Spherical Patches Research Articles

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

A Cross-matching Service for Data Center of Xinjiang Astronomical Observatory

Cross-matching is a key technique to achieve fusion of multi-band astronomical catalogs. Due to different equipment such as various astronomical telescopes, the existence of measurement errors, and proper motions of the celestial bodies, the same celestial object will have different positions in different catalogs, making it difficult to integrate multi-band or full-band astronomical data. In this study, we propose an online cross-matching method based on pseudo-spherical indexing techniques and develop a service combining with high performance computing system (Taurus) to improve cross-matching efficiency, which is designed for the Data Center of Xinjiang Astronomical Observatory. Specifically, we use Quad Tree Cube to divide the spherical blocks of the celestial object and map the 2D space composed of R.A. and decl. to 1D space and achieve correspondence between real celestial objects and spherical patches. Finally, we verify the performance of the service using Gaia 3 and PPMXL catalogs. Meanwhile, we send the matching results to VO tools-Topcat and Aladin respectively to get visual results. The experimental results show that the service effectively solves the speed bottleneck problem of cross-matching caused by frequent I/O, and significantly improves the retrieval and matching speed of massive astronomical data.

Analytical formulae for the projected solid angle and differential configuration factor of arbitrary polygons

Radiative heat transfer often requires the calculation of projected solid angle or configuration (view) factors, which are closely related. Closed form solutions for the computation of the solid angle from polygonal cross-sections are well known, however similar formulae for computation of projected solid angle are not generally available. Formulae for computing the projected solid angle from arbitrarily shaped polygons are derived using the Gauss-Bonnet theorem. This is accomplished by transforming the projected solid angle integral to an integral over a spherical patch, which is then reduced by Gauss-Bonnet to a simple summation over its edges, allowing the projected solid angle to be computed exactly. Application of the formulae allows exact calculation of projected solid angle over discrete intervals which may be used for computing radiative flux to surfaces or view factors from differential elements.

Non-Rational and Rational Transfinite Interpolation Using Bernstein Polynomials

It is shown that the standard transfinite interpolation in quadrilateral patches may be written in terms of two sets of Bernstein polynomials. The former set is of the same degree with the corresponding blending functions while the latter is of the same degree with the Lagrange polynomials which operate like trial functions along each of the four edges as well as the additional inter-boundaries of the patch. The replacement of the Lagrange polynomials by the Bernstein ones allows the use of control points useful for design. Also it allows the determination of weights that may ensure accurate representation of quadric surfaces including those of revolution (spheres, ellipsoids, hyperboloids, etc). The presentation restricts to the standard Gordon formulation, which refers to structured stencils, similar to rectangular plates reinforced with longitudinal and transverse stiffeners. As an example, the proposed formula is applied to the geometric representation of a cylindrical and a spherical patch. In the latter case a nonlinear programming optimization technique was applied, starting with initial data pertinent to a tensor product surface, from which original weights and control points were obtained for the accurate representation of a spherical cap. In addition, the involved shape functions were successfully applied to the numerical solution of a boundary value problem.

Spherical Transformer on Cortical Surfaces.

Motivated by the recent great success of attention modeling in computer vision, it is highly desired to extend the Transformer architecture from the conventional Euclidean space to non-Euclidean spaces. Given the intrinsic spherical topology of brain cortical surfaces in neuroimaging, in this study, we propose a novel Spherical Transformer, an effective general-purpose backbone using the self-attention mechanism for analysis of cortical surface data represented by triangular meshes. By mapping the cortical surface onto a sphere and splitting it uniformly into overlapping spherical surface patches, we encode the long-range dependency within each patch by the self-attention operation and formulate the cross-patch feature transmission via overlapping regions. By limiting the self-attention computation to local patches, our proposed Spherical Transformer preserves detailed contextual information and enjoys great efficiency with linear computational complexity with respect to the patch size. Moreover, to better process longitudinal cortical surfaces, which are increasingly popular in neuroimaging studies, we unprecedentedly propose the spatiotemporal self-attention operation to jointly extract the spatial context and dynamic developmental patterns within a single layer, thus further enlarging the expressive power of the generated representation. To comprehensively evaluate the performance of our Spherical Transformer, we validate it on a surface-level prediction task and a vertex-level dense prediction task, respectively, i.e., the cognition prediction and cortical thickness map development prediction, which are important in early brain development mapping. Both applications demonstrate the competitive performance of our Spherical Transformer in comparison with the state-of-the-art methods.

Smooth multifocal wavefronts with a prescribed mean curvature for visual optics applications.

Multifocal lenses comprising progressive power surfaces are commonly used in contact and intraocular lens designs. Given a visual performance metric, a wavefront engineering approach to design such lenses is based on searching for the optimal wavefront at the exit pupil of the eye. Multifocal wavefronts distribute the energy along the different foci thanks to having a varying mean curvature. Therefore, a fundamental step in the wavefront engineering approach is to generate the wavefront from a prescribed mean curvature function. Conventionally, such a thing is done by superimposing spherical wavefront patches and maybe adding a certain component of spherical aberration to each spherical patch in order to increase the depth-of-field associated with each focus. However, such a procedure does not lead to smooth wavefront solutions and also restricts the type of available multifocal wavefronts. We derive a new, to the best of our knowledge, mathematical method to uniquely construct multifocal wavefronts from mean curvature functions (depending on radial and angular coordinates) under certain numerically justified approximations and restrictions. Additionally, our procedure leads to a particular family of wavefronts (line-umbilical multifocal wavefronts) described by 2 conditions: (1) to be smooth multiplicative separable functions in the radial and angular coordinates; (2) to be umbilical along a specific segment connecting the circle center with its edge. We provide several examples of multifocal wavefronts belonging to this family, including a smooth variant of the so-called light sword element.

Design and implementation of I‐slot conformal circular patch antenna on sphere structure

In this article, an I-slot circular patch antenna conformed on a spherical structure is proposed. For the implementation of the proposed antenna, the polylactic acid (PLA) material is used with the help of a 3D printer. In accordance with this purpose, we determine the permittivity value of the PLA with a T-resonator method before the design process. Then, the geometrical design of the spherical patch antenna is optimized by means of a differential evolution algorithm. After the design and optimization process, the conformal spherical patch antenna is fabricated on a sphere cap. The main characteristics of the proposed antenna are given for both the simulations and the measurements. It is seen that the simulation results agree well with the measurement results. Furthermore, an experiment is performed in order to examine the application possibility of the presented spherical antenna. For this experiment, a spherical communication module with the proposed antenna and an inner wireless transmitter system is built. In this experiment, the spherical module is rolled down a slope while it continuously transmits pseudo-random numbers. The data are received by a receiver station at the top of the slope. The received data are evaluated in terms of the amount of received packets per unit of time. The numerical results indicate that the proposed antenna can be used for dynamic spherical systems such as throwable objects and spherical robots.

Spherical-Patches Extraction for Deep-Learning-Based Critical Points Detection in 3D Neuron Microscopy Images.

Digital reconstruction of neuronal structures is very important to neuroscience research. Many existing reconstruction algorithms require a set of good seed points. 3D neuron critical points, including terminations, branch points and cross-over points, are good candidates for such seed points. However, a method that can simultaneously detect all types of critical points has barely been explored. In this work, we present a method to simultaneously detect all 3 types of 3D critical points in neuron microscopy images, based on a spherical-patches extraction (SPE) method and a 2D multi-stream convolutional neural network (CNN). SPE uses a set of concentric spherical surfaces centered at a given critical point candidate to extract intensity distribution features around the point. Then, a group of 2D spherical patches is generated by projecting the surfaces into 2D rectangular image patches according to the orders of the azimuth and the polar angles. Finally, a 2D multi-stream CNN, in which each stream receives one spherical patch as input, is designed to learn the intensity distribution features from those spherical patches and classify the given critical point candidate into one of four classes: termination, branch point, cross-over point or non-critical point. Experimental results confirm that the proposed method outperforms other state-of-the-art critical points detection methods. The critical points based neuron reconstruction results demonstrate the potential of the detected neuron critical points to be good seed points for neuron reconstruction. Additionally, we have established a public dataset dedicated for neuron critical points detection, which has been released along with this article.

Energy harvesting for electrochemical OER and solar photocatalysis via dual functional GO/TiO2-NiO nanocomposite

Highly efficient dual functional electrochemical and photocatalysts by incorporation of GO and TiO2 doped NiO composites have been developed. The prepared catalysts were subjected toward oxygen evolution reaction (OER) in electro catalytic water splitting and photocatalytic degradation of bromophenol blue (BPB). TiO2 doped NiO having monoclinic phase with GO is confirmed by XRD profile. The introduction of TiO2 into NiO matrix and addition of GO has reduced intensity of XRD peaks in composites. The morphological structures confirm fluffy NiO with small spherical dark patches of TiO2 nanoparticles which are dispersed randomly on broken sheets of GO in composite structure. UV–Vis spectra illustrated obvious band gap reduction (∼2.0–2.6 eV) in various GO/Ti-NiO composites with improvement in visible light absorption. The generation of Ti−Ni−C junction in composite samples is examined by FTIR and Raman analysis. In OER (1M KOH) 4.0 wt. % GO/Ti-NiO generated the small Tafel slope of ∼60 mVdec−1, low Rct∼4.1 Ω and high current density ∼70 mAcm−2 as compared to other samples. The composite materials govern OER process by Volmer-Heyrovsky mechanism. GO/Ti-NiO composite with 4.0 wt. % Ti shows the highest degradation (∼95 % )of BPB due to active surface sites generated by Ti−Ni−C linkages and interconnected conductive carbon materials of GO. Defective structure of GO provides favorable kinetics to the electro catalytic OER and supportive in enhancement of the electronic conduction. Ni2+ offers more active sites along with Ti4+ for OH−adsorption. Intermediate species of BPB have been identified by HPLC analysis. This work brings new insight for designing multifunctional materials for energy conversion and environmental remediation applications.

Computing interface curvature from volume fractions: A machine learning approach

The volume of fluid (VOF) method is widely used to simulate the flow of immiscible fluids. It uses a discrete and sharp volume fractions field to represent the fluid-fluid interface on a Eulerian grid. The most challenging part of the VOF method is the accurate computation of the local interface curvature which is essential for evaluation of the surface tension force at the interface. In this paper, a machine learning approach is used to develop a model which predicts the local interface curvature from neighbouring volume fractions. A novel data generation methodology is devised which generates well-balanced randomized data sets comprising of spherical interface patches of different configurations/orientations. A two-layer feed-forward neural network with different network parameters is trained on these data sets and the developed models are tested for different shapes i.e. ellipsoid, 3D wave and Gaussian. The best model is selected on the basis of specific criteria and subsequently compared with conventional curvature computation methods (convolution and height function) to check the nature and grid convergence of the model. The model is also coupled with a multiphase flow solver to evaluate its performance using standard test cases: i) stationary bubble, ii) oscillating bubble and iii) rising bubble under gravity. Our results demonstrate that machine learning is a feasible approach for fairly accurate curvature computation. It easily outperforms the convolution method and even matches the accuracy of the height function method for some test cases.

Noise reduction in diffusion MRI using non-local self-similar information in joint x-q space.

Diffusion MRI affords valuable insights into white matter microstructures, but suffers from low signal-to-noise ratio (SNR), especially at high diffusion weighting (i.e., b-value). To avoid time-intensive repeated acquisition, post-processing algorithms are often used to reduce noise. Among existing methods, non-local means (NLM) has been shown to be particularly effective. However, most NLM algorithms for diffusion MRI focus on patch matching in the spatial domain (i.e., x-space) and disregard the fact that the data live in a combined 6D space covering both spatial domain and diffusion wavevector domain (i.e., q-space). This drawback leads to inaccurate patch matching in curved white matter structures and hence the inability to effectively use recurrent information for noise reduction. The goal of this paper is to overcome this limitation by extending NLM to the joint x-q space. Specifically, we define for each point in the x-q space a spherical patch from which we extract rotation-invariant features for patch matching. The ability to perform patch matching across q-samples allows patches from differentially orientated structures to be used for effective noise removal. Extensive experiments on synthetic, repeated-acquisition, and HCP data demonstrate that our method outperforms state-of-the-art methods, both qualitatively and quantitatively.

Effects of ellipsoidal heterogeneities on wave propagation in partially saturated double-porosity rocks

Reservoir rocks are heterogeneous porous media saturated with multiphase fluids, in which strong wave dissipation and velocity dispersion are closely associated with fabric heterogeneities and patchy saturation at different scales. The irregular solid inclusions and fluid patches are ubiquitous in nature, whereas the impact of geometry on wave dissipation is still not well-understood. We have investigated the dependence of wave attenuation and velocity on patch geometry. The governing equations for wave propagation in a porous medium, containing fluid/solid heterogeneities of ellipsoidal triple-layer patches, are derived from the Lagrange equations on the basis of the potential and kinetic energies. Harmonic functions describe the wave-induced local fluid flow of an ellipsoidal patch. The effects of the aspect ratio on wave velocity are illustrated with numerical examples and comparisons with laboratory measurements. The results indicate that the P-wave velocity dispersion and attenuation depend on the aspect ratio of the ellipsoidal heterogeneities, especially in the intermediate frequency range. In the case of Fort Union sandstone, the P-wave velocity increases toward an upper bound as the aspect ratio decreases. The example of a North Sea sandstone clearly indicates that introducing ellipsoidal heterogeneities gives a better description of laboratory data than that based on spherical patches. The unexpected high-velocity values previously reported and ascribed to sample heterogeneities are explained by varying the aspect ratio of the inclusions (or patches).

3D seismic modeling in geothermal reservoirs with a distribution of steam patch sizes, permeabilities and saturations, including ductility of the rock frame

Seismic propagation in the upper part of the crust, where geothermal reservoirs are located, shows generally strong velocity dispersion and attenuation due to varying permeability and saturation conditions and is affected by the brittleness and/or ductility of the rocks, including zones of partial melting. From the elastic-plastic aspect, the seismic properties (seismic velocity, quality factor and density) depend on effective pressure and temperature. We describe the related effects with a Burgers mechanical element for the shear modulus of the dry-rock frame. The Arrhenius equation combined to the octahedral stress criterion define the Burgers viscosity responsible of the brittle-ductile behaviour.The effects of permeability, partial saturation, varying porosity and mineral composition on the seismic properties is described by a generalization of the White mesoscopic-loss model to the case of a distribution of heterogeneities of those properties. White model involves the wave-induced fluid flow attenuation mechanism, by which seismic waves propagating through small-scale heterogeneities, induce pressure gradients between regions of dissimilar properties, where part of the energy of the fast P-wave is converted to slow P (Biot)-wave. We consider a range of variations of the radius and size of the patches and thin layers whose probability density function is defined by different distributions. The White models used here are that of spherical patches (for partial saturation) and thin layers (for permeability heterogeneities). The complex bulk modulus of the composite medium is obtained with the Voigt-Reuss-Hill average. Effective pressure effects are taken into account by using exponential functions.We then solve the 3D equation of motion in the space-time domain, by approximating the White complex bulk modulus with that of a set of Zener elements connected in series. The Burgers and generalized Zener models allows us to solve the equations with a direct grid method by the introduction of memory variables. The algorithm uses the Fourier pseudospectral method to compute the spatial derivatives. It is tested against an analytical solution obtained with the correspondence principle. We consider two main cases, namely the same rock frame (uniform porosity and permeability) saturated with water and a distribution of steam patches, and water-saturated background medium with thin layers of dissimilar permeability. Our model indicates how seismic properties change with the geothermal reservoir temperature and pressure, showing that both seismic velocity and attenuation can be used as a diagnostic tool to estimate the in situ conditions.

Registration-Free Infant Cortical Surface Parcellation using Deep Convolutional Neural Networks.

Automatic parcellation of infant cortical surfaces into anatomical regions of interest (ROIs) is of great importance in brain structural and functional analysis. Conventional cortical surface parcellation methods suffer from two main issues: 1) Cortical surface registration is needed for establishing the atlas-to-individual correspondences; 2) The mapping from cortical shape to the parcellation labels requires designing of specific hand-crafted features. To address these issues, in this paper, we propose a novel cortical surface parcellation method, which is free of surface registration and designing of hand-crafted features, based on deep convolutional neural network (DCNN). Our main idea is to formulate surface parcellation as a patch-wise classification problem. Briefly, we use DCNN to train a classifier, whose inputs are the local cortical surface patches with multi-channel cortical shape descriptors such as mean curvature, sulcal depth, and average convexity; while the outputs are the parcellation label probabilities of cortical vertices. To enable effective convolutional operation on the surface data, we project each spherical surface patch onto its intrinsic tangent plane by a geodesic-distance-preserving mapping. Then, after classification, we further adopt the graph cuts method to improve spatial consistency of the parcellation. We have validated our method based on 90 neonatal cortical surfaces with manual parcellations, showing superior accuracy and efficiency of our proposed method.

An algorithm for finding intersection between ball B-spline curves

A ball B-spline curve (BBSC) is a skeleton based solid model representation, which consists of a B-spline curve and a B-spline function serving as the (varying) radius of a ball moving along the B-spline curve. The surface shape specified by the BBSC is the envelope of the moving ball and thus the BBSC is very suitable for representing vascular shapes. To apply this good representation in various fields as CSG-to-boundary conversion, Boolean operations, geometric design, pattern recognition, scientific visualization, this paper proposes an efficient algorithm to solve their common fundamental intersection problem. The surface of each BBSC is decomposed into a starting spherical patch, a set of lateral surfaces defined by ball Bézier curves (BBC), and an ending spherical patch. The intersection algorithm proceeds in four steps: (1) tessellating both BBSCs into triangular meshes and computing the intersection curves between the triangular meshes, which serve as the initial intersections for later refinement; (2) parameterizing the surfaces/patches of one BBSC and implicitizing the surfaces/patches of the other BBSC; (3) calculating the intersection curves between a parametric surface and an implicit surface numerically by using the Newton’s method; and (4) tracing the intersection curves for the construction of intersection solid regions. In particular, we present simple methods for parameterization and implicitization of BBSCs, which are the core of the proposed algorithm. And BBSC is a solid model, therefore the intersection of BBSCs is the intersection of solid models, and the final intersection result is also a solid object. Experimental results show that our method is able to efficiently find the intersections of BBSCs with high accuracy.

Utility of Slepian basis functions for modeling near-surface and satellite magnetic anomalies of the Australian lithosphere

The utility of frequency- and space-limited spherical harmonic Slepian basis functions for magnetic anomaly modeling over restricted spherical patches of the Earth was investigated using combined near-surface scalar and CHAMP satellite vector observations from Australia and adjacent marine areas. In particular, Slepian spherical harmonic models up to degree 360 were studied for modeling anomaly features of 1° (~111 km) and longer over a 25°-radius cap centered on Australia. Relative to the roughly 130,000 coefficients required for global spherical harmonic modeling, less than 5% of this number of coefficients is sufficient for effective localized Slepian modeling. Slepian coefficients have maximum power over the spherical cap and may be exploited for estimating the magnetic anomaly vectors and gradients to all orders within the working precision of the observations. The Earth cap modeled by Slepian coefficients is also more efficient in accommodating local crustal constraints from drilling and other geological and geophysical studies for interpreting the associated magnetic anomaly data registered in spherical coordinates. In general, Slepian spherical harmonic modeling is well suited for combining spectrally diverse compilations of near-surface and satellite magnetic observations over any spatially restricted spherical cap of the Earth or other planetary body.Graphical The utility of frequency- and space-limited Slepian spherical harmonic basis functions up to degree 360 was studied for modeling near-surface scalar and CHAMP satellite magnetic anomalies of 1° (~111 km) and longer over a 25°-radius cap centered on Australia. Slepian spherical harmonic modeling is well suited for combining spectrally divorce compilations of near-surface and satellite magnetic observations. It is also very efficient for updating global spherical harmonic models for new regional data and providing perspectives on how magnetic lithospheric anomalies vary up to satellite altitudes that are not available from standard upward and downward anomaly continuations. For example, Map A shows the Australians magnetic anomaly estimates at 10 km altitude from the Slepian model jointly constrained by near-surface and CHAMP satellite magnetic observations that minimize the differences between Maps B and C of upward continued near-surface and downward continued CHAMP data, respectively. Map D, on the other hand, shows the Slepian model estimates at 275 km altitude that minimize differences between Maps E and F of downward continued CHAMP and upward continued near-surface data, respectively. Any continuous, however, is not unique and subject to measurement and modeling errors so that its interpretation at location lacking observations requires considerable care.

MEG and EEG dipole clusters from extended cortical sources.

Data from magnetoencephalography (MEG) and electroencephalography (EEG) suffer from a rather limited signal-to-noise-ratio (SNR) due to cortical background activities and other artifacts. In order to study the effect of the SNR on the size and distribution of dipole clusters reconstructed from interictal epileptic spikes, we performed simulations using realistically shaped volume conductor models and extended cortical sources with different sensor configurations. Head models and cortical surfaces were derived from an averaged magnetic resonance image dataset (Montreal Neurological Institute). Extended sources were simulated by spherical patches with Gaussian current distributions on the folded cortical surface. Different patch sizes were used to investigate cancellation effects from opposing walls of sulcal foldings and to estimate corresponding changes in MEG and EEG sensitivity distributions. Finally, white noise was added to the simulated fields and equivalent current dipole reconstructions were performed to determine size and shape of the resulting dipole clusters. Neuronal currents are oriented perpendicular to the local cortical surface and show cancellation effects of source components on opposing sulcal walls. Since these mostly tangential aspects from large cortical patches cancel out, large extended sources exhibit more radial components in the head geometry. This effect has a larger impact on MEG data as compared to EEG, because in a spherical head model radial currents do not yield any magnetic field. Confidence volumes of single reconstructed dipoles from simulated data at different SNRs show a good correlation with the extension of clusters from repeated dipole reconstructions. Size and shape of dipole clusters reconstructed from extended cortical sources do not only depend on spike and timepoint selection, but also strongly on the SNR of the measured interictal MEG or EEG data. In a linear approximation the size of the clusters is proportional to the inverse SNR.

Steering patchy particles using multivalent electrolytes.

Proteins and many recently designed colloids can be regarded as patchy particles where directional interactions strongly influence and govern assembly behavior. Using explicit ion implicit solvent Metropolis Monte Carlo simulations, we investigate spherical model particles, carrying both charge and electric patches, in dilute aqueous 1 : 1, 1 : 3, and 3 : 1 electrolyte solutions. Striking differences in pair interaction free energies and orientational correlations are induced by three different salts which are discussed and rationalized in terms of ion-binding to surface groups, ion-ion correlations, and double layer forces. These findings suggest a general strategy where directional, intermolecular interactions can be invoked and tuned via small amounts of a carefully chosen electrolyte.

4D Infant Cortical Surface Atlas Construction using Spherical Patch-based Sparse Representation.

The 4D infant cortical surface atlas with densely sampled time points is highly needed for neuroimaging analysis of early brain development. In this paper, we build the 4D infant cortical surface atlas firstly covering 6 postnatal years with 11 time points (i.e., 1, 3, 6, 9, 12, 18, 24, 36, 48, 60, and 72 months), based on 339 longitudinal MRI scans from 50 healthy infants. To build the 4D cortical surface atlas, first, we adopt a two-stage groupwise surface registration strategy to ensure both longitudinal consistency and unbiasedness. Second, instead of simply averaging over the co-registered surfaces, a spherical patch-based sparse representation is developed to overcome possible surface registration errors across different subjects. The central idea is that, for each local spherical patch in the atlas space, we build a dictionary, which includes the samples of current local patches and their spatially-neighboring patches of all co-registered surfaces, and then the current local patch in the atlas is sparsely represented using the built dictionary. Compared to the atlas built with the conventional methods, the 4D infant cortical surface atlas constructed by our method preserves more details of cortical folding patterns, thus leading to boosted accuracy in registration of new infant cortical surfaces.

Nasal Patches and Curves for Expression-Robust 3D Face Recognition.

The potential of the nasal region for expression robust 3D face recognition is thoroughly investigated by a novel five-step algorithm. First, the nose tip location is coarsely detected and the face is segmented, aligned and the nasal region cropped. Then, a very accurate and consistent nasal landmarking algorithm detects seven keypoints on the nasal region. In the third step, a feature extraction algorithm based on the surface normals of Gabor-wavelet filtered depth maps is utilised and, then, a set of spherical patches and curves are localised over the nasal region to provide the feature descriptors. The last step applies a genetic algorithm-based feature selector to detect the most stable patches and curves over different facial expressions. The algorithm provides the highest reported nasal region-based recognition ranks on the FRGC, Bosphorus and BU-3DFE datasets. The results are comparable with, and in many cases better than, many state-of-the-art 3D face recognition algorithms, which use the whole facial domain. The proposed method does not rely on sophisticated alignment or denoising steps, is very robust when only one sample per subject is used in the gallery, and does not require a training step for the landmarking algorithm.