this sparse and ill-posed problem either produce low quality results, or rely on additional priors such as target topology, appearance information, or input normal directions. In this paper, we present OReX, a method for 3D shape reconstruction from slices alone, featuring a Neural Field as the interpolation prior. A simple neural network is trained on the input planes to receive a 3D coordinate and return an inside/outside estimate for the query point. This prior is powerful in inducing smoothness and self-similarities. The main challenge for this approach is high-frequency details, as the neural prior is overly smoothing. To alleviate this, we offer an iterative estimation architecture and a hierarchical input sampling scheme that encourage coarse-to-fine training, allowing focusing on high frequencies at later stages. In addition, we identify and analyze a common ripple-like effect stemming from the mesh extraction step. We mitigate it by regularizing the spatial gradients of the indicator function around input in/out boundaries, cutting the problem at the root. Through extensive qualitative and quantitative experimentation, we demonstrate our method is robust, accurate, and scales well with the size of the input. We report state-of-the-art results compared to previous approaches and recent potential solutions, and demonstrate the benefit of our individual contributions through analysis and ablation studies

BPM: Blended Piecewise Moebius Maps

We propose a novel Moebius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a smooth map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi-conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed-form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi-conformal errors

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes

Single breath-hold 3D measurement of left atrial volume using compressed sensing cardiovascular magnetic resonance and a non-model-based reconstruction approach

Background:Left atrial (LA) dilatation is associated with a large variety of cardiac diseases. Current cardiovascular magnetic resonance (CMR) strategies to measure LA volumes are based on multi-breath-hold multi-slice acquisitions, which are time-consuming and susceptible to misregistration.Aim:To develop a time-efficient single breath-hold 3D CMR acquisition and reconstruction method to precisely measure LA volumes and function.Methods:A highly accelerated compressed-sensing multi-slice cine sequence (CS-cineCMR) was combined with a non-model-based 3D reconstruction method to measure LA volumes with high temporal and spatial resolution during a single breath-hold. This approach was validated in LA phantoms of different shapes and applied in 3 patients. In addition, the influence of slice orientations on accuracy was evaluated in the LA phantoms for the new approach in comparison with a conventional model-based biplane area-length reconstruction. As a reference in patients, a self-navigated high-resolution whole-heart 3D dataset (3D-HR-CMR) was acquired during mid-diastole to yield accurate LA volumes.Results:Phantom studies. LA volumes were accurately measured by CS-cineCMR with a mean difference of −4.73 ± 1.75 ml (−8.67 ± 3.54 %, r² = 0.94). For the new method the calculated volumes were not significantly different when different orientations of the CS-cineCMR slices were applied to cover the LA phantoms. Long-axis “aligned” vs “not aligned” with the phantom long-axis yielded similar differences vs the reference volume (−4.87 ± 1.73 ml vs −4.45 ± 1.97 ml, p = 0.67) and short-axis “perpendicular” vs “not-perpendicular” with the LA long-axis (−4.72 ± 1.66 ml vs −4.75 ± 2.13 ml; p = 0.98). The conventional bi-plane area-length method was susceptible for slice orientations (p = 0.0085 for the interaction of “slice orientation” and “reconstruction technique”, 2-way ANOVA for repeated measures). To use the 3D-HR-CMR as the reference for LA volumes in patients, it was validated in the LA phantoms (mean difference: −1.37 ± 1.35 ml, −2.38 ± 2.44 %, r² = 0.97). Patient study: The CS-cineCMR LA volumes of the mid-diastolic frame matched closely with the reference LA volume (measured by 3D-HR-CMR) with a difference of −2.66 ± 6.5 ml (3.0 % underestimation; true LA volumes: 63 ml, 62 ml, and 395 ml). Finally, a high intra- and inter-observer agreement for maximal and minimal LA volume measurement is also shown.Conclusions:The proposed method combines a highly accelerated single-breathhold compressed-sensing multi-slice CMR technique with a non-model-based 3D reconstruction to accurately and reproducibly measure LA volumes and function

Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches

In the present paper, crack initiation in mixed-mode (I+II) fracture has been studied using notched circular ring specimens. A new criterion of brittle mixed-mode (I+II) fracture based on the notch tangential stress and the volumetric approach has been developed. The critical value of the equivalent notch stressintensity factor has been considered as fracture toughness in mixed-mode (I+II) fracture.Исследуется зарождение трещины по смешанному механизму разрушения (типа I+II) в образцах кольцевого типа с внутренним надрезом. Предложен новый критерий для описания хрупкого разрушения смешанного типа I+II, в основу которого положен объемный подход, а базовым параметром служит касательное напряжение в надрезе. Предлагается в качестве параметра вязкости разрушения для смешанного механизма разрушения по типу I+II использовать предельное значение эквивалентного коэффициента интенсивности напряжений в надрезе.Досліджується зародження тріщини за змішаним механізмом руйнування (типу І+ІІ) в зразках кільцевого типу з внутрішнім надрізом. Запропоновано новий критерій для описання крихкого руйнування змішаного типу І+ІІ, в основу якого покладено об’ємний підхід, а базовим параметром є дотичне напруження у надрізі. За параметр в ’язкості руйнування для змішаного механізму руйнування за типом І+ІІ пропонується використовувати граничне значення еквівалентного коефіцієнта інтенсивності напружень у надрізі