Offset surfaces, defined as the Minkowski sum of a base surface and a rolling ball, play a crucial role in geometry processing, with applications ranging from coverage motion planning to brush modeling. While considerable progress has been made in computing constant radius offset surfaces, computing variable-radius offset surfaces re mains a challenging problem. In this paper, we present OffsetCrust, a novel framework that efficiently addresses the variable-radius off setting problem by computing a power diagram. Let R denote the radius function defined on the base surface S. The power diagram is constructed from contributing sites, consisting of carefully sampled base points on S and their corresponding off-surface points, displaced along R-dependent directions. In the constant-radius case only, these displacement directions align exactly with the surface normals of S. Moreover, our method mitigates the misalignment issues commonly seen in crust-based approaches through a lightweight fine-tuning procedure. We validate the accuracy and efficiency of OffsetCrust through extensive experiments, and demonstrate its practical utility in applications such as reconstructing original boundary surfaces from medial axis transform (MAT) representations.

Instrumental methods for measuring hair volume can detect physical differences that are not perceptible to consumers, creating a disconnect between in-vitro measurements and real-world perception. This study aimed to establish the Just-Noticeable Difference (JND) for hair volume and to develop a predictive model incorporating hair waviness to better align instrumental data with sensory evaluation. Bulk volume and waviness were quantified from hair swatches using a custom image-analysis framework. Volume was determined via pixel intensity-based fibre density classification, while waviness was derived from an amplitude-frequency analysis of the detrended medial axis. Sensory evaluation was conducted by an expert panel across controlled four-swatch sets with defined volume differences (ΔV ≈ 5, 3 and 2 cm2). Mean sensory rank was used as the primary perceptual metric. A multivariable Ordinary Least Squares regression model was developed and validated using Leave-One-Group-Out Cross-Validation across 35 independent sets. A perceptual threshold of ΔV ≈ 3 cm2 was identified as the JND for hair volume. At ΔV ≈ 5 cm2, instrumental and sensory rankings were in strong agreement, whereas at ΔV ≈ 2 cm2, perceptual discrimination failed. Regression analysis confirmed bulk volume as the dominant predictor of perceived volume (β = 0.798, p < 0.001), with waviness providing a significant secondary contribution (β = 0.135, p = 0.005; Adjusted R2 = 0.660). A waviness correction factor (k ≈ 0.169) was derived, and cross-validation demonstrated robust predictive performance (66.67% top-rank accuracy; 94.87% top-two accuracy). This study establishes the quantitative perceptual threshold for hair volume and introduces a morphology-aware correction model that aligns instrumental measurements with human perception. Together, these tools provide a practical framework for ensuring that volumising product claims are both physically measurable and perceptually meaningful.

A predictive model for birth delivery mode utilizing shape characteristics of uterine contractions from electrohysterographic signals and obstetric parameters

Cystodiscus nigromaculatus sp. n. (Myxozoa: Myxidiidae) infecting the gallbladder of the frog Pelophylax nigromaculatus in China.

Lifestyle variables, including physical activity and diet, are key determinants of bone remodelling. However, remodelling is also spatially limited by the anatomical form of bone. The extent to which these macroscopic, microscopic and lifestyle variables co-vary allometrically in humans is subject to ongoing investigation. We hypothesised that femur midshaft size and biomechanical properties would have a dimensional effect on the size and density of secondary osteons produced during remodelling, independently of age and sex, and that this effect would manifest under different lifestyle conditions. We examined femur midshaft microradiographs in 73 samples part of the Melbourne Femur Research Collection (Australia). We measured cortical/total area (CA/TA) and the ratio of axes of the largest and smallest femur rigidity areas (Imax/Imin); secondary osteon area (On.Ar), the ratio of Haversian canal to On.Ar (H.Ar/On.Ar), and total population density of intact and fragmentary (partially remodelled) secondary osteons (OPD) from the anterior, posterior, medial and lateral anatomical femur axes. Out of all the bone parameters, CA/TA and H.Ar/On.Ar showed low to high negative correlations (p < 0.001, r range - 0.323 to -0.752) and negative allometry relationships in the entire sample, in males and in a sub-group that consisted of individuals who were sedentary but well nourished. In these groups, higher log values of CA/TA were associated with smaller log values of H.Ar/On.Ar, which translated to femoral midshafts with thicker cortices having less porous secondary osteons (smaller Haversian canals relative to thicker lamellar bone). However, this relationship was not evident in females and other age and lifestyle groupings. We suggest that effects of allometry on bone histology, or even basic variables such as cortical thickness, are included in future remodelling and lifestyle assessments to ensure that histological interpretations are not confounded by bone size.

Medial epicanthoplasty revision is required when prior surgery results in caruncle overexposure or an unnaturally harsh appearance. Achieving a natural medial canthal shape with minimal visible scar remains a surgical challenge. We introduce the Epicanthal mini-redraping reconstruction (EC mini-recon) technique, which involves V-Y advancement of posterior skin lining of the epicanthal fold followed by anterolateral redraping of overlying skin. This retains natural canthal angulation and minimizes the final scar created along relaxed skin tension line (RSTL). From October 2018 to February 2025, 32 patients underwent EC mini-recon, with interepicanthal distance (IeCD) or midline-to-medial canthus distance (MeCD) measured preoperatively and postoperatively using standardized photographs. In bilateral cases (n = 16), mean IeCD ratio increased from 2.95 to 3.11 (p < 0.01). In unilateral cases (n = 16), mean MeCD ratio increased from 1.37 to 1.45 (p < 0.01). No major complications were observed. Two patients (6.25%) required reoperation, and 90.6% of patients expressed satisfaction with the results. Scarring was minimal. In patients with hook-shaped medial canthi, the medial canthal axis was successfully rotated by modifying suture point during the V-Y advancement. EC mini-recon can be an intuitive, technically straightforward approach for medial epicanthoplasty revision which maintains the natural canthal angulation while resulting in acceptable scar. This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

Transferable and undefendable point cloud attacks via medial axis transform

The medial axis, a lower-dimensional descriptor that captures the extrinsic structure of a shape, plays an important role in digital geometry processing. Despite its importance, computing the medial axis transform robustly from diverse inputs, especially point clouds with defects, remains a challenging problem. In this paper, we propose a new implicit method that deviates from traditional explicit medial axis computation. Our key technical insight is that the difference between the signed distance field (SDF) and the medial field (MF) of a solid shape relates to the unsigned distance field (UDF) of the shape’s medial axis. This observation allows us to formulate medial axis extraction as an implicit reconstruction problem. By employing a modified double covering strategy, we recover the medial axis as the zero level-set of the UDF. Extensive experiments demonstrate that our method achieves higher accuracy and robustness in learning compact medial axis transforms from challenging meshes and point clouds, outperforming existing approaches.

Abstract We show that the (multi)function of the closest points is locally Lipschitz outside of the medial axis of a closed set $$X\subset \mathbb {R}^n$$ X ⊂ R n . With this result, we prove that the medial axis of X approaches every point where X is not Lipschitz normally embedded.

Branches of the tibial nerve in the foot of fetal cadavers.

BackgroundAcetabular impaction bone grafting (IBG) is used to address bone loss in revision total hip arthroplasty (rTHA). We evaluated graft incorporation and cup migration after acetabular IBG in rTHA.MethodsSystematic search of MEDLINE, EMBASE, and Scopus from inception to June 30, 2024 (PROSPERO CRD42024557047). Studies of acetabular IBG in rTHA with ≥12-month follow-up were included. Outcomes were graft incorporation and horizontal (i.e., lateral to medial axis) and vertical cup migration. Prespecified subgroup analyses assessed bone-loss severity, graft type, additional fixation, and age. Random-effects meta-analyses were used; heterogeneity was quantified with I2. Risk of bias was assessed with the Methodological Index for Non-Randomized Studies.ResultsNineteen studies (1093 hips) were included; weighted follow-up was 8.0 years (range 2.0-16.9). Pooled graft incorporation was 89% (95% CI [confidence interval] 79-96; I2 85%). Mean lateral migration was 2.4 mm (95% CI 0.53-4.27) and mean superior migration 4.2 mm (95% CI 1.61-6.75); heterogeneity was high (I2 100% for both). Lateral migration was greater in cohorts with ≥5 years’ follow-up than <5 years (3.65 vs 1.16 mm; P = .018). No significant differences in incorporation were detected by bone-loss severity, graft type, or age; however, subgroup analyses were frequently underpowered and exploratory.ConclusionsAcetabular IBG can achieve mid- to long-term graft incorporation of 89%, but estimates are imprecise due to heterogeneous (I2 = 85%), predominantly retrospective evidence. Migration occurs—most commonly cranial—and may accrue over time; values often remain within commonly accepted surveillance thresholds but warrant structured radiographic follow-up. Given high heterogeneity, variable outcome definitions, and potential small-study/publication bias, recommendations should be interpreted cautiously.

In the manufacturing of electric motors, the quality of wire windings is a critical factor determining motor performance and reliability. Defective windings typically result in motor rejection during production, necessitating effective quality control methods. Non-destructive evaluation via computed tomography (CT) offers a promising approach for assessing internal components, although data analysis presents significant challenges. In this study, we introduce a novel method to extract the complete 6D pose of stator windings from CT measurements. The method applies a topology-preserving thinning algorithm to reduce segmented wires to their medial axis, from which positional and curvature data are derived. Local twist about the wire’s axis is quantified by analysing image moments in orthogonal cross-sections along the winding. Validation with synthetic CT data and physical motor replicas demonstrates high precision in estimating wire positions and rotations, including intricate bending and twisting behaviours. Future work will focus on optimizing the method to enhance its robustness against the noise inherent in real-world CT scans.

A distance-based algorithm approximating the medial axis transform for connected, open, bounded regions in the complex plane

To evaluate the performance of a novel orthogonal minimum joint space width, defined as the narrowest distance measured orthogonally to the tibial joint line within a standardised medial axis, for detecting and monitoring knee osteoarthritis progression. This artificial intelligence-derived metric, obtained through deep learning, was compared with conventional fixed-location joint space width measures. A total of 15,313 knee radiographs from the Osteoarthritis Initiative (OAI), spanning baseline to 72-month follow-up, were analysed in this retrospective cohort study. Orthogonal minimum joint space width was automatically measured using a deep learning model. Discriminative ability for cross-sectional joint space narrowing severity was assessed using the area under the receiver operating characteristic curve. Longitudinal responsiveness was evaluated with the standardised response mean and relative standardised response mean, computed from pooled 12-month intervals and stratified by baseline narrowing grade. Variability was estimated through 1000 bootstrap resamples. Performance was compared with fixed-location measures at 22.5% and 25.0% from the medial edge of the tibial plateau. Orthogonal minimum joint space width demonstrated strong discriminative performance, with area under the curve values of 0.86, 0.95 and 0.97 for joint space narrowing greater than grades 0, 1 and 2, respectively - exceeding the corresponding values for fixed-location measures by absolute margins of 0.04-0.08. Pooled relative standardised response mean analysis showed consistently higher responsiveness for orthogonal minimum joint space width, with median values of 0.97, 0.95 and 0.91 for baseline narrowing grades 0, 1 and 2, representing approximately 2%-4% greater responsiveness than fixed-location measures across all strata. Orthogonal minimum joint space width, automatically quantified by deep learning, enhances both sensitivity and specificity for assessing knee osteoarthritis severity and progression. This metric may serve as a robust and reproducible structural imaging endpoint for clinical trials and longitudinal research. Level II.

Pore network models are useful for studying transport in porous materials in a computationally efficient way. Extraction of networks from volumetric images has evolved over the years, starting with medial axis-based approaches to more recent watershed segmentation. This paper reconsiders the classic medial axis method, which offers several advantages such as speed and topological correctness, and develops a modernized, updated, and improved version. The new method is named Medial Axis Guided Network Extraction Tool (MAGNET). It works by analyzing the skeleton of a porous material to identify pore centers at junctions and endpoints. Additional pore bodies are found on long throats using two different approaches. This work includes an efficient tool for calculating the cross-sectional area of throats with irregular shape by using walkers with an infinite mean-free path to probe the geometry orthogonal to the medial axis at the point of the throat constriction. This extra step was critical for obtaining an equivalent diameter needed to calculate the permeability. Lastly, MAGNET was written with computational efficiency in mind. The skeletonization approach was itself 4.2X faster than the SNOW watershed segmentation for a 10003 image. Additionally, a parallelized skeletonization was applied by processing the image in blocks with sufficient overlap which resulted in a 5.5X speed-up compared to the serial approach. To validate the output, MAGNET was tested on a 4003 voxel image of a Berea sandstone, and the flow and capillary properties of the extracted network were compared to the results from SNOW and the lattice-Boltzmann method. Structural information such as pore and throat size distribution and mercury intrusion curves was compared, and noticeable similarity was achieved. Crucially, the permeability predicted by MAGNET was within 5% of the lattice-Boltzmann prediction on the same image.

Recent advances in 3D X-ray Computed Tomography (CT) sensors have stimulated research efforts to unveil the extremely complex micro-scale processes that control the activity of soil microorganisms. Classical methods for the numerical simulation of biological dynamics using meshes of voxels, such as the Lattice Boltzmann Method (LBM), tend to require long computation times. The use of more compact geometrical representations of the pore space can drastically decrease the computational cost of simulations. Recent research has introduced basic analytic volume primitives to define piece-wise approximations of the pore space to simulate drainage, diffusion, and microbial mineralization of organic matter in soils. Such approaches work well but a drawback is that they give rise to significant approximation errors caused by imposing a priori shapes to represent the pores. In the present article, another alternative is proposed, where pore space is described by means of geometrically relevant connected subsets of voxels (regions) regrouped on the basis of the curve skeleton (3D medial axis). The curve skeleton has been adopted to characterize 3D shapes in various fields (e.g., medical imaging, material sciences, etc.). The few publications that have used it in the context of soils have dealt exclusively with the determination of pore throats. This technique is used mostly to describe shape and not to partition it into connected subsets like in the present work. Here, the pore space is partitioned by using the branches of the curve skeleton, then an Attributed Relational Graph (ARG) is created in order to simulate numerically the microbial mineralization of organic matter, including the diffusion of by-products. Each node of the ARG is attached to an element of the partition (pore) and each arc to an adjacency relationship between pores (connectivity). The graph is valuated in the sense that the attributes related both to geometry and dynamic are linked to nodes and arcs. This new representation can be used for graph-based simulations, which are different from voxel-based simulations.

This study presents a computational framework for optimising a hip implant through a functionally graded biomimetic lattice structure, designed to reduce stress shielding. The optimisation technique, inspired by an inverse bone remodelling algorithm, promotes even stress distribution by reducing density and stiffness in regions with high strain energy compared to a reference level. The resulting non-uniform density distribution showed lower density levels along the implant stem's sides and higher density around its medial axis. This optimised material distribution was captured using mapping of a triply periodic minimal surface lattice structure on the implant, creating porous lattice surfaces within the solid structure. The porous implant's performance was evaluated using a finite element bone remodelling algorithm, comparing its bone response to a femur with a fully solid implant model, in terms of stress distribution and mass change. Results demonstrated improved bone formation at the bone-implant interface and enhanced stress transmission to the surrounding bone.

Extraction of a high-fidelity 3D medial axis is a crucial operation in CAD. When dealing with a polygonal model as input, ensuring accuracy and tidiness becomes challenging due to discretization errors inherent in the mesh surface. Commonly, existing approaches yield medial-axis surfaces with various artifacts, including zigzag boundaries, bumpy surfaces, unwanted spikes, and non-smooth stitching curves. Considering that the surface of a CAD model can be easily decomposed into a collection of surface patches, its 3D medial axis can be extracted by computing the Voronoi diagram of these surface patches, where each surface patch serves as a generator. However, no solver currently exists for accurately computing such an extended Voronoi diagram. Under the assumption that each generator defines a linear distance field over a sufficiently small range, our approach operates by tetrahedralizing the region of interest and computing the medial axis within each tetrahedral element. Just as SurfaceVoronoi computes surface-based Voronoi diagrams by cutting a 3D prism with 3D planes (each plane encodes a linear field in a triangle), the key operation in this paper is to conduct the hyperplane cutting process in 4D, where each hyperplane encodes a linear field in a tetrahedron. In comparison with the state-of-the-art, our algorithm produces better outcomes. Furthermore, it can also be used to compute the offset surface.

Ribbon-based multi-sided parametric patches represent a special group of free-form surfaces. While ribbons with boundary and cross-derivative constraints can sufficiently define patches, modifying their interior – in addition – may be required, as well. We introduce a new technique to add interior control structures for Generalized B-spline (GBS) patches defined over multi-connected curved domains, where the sum of the blending functions produces a weight deficiency. A local parameterization and a parametric MAT (medial axis transform) structure is computed within the domain, which leads to a family of quad structures, called templates. We define blend functions by distributing weight deficiency amongst the vertices of the template and describe algorithms to locate related control points in 3D, yielding Template GBS patches that tightly approximate the input ribbons. As a final step, a modified representation is created, called Hybrid GBS, that exactly interpolates the given input ribbons and inherits the interior control structure of an arbitrarily chosen Template GBS patch. • Interior control structure for multi-sided Generalized B-spline patches. • Support for multiply connected domains. • Quadrilateral templates from parametric medial axis. • Special blending functions associated with the template geometry. • Surfaces based on approximating and exact ribbons.

PurposeThe aim of this study was to quantify the difference in posterior tibial slope (PTS) in patients with anterior cruciate ligament (ACL) deficiency compared to those with a degenerative medial meniscus posterior horn tear (dMMPHT). We hypothesised that patients with ACL deficiency would have a greater PTS compared to patients with dMMPHT.MethodsAll consecutive patients diagnosed with ACL deficiency (ACL group) or a dMMPHT with an intact ACL (MM group) from 2016 to 2022 at a single centre were reviewed. PTS was measured using the medial tibial plateau and tibial mechanical axis on standard lateral knee radiographs. Linear regression analysis was completed to assess the correlation between PTS and age in both groups. Further sub‐group analysis was undertaken according to age and sex of the study population.ResultsIn total, 294 patients in the ACL group and 250 patients in the MM group were analysed. Age differed between the groups (ACL group: 29.5 ± 10.4 years vs. MM group: 38.7 ± 8.5 years, p < 0.001). Radiological evaluation demonstrated increased mean PTS in the ACL group compared to the MM group (10.1° ± 3.0 vs. 6.7° ± 2.6, p < 0.001). Regression analysis showed no significant correlation between the PTS and age in each group. When patients were subdivided into those younger than 30 years of age and those 30 years or older, there was no significant difference in the PTS in the ACL group. In contrast, younger patients had a steeper PTS compared to those 30 years or older in the MM group (7.5° ± 1.9 vs. 6.5° ± 2.7, p = 0.005).ConclusionsPatients with an ACL tear had a significantly higher PTS on standard lateral knee radiographs compared to those with dMMPHTs and a normal ACL. Elevated PTS may be a risk factor for ACL rupture.Level of EvidenceLevel III, retrospective cohort study.