Geometric Interpolation Research Articles

Geometric interpolation by PH curves with quadratic or quartic rational normals

Pythagorean-hodograph (PH) curves have nice properties and have found important applications in geometric modeling and CNC machining. While the unit normals of PH curves of degree n are generally rational curves of degree n−1, this paper investigates PH curves of arbitrary degrees but with only quadratic rational unit normals when the curves are convex or with quartic rational unit normals when the curves have single inflection points. PH curves with quadratic or quartic rational normals have simple Gauss maps and hodographs of the curves are given by low degree tangent vector fields together with simple real scaling functions. Practical algorithms for interpolation of point-normal pairs or point-normal-curvature pairs together with unit normals at selected parameter coordinates or at inflection points by the investigated PH curves without or with the constraint of arc lengths have been given. The parameters for defining the interpolating PH curves are either obtained directly from the input data or by solving simple linear systems. This method of PH curve interpolation has unique solutions and the shapes of the interpolating PH curves are controlled well by the interpolated data. Even though the interpolating curves may have cusps, the regularity of the PH curves can be checked easily based on the signs of the real scaling functions within the hodographs.

Geometric realization of algebraic conformal field theories

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory, and that the conformal net may be thought of as a boundary value of the Segal CFT. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that ‘geometrically mollified’ versions of the fields yield bounded, local observables on the Hilbert space completion of the vertex algebra. These are the first results which unite the three major definitions of chiral conformal field theory. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.

Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things.

A full design of the Internet of Underwater Things (IoUT) with a high data rate is one of the greatest underwater communication difficulties due to the unavailability of a sustainable power source for the battery supplies of sensor nodes, electromagnetic spread weakness, and limited acoustic waves channel bandwidth. This paper presents a new energy-efficient communication scheme named Enhanced Fully Generalized Spatial Modulation (EFGSM) for the underwater acoustic channel, where the different number of active antennas used in Fully Generalized Spatial Modulation (FGSM) is combined with multiple signal constellations. The proposed EFGSM enhances energy efficiency over conventional schemes such as spatial modulation, generalized spatial modulation, and FGSM. In order to increase energy and spectral performance, the proposed technique conveys data bits not just by the number of active antenna’s index as in the existing traditional FGSM, but also using the type of signal constellation to increase the data bit rate and improve power saving without increasing the receiver’s complexity. The proposed EFGSM uses primary and secondary constellations as indexes to carry information, they are derived from others by geometric interpolation signal space. The performance of the suggested EFGSM is estimated and demonstrated through Monte Carlo simulation over an underwater acoustic channel. The simulation results confirm the advantage of the suggested EFGSM scheme not just regarding energy and spectral efficiency but also concerning the average bit error rate (ABER).

Area-preserving geometric Hermite interpolation

In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic Bézier polynomials. We show there exists a family of such curves which are 5th order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is 4th order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.

Fingerprint Smoothing Using Different Interpolation Techniques

Classical fingerprint analysts use binary images in the minutiae recognition and extraction processes. These obtained images do not have a sufficient quality that allows the extraction of the robust primitives, either during the contours detection or during skeletonization. In this study, a mathematical approach for the fingerprint curves modeling have been performed. The adopted smoothing technique is based on two geometric interpolation types adapted to fingerprint images. The obtained results reveal that the Bezier curve method has an error lower than that by the cubic spline method. The Bézier curve method has a RMS value of 0.035 pixels and an average maximum error of the order of 0.33 pixels. On the other hand, the cubic spline method has a RMS value about of 0.043 pixels and an average maximum error about of 0.37 pixels. We can see that the proposed method facilitates the design process of real-world objects and makes the fingerprint curves smooth.

A method of establishing an instantaneous water level model for tide correction

Tide correction is important in ship-board bathymetric data. Currently, most tide correction algorithms use a space-time interpolation method to regain the composite sea surface morphology. These interpolation algorithms are mostly based on geometric interpolation. However, when the tide stations are insufficient in the survey region, the spatial pattern of tide change may not comply with the geometric trend. Some of the algorithms use tide simulation to obtain a space-time astronomical tide model for tide correction in tide-station-insufficient regions, because the instantaneous water level changes are mainly caused by astronomical tides in normal conditions. However, in some cases, the instantaneous water level effect of the short-period stochastic meteorological factors results in short-term water level anomalies, which can be difficult to simulate using a tide simulation method. Thus, in this paper, an instantaneous water level model for tide correction in tide-station-insufficient regions is proposed. The model includes the simulation of astronomical tide and deviation-tidal components. We first simulate the astronomical tidal model using a two-dimensional MIKE21 Flow Model. Then, we propose a deviation correction method to mitigate the deviation-tidal components. Using the revised instantaneous water level model, we present the instantaneous tide correction (ITC) algorithm. Then, we compare the ITC algorithm with the two commonly used algorithms of Discrete Tidal Zoning and TCARI. The results show that the ITC algorithm is superior to the common algorithms in terms of accuracy and applicability with respect to a tide-station-insufficient survey region. Our conclusion is that ITC algorithm is a feasible tide correction algorithm when tide stations are not sufficiently measured.

An Adaptive Geometry Transformation and Repair Method for Hybrid Manufacturing

Hybrid manufacturing has become particularly attractive for refurbishing of high-value freeform components. Components may experience unique geometric distortions and/or wear-driven material loss in service, which require the use of part-specific, adaptive repair strategies. The current work presents an integrated adaptive geometry transformation method for additive/subtractive hybrid manufacturing based on rigid and nonrigid registrations of parent region material and geometric interpolation of the repair region material. In this approach, rigid registration of nominal part geometry to actual part geometry is accomplished using iterative alignment of profiles in the parent material. Nonrigid registration is used to morph nominal part geometry to actual part geometry by transformation of the profile mean line. Adaptive additive and subtractive tool paths are then used to add material based on constant stock margin requirements, as well as to produce blend repairs with smooth transition between parent and repair regions. A range of part deformation conditions due to profile twist and length changes are evaluated for the case of a compressor blade/airfoil geometry. Accuracy of the resulting adaptive geometry transformation method were quantified by (1) surface comparisons of actual and transformed nominal geometry and (2) blend region surface accuracy. Performance of the adaptive repair strategy relative to a naïve strategy is evaluated by the consideration of material efficiency and process cycle time. It is shown that the adaptive repair strategy resulted in an increase in material efficiency by 42.2% and a decrease in process time by 17.8%, depending on the initial deformation imposed on the part geometry.

Development of generalized technique for formation of characteristic functions and Balyuba-Naidysh coordinates in the composition method of geometrical modeling

The object of the study is the technique for the formation of characteristic functions and Balyuba-Naidysh coordinates (BN-coordinates) in the composite method of geometric modeling. The existing methods of modeling economic, technological and any other processes on real objects are rather complicated, with significant limitations on the number of incoming factors. One of the most problematic places is the complexity and narrow scope of each of the existing modeling methods, restrains their distribution and practical implementation on real business entities. Hence the need to develop a universal method for modeling multifactor systems. The closest to this is the composite method of geometric modeling (CMGM), the universality of which is ensured, first of all, thanks to the use of the intrinsic technique of generating characteristic functions and BN-coordinates. The use of the Balyuba curves (B-curves) constructed in BN-coordinates in KMGM gives significant advantages to CMGM. One-, two-, three-parameter B-curve can be considered in the n-dimensional Euclidean space E n . As a result, CMGM can be used to solve problems in n-dimensional space, and the result of the solution can be decomposed into n one-dimensional projections, on which it is easy to analyze the solution. This can be used, in particular, in information systems to support management decisions. A feature of the B-curves is that the BN-coordinates p(t); q(t); r(t) are its parametric model, which is a constant. Applying a variety of variants of changing points, it is possible to obtain a lot of variants of B-curves, which is important for carrying out computer experiments in order to increase the adequacy of the constructed geometric model. The principle of the formation of characteristic functions is the operation of multiplying parameters and artificially assigned coefficients. As a result of the definition the product at the node points becomes zero or one, and in the intervals between nodal points it changes from zero to one. The number of factors of the characteristic function is equal to the number of nodal points that the characteristic function interpolates. The BN-coordinates of one B-curve form a system of interrelated fractional-rational functions. Thus, a generalizing technique for the algebraic formation of characteristic functions has been developed, and a transition from characteristic functions to BN-coordinates for the interpolation of three points has been determined. The technique used here can also be used for geometric interpolation of four or more points. The possibility of increasing the number of initial points of a geometric figure for BN-interpolation, extends the capabilities of models of multifactor processes and systems.

Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction

We consider the problem of optimality in manifold reconstruction. A random sample \(\mathbb {X}_n = \{X_1,\ldots ,X_n\}\subset \mathbb {R}^D\) composed of points close to a d-dimensional submanifold M, with or without outliers drawn in the ambient space, is observed. Based on the tangential Delaunay complex (Discrete Comput Geom 51(1):221–267 2014), we construct an estimator \(\widehat{M}\) that is ambient isotopic and Hausdorff-close to M with high probability. The estimator \(\widehat{M}\) is built from existing algorithms. In a model with additive noise of small amplitude, we show that this estimator is asymptotically minimax optimal for the Hausdorff distance over a class of submanifolds satisfying a reach constraint. Therefore, even with no a priori information on the tangent spaces of M, our estimator based on Tangential Delaunay Complexes is optimal. This shows that the optimal rate of convergence can be achieved through existing algorithms. A similar result is also derived in a model with outliers. A geometric interpolation result is derived, showing that the Tangential Delaunay Complex is stable with respect to noise and perturbations of the tangent spaces. In the process, a decluttering procedure and a tangent space estimator both based on local principal component analysis (PCA) are studied.

Elastic Splines II: Unicity of Optimal s-Curves and Curvature Continuity

Given points $$P_1,P_2,\ldots ,P_m$$ in the complex plane, we are concerned with the problem of finding an interpolating curve with minimal bending energy (i.e., an optimal interpolating curve). It was shown previously that existence is assured if one requires that the pieces of the interpolating curve be s-curves. In the present article, we also impose the restriction that these s-curves have chord angles not exceeding $$\pi /2$$ in magnitude. With this setup, we have identified a sufficient condition for the curvature continuity of optimal interpolating curves. This sufficient condition relates to the stencil angles $$\{\psi _j\}$$ , where $$\psi _j$$ is defined as the angular change in direction from segment $$[P_{j-1},P_j]$$ to segment $$[P_j,P_{j+1}]$$ . An angle $$\Psi $$ ( $$\approx 37^\circ $$ ) is identified, and we show that if the stencil angles satisfy $$\vert {\psi _j} \vert <\Psi $$ , then optimal interpolating curves are curvature continuous. We also prove that the angle $$\Psi $$ is sharp. As with the previous article (Borbely and Johnson in Constr Approx 40:189–218, 2014), much of our effort is concerned with the geometric Hermite interpolation problem of finding an optimal s-curve $$c_1(\alpha ,\beta )$$ that connects $$0+i0$$ to $$1+i0$$ with prescribed chord angles $$(\alpha ,\beta )$$ . Whereas existence was previously shown, and sometimes uniqueness, the present article begins by establishing uniqueness when $$\vert {\alpha } \vert ,\vert {\beta } \vert \le \pi /2$$ and $$\vert {\alpha -\beta } \vert <\pi $$ . We also prove two fundamental identities involving the initial and terminal signed curvatures of $$c_1(\alpha ,\beta )$$ and partial derivatives, with respect to $$\alpha $$ or $$\beta $$ , of the bending energy of $$c_1(\alpha ,\beta )$$ .

A large deformation hybrid isogeometric-finite element method applied to cohesive interface contact/debonding

A hybrid NURBS-based isogeometric-finite element discretization technique is presented to take advantages of IGA for problems with regions demanding higher geometric accuracy or interpolation order. The validity of this coupled discretization is tested through standard patch test. Also, for demonstrative purposes, the proposed methodology is then applied to 2D interface contact/debonding of fiber–matrix as well as a 2D double cantilever beam peeling problem. Adopting a novel approach, the mortar method is used to account for both cohesive behavior and non-penetration constraint in the interface. Non-penetration constraint is enforced via Lagrange multiplier technique along with a mixed-mode cohesive law. The validity of the proposed unified contact/debonding formulation is tested against a contact/tension patch test. The stress contours after simulations show a smooth variation across IG and FE domains further demonstrating the coupling’s eligibility. Moreover, smooth contact pressure distributions are obtained along the interfaces resulting from higher-order NURBS basis functions. Having significantly less DOFs, the hybrid IG–FE discretization also behaved more robustly compared with linear FEs.

Signal constellations employing multiplicative groups of Gaussian and Eisenstein integers for Enhanced Spatial Modulation

In this paper, we propose two new signal constellation designs employing Gaussian and Eisenstein Integers for Enhanced Spatial Modulation (ESM). ESM is a novel technique which was propounded by Cheng et al. The advantage of ESM over other Spatial Modulation (SM) schemes lies in its ability to enhance spectral efficiency while keeping the energy efficiency intact. This is done by activating either one or two antennas judiciously depending upon the required trade-off. In ESM, information radiated from the antennas depends upon index of the active transmit antenna combination(s) and also on the set of constellation points chosen, which may include points from multiple constellations. In this paper, we propose signal constellations based on multiplicative groups of Gaussian and Eisenstein integers. The set comprising of Gaussian and Eisenstein integers serves as primary and secondary constellation points for Gaussian Enhanced Spatial Modulation (GESM) scheme. The secondary constellation points are deduced from a single geometric interpolation from the primary constellation points. The Monte Carlo simulation results indicate that the proposed nonuniform constellations achieve impressive SNR gains compared to conventional constellation points used in the design of ESM. This new design has been described for MIMO employing 4 × 4 and 8 × 8 antenna configurations with only two active antennas. Furthermore, a closed form expression for the pairwise error probability (PEP) for the GESM scheme has been deduced. The PEP is utilized to determine the upper bound on the average bit error probability (ABEP). Our simulations indicate that the proposed GESM from Gaussian and Eisenstein integers scheme outperforms all the other variants of SM including conventional ESM by at least 2.5 dB at an average bit error ratio (ABER) of 10−5. Close correspondence between the theoretical analysis and the Monte Carlo simulation results are observed.

Interpolation with spatial rational Pythagorean-hodograph curves of class 4

The paper presents a construction of rational Pythagorean-hodograph curves of class 4 and reveals their properties. In particular, it is shown that each such curve depends on twelve free parameters and has a piecewise rational arc-length function. Geometric interpolation of two data points and two tangent directions is considered in detail and a closed form solution that depends on two shape parameters is given. Regions of shape parameters are derived that imply the interpolant to be regular and admissible. Further, it is shown that one of the shape parameters can be fixed by additionally prescribing also the length of the interpolant. Theoretical results and the construction of G1 Hermite interpolation splines are illustrated with numerical examples.

Spatiotemporal predictions of soil properties and states in variably saturated landscapes

Understanding greenhouse gas (GHG) fluxes from landscapes with variably saturated soil conditions is challenging given the highly dynamic nature of GHG fluxes in both space and time, dubbed hot spots, and hot moments. On one hand, our ability to directly monitor these processes is limited by sparse in situ and surface chamber observational networks. On the other hand, remote sensing approaches provide spatial data sets but are limited by infrequent imaging over time. We use a robust statistical framework to merge sparse sensor network observations with reconnaissance style hydrogeophysical mapping at a well-characterized site in Ohio. We find that combining time-lapse electromagnetic induction surveys with empirical orthogonal functions provides additional environmental covariates related to soil properties and states at high spatial resolutions (~5 m). A cross-validation experiment using eight different spatial interpolation methods versus 120 in situ soil cores indicated an ~30% reduction in root-mean-square error for soil properties (clay weight percent and total soil carbon weight percent) using hydrogeophysical derived environmental covariates with regression kriging. In addition, the hydrogeophysical derived environmental covariates were found to be good predictors of soil states (soil temperature, soil water content, and soil oxygen). The presented framework allows for temporal gap filling of individual sensor data sets as well as provides flexible geometric interpolation to complex areas/volumes. We anticipate that the framework, with its flexible temporal and spatial monitoring options, will be useful in designing future monitoring networks as well as support the next generation of hyper-resolution hydrologic and biogeochemical models.

Fast and Accurate Estimation of the HARDI Signal in Diffusion MRI Using a Nearest-Neighbor Interpolation Approach

In the diffusion MRI domain, the HARDI methods were proposed to better characterize the complex biological tissues such as the white matter. In fact, they allow to overcome the problem of crossing fibers detection in the case of the Diffusion Tensor Imaging (DTI). However, the HARDI technique requires a very large number of Diffusion Weighted (DW) magnetic resonance images and thus it takes a long acquisition time, restricting its use in the clinical practice. We propose, in this paper, to develop a novel method for accelerating the reconstruction of the HARDI signal from a few number of DW images of the brain. The approach is a triangulation-based geometrical interpolation of existing signals. It consists on recovering non-acquired data according to their neighborhood from a reduced set of diffusion orientations on the sphere of the q-space. The accuracy of the proposed method was performed on two different phantom datasets for a qualitative and quantitative evaluation against known ground truths. We test also the robustness of the proposed approach under the noise. The obtained results demonstrate that the novel method can approximately halve the scan time and simultaneously obtain a proper fiber orientation estimation. In fact, the estimated FOD (e.g. FOD, Fiber Orientation Distribution) function based on only 34 directions is nearly identical to the one calculated with the full HARDI acquisitions (65 directions). Comparative evaluation on standard phantoms show that the proposed approach outperforms a large selection of methods of state-of-the-art according to the success rate and the angular error criteria.

Geodesic Video Stabilization in Transformation Space.

We present a novel formulation of video stabilization in the space of geometric transformations. With the setting of the Riemannian metric, the optimized smooth path is cast as the geodesics on the Lie group embedded in transformation space. While solving the geodesics has a closed-form expression in a certain space, path smoothing can be easily implemented by using geometric interpolation, rather than optimizing any space-time energy function. Specially, by using the geodesic solution in the space of rigid transformations, our approach even gains speedup 10× faster than state-of-the-art methods for path smoothing and motion compensation, and guarantees no extra distortion drawn into the stabilized frames. The experiments demonstrate the efficiency and effectiveness of our algorithm on stabilizing a variety of shaky videos.

A simulation-based product diffusion forecasting method using geometric Brownian motion and spline interpolation

This study addresses the problem of stochasticity in forecasting diffusion of a new product with scarce historical data. Demand uncertainties are calibrated using a geometric Brownian motion (GBM) ...

Improving the Accuracy of Geometric Interpolation for Determining Fundamental Frequency of Parallelogram Plates Vibration

The paper considers a method of geometric interpolation of basic solutions for two-dimensional problems in the theory of elasticity and structural mechanics, particularly as applied to mechanical engineering. The scope of the study is the vibrations of thin elastic parallelogram plates of constant thickness. To determine the fundamental frequency of vibration it is suggested to use an interpolation method with a newly introduced geometric characteristic of plates which is a ratio of inner conformal radius to outer conformal radius. Taken from the theory of conformal mapping, the conformal radii of domains are obtained by mapping the plates onto the interior and exterior of a unit circle. The studies have shown that the use of the suggested ratio ​​gives up to twice more accurate results of the geometric interpolation. The paper presents the basic terms and formulas of the method with comparative analysis of the curve diagrams using a shape factor and a conformal radii ratio.

New Signal Designs for Enhanced Spatial Modulation

In this paper, we present three new signal designs for enhanced spatial modulation (ESM), which was recently introduced by the present authors. The basic idea of ESM is to convey information bits not only by the index(es) of the active transmit antenna(s) as in conventional SM, but also by the types of the signal constellations used. The original ESM schemes were designed with reference to single-stream SM and involved one or more secondary modulations in addition to the primary modulation. Compared with single-stream SM, they provided either higher throughput or improved signal-to-noise ratio (SNR). In this paper, we focus on multi-stream SM (MSM) and present three new ESM designs leading to increasing SNR gains when they are operated at the same spectral efficiency. The secondary signal constellations used in the first two designs are derived through a single geometric interpolation step in the signal constellation plane, while the third design also makes use of additional constellations derived through a second interpolation step. The new ESM signal designs are described for MIMO systems with four transmit antennas out of which two are active, but we also briefly present extensions to higher numbers of antennas. Theoretical analysis and simulation results indicate that the proposed designs provide a significant SNR gain over MSM.

3D freehand ultrasound reconstruction using a piecewise smooth Markov random field

In this paper, we introduce a novel three-dimensional (3D) reconstruction framework for ultrasound images using a piecewise smooth Markov random field (MRF) model from irregularly spaced B-scan images obtained by freehand scanning. Freehand 3D ultrasound imaging is a useful system for various clinical applications, including image-guided surgeries and interventions, as well as diagnoses, due to the variety of its scan ranges and relatively low cost. The reconstruction process performs a key role in this system because its sampling irregularities may cause undesired artifacts, and ultrasound images generally suffer from noise and distortions. However, traditional approaches are based on simple geometric interpolations, such as pixel-based or distance-weighted methods, which are sensitive to sampling density and speckle noise. These approaches generally have an additional limitation of smoothing objects boundaries. To reduce speckle noise and preserve boundaries, we devised a piecewise smooth (PS) MRF model and developed its optimization algorithm. In our framework, we can easily apply an individual noise level for each image pixel, which is specified by the characteristics of an ultrasound probe, and possibly, the lateral and axial positions of an image. As a result, the reconstructed volume has sharp object boundaries with reduced speckle noise and artifacts. Our PS-MRF model provides simple segmentation results within a reconstruction framework that is useful for various purposes, such as clear visualization. The corresponding optimization methods have also been developed, and we tested a virtual phantom and a physical phantom model. Experimental results show that our method outperforms existing methods in terms of interpolation and segmentation accuracy. With this method, all computations can be performed with practical time consumption and with an appropriate resolution, via parallel computing using graphic processing units.