Quadratic Surface Research Articles

Response Surface Models Using the Wavelet Technique for Reservoir Inflow Prediction

Reliable streamflow prediction is vital to improving river operations, flood avoidance, water supply, and water resources management. Recently, response surface models have been launched in reservoir inflow prediction due to their potential to model composite nonlinear behaviour. Authors develop a hybrid model, wavelet quadratic response surface for reservoir inflow prediction in Chenab river basin, Pakistan. Wavelet transform has extensive applications in the field biomedical, engineering, and hydrology. Discrete wavelet transform technique discloses the structure of nonstationary signals. A proper and careful selection of mother wavelet ensure the best performance of wavelet transform. The choice of a suitable wavelet function participates in implementing the wavelet function used in response surface based models for reliable prediction. The performance of the proposed model is checked on different performance indices for model evaluation. The new developed model, wavelet quadratic response surface, depicts excellent results than other studied models.

A modified fuzzy dual-local information c-mean clustering algorithm using quadratic surface as prototype for image segmentation

Single-valued point can not effectively describe the gray characteristic of the image region, which makes existing fuzzy clustering algorithms difficult to segment image with uneven illumination, weak edges and strong noise. Hence, fuzzy c-means clustering with quadratic polynomial surface as clustering center has become a new method to solve complex image segmentation. Aiming at the shortcoming of current latest spatial fuzzy c-means clustering algorithm with quadratic surface as clustering center, quadratic polynomial surface is firstly introduced into existing widely used fuzzy local information c-means clustering algorithm, and further modify the membership degree of fuzzy local information c-means clustering by means of local membership of neighborhood pixels, a novel fuzzy dual-local information c-means clustering with quadratic polynomial-center clusters is proposed to solve the segmentation problem of complex image in the presence of high noise. Then the convergence of the proposed algorithm is strictly proved by Zangwill theorem and bordered Hessian matrix. Experimental results show that the proposed algorithm has better segmentation performance and anti-noise robustness than existing quadratic polynomial-related QPFCM, widely used FLICM, and multiple state-of-the-art algorithms, meanwhile, it also promotes the development of robust fuzzy clustering-related image segmentation theory.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

Bathymetric LiDAR and multibeam echo-sounding data registration methodology employing a point cloud model

Airborne LiDAR bathymetry (ALB) and multibeam echosounders (MBESs) are the most commonly used technologies for underwater topographical measurements. However, ALB cannot obtain underwater information in deep water. Due to concerns about the safety of the survey vessels, it is difficult to take measurements in shallow water using an MBES. To maximize the complementary advantages of these two technologies, this paper proposes a point cloud registration algorithm for ALB and MBES data based on azimuth angle features. First, the topographical curvature features of ALB and the MBES point cloud data are extracted by using a cylindrical neighborhood model and a quadric surface fitting algorithm. Then, the azimuth angle similarity of the curvature feature points is used to match the corresponding points. Finally, a truncated least-squares-based iterative nearest-adjacent point (TrICP) method is utilized to realize multisource data registration. To verify the performance of the proposed registration algorithm, four areas with typical ALB and MBES point cloud data from Xisha Island, South China Sea, are used as the study areas. Compared to the traditional and typical curvature-based registration (CR) algorithm for point cloud data registration, the proposed algorithm obtains average RMSE values of 0.278 m, 0.252 m, 0.214 m and 0.177 m in the four typical study areas. The corresponding registration accuracy is improved by 91%, 73%, 84% and 49%, respectively. The experimental results show that the proposed algorithm is more reliable and effective for ALB and MBES point cloud registration than the previous algorithms.

Single-view calibration of a catadioptric camera based on a theodolite model.

The catadioptric panoramic imaging system may provide 360° panoramic imaging by employing the convex surface of a quadric surface with rotational symmetry as the reflector, which effectively compensates for the disadvantages of the narrow field of view in typical camera systems. First, this paper proposes a theodolite-based catadioptric camera image model based on the rotational symmetry of a catadioptric camera mirror, which simplifies the 2D modeling problem to a 1D problem. Simultaneously, the equivalence of the theodolite imaging model and the standard spherical imaging model also is demonstrated in this work. Second, this paper presents a method to calibrate the theodolite model parameters using only a single view and explains the calculation of model parameter initialization and iterative optimization steps in detail. Then, this paper demonstrates how to calibrate the theodolite model parameters using only a single view, as well as how to calculate the model parameter initialization and iterative optimization steps. Finally, simulation experiments and actual experiments have confirmed the correctness and stability of the method. The experimental results show that the average and standard deviation of the back-projection error are, respectively, 0.1983125 pixels and 0.0006153 pixels, in this model. We believe the theodolite model suggested in this paper offers a viable approach to catadioptric camera image modeling.

Interactive Geological Data Visualization in an Immersive Environment

Underground flow paths (UFP) often play an important role in the illustration of geological data by geologists, especially in illustrating geological data and revealing stratigraphic structures, which can help domain experts in their exploration of petroleum information. In this paper, we present a new immersive visualization tool to help domain experts better illustrate stratigraphic data. We use a visualization method based on bit-array-based 3-D texture to represent stratigraphic data. Our visualization tool has three major advantages: it allows for flexible interaction at the immersive device, it enables domain experts to obtain their desired UFP structure through the execution of quadratic surface queries, and supports different stratigraphic display modes, as well as switching and integration geological information flexibly. Feedback from domain experts has shown that our tool can contribute more for domain experts in the scientific exploration of stratigraphic data, compared to the existing UFP visualization tools in the field. Thus, experts in geology can have a more comprehensive understanding and more effective illustration of the structure and distribution of UFPs.

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.

Methodology for predicting the washability of apatite ores (Kirovsky mine, Kola Peninsula)

A statistical approach has been proposed that allows strictly defining the structure of a rock (ore), isolating fields of various structures in barycentric diagrams, and comparing them with the processing results. An original method is proposed for predicting the washability defined as the separation of apatite grains from polymineral aggregations at the crushing and flotation stages. It develops and formalizes the current approach that combines the mineral composition of ores, their structures and textures into a multifactor semiquantitative model. The method is based on the calculation of the probabilities (frequencies) of all types of intergranular contacts recorded in a symmetric matrix under a microscope in petrographic thin sections. Its canonical diagonal shape allows identifying the rock (ore) structure type. The algebraic theory of quadratic forms is used to demonstrate that the structure identified using this method corresponds to one of the types of n-dimensional nondegenerate quadratic surfaces, where n is the number of minerals in the rock (ore). The structure is invariant for a certain fluctuation in the probabilities of intergranular contacts. The calculated probabilities uniquely identify the imaging point in the corresponding barycentric diagram. The Hardy–Weinberg boundary is important as it corresponds to the ideal uniform mixing of minerals and the balance of various intergranular contacts for any modal composition of rocks (ores). It allows distinguishing fields of different structures in barycentric diagrams. The method proposed offers a means to identify ores with different washability characteristics at early exploration stages.

Revisit to the THINC/QQ scheme: Recent progress to improve accuracy and robustness

AbstractIn this article, we revisit to the THINC/QQ (tangent of hyperbola interface capturing method with quadratic surface representation and Gaussian quadrature) scheme to make thorough investigations on its numerical property. Furthermore, it presents some recent progress made extensively on THINC/QQ algorithm to improve numerical accuracy and robustness which include (1) construct the interface equation on the global coordinate so as to apply THINC/QQ on polygonal/polyhedral elements, which ensures the boundedness of numerical solutions as well; (2) determine the coefficients of reconstruction polynomial from the unit normal vector of interface to reduce computational cost and alleviate numerical diffusion; (3) represent the interface with a cubic polynomial rather than quadratic representation used in THINC/QQ to obtain better geometric fidelity; (4) provide an appropriate choice considering different combinations, such as the number of quadrature points and temporal discretization methods. The verification benchmark tests are carried out on different types of grids, which convinces that THINC/QQ scheme with normalization on global coordinate delivers higher accuracy and robustness than the original one, and demonstrates effective competition compared with other existing interface capturing methods.

Formula omitted]-Kernel-free soft quadratic surface support vector regression

In this paper, we propose a new regression method called the ∊-kernel-free soft quadratic surface support vector regression (∊-SQSSVR). After converting the n-dimensional regression problem into the (n+1)-dimensional classification problem, the principle of maximizing the sum of relative geometrical margin of each training point is used to construct our optimization problem, where the quadratic surface is restricted to be a hyperparaboloid by setting both the (n+1)-th row and (n+1)-th column of the corresponding matrix to be zero. The existence and uniqueness of the optimal solution to both primal and dual problems are also addressed. It should be pointed out that our model is nonlinear and kernel-free, so it does not need to select kernel function and corresponding parameters. At the same time, it is highly interpretable. In addition, our model is still a quadratic convex programming problem similar to the standard SVR. To visualize the effectiveness of our ∊-SQSSVR, 6 artificial datasets and 15 benchmark datasets are implemented in numerical experiments. The results show that our method is less time-consuming and as good as the nonlinear standard SVR with kernel function in comprehensive performances.

A New Approach to Rotational Weingarten Surfaces

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

Quadratic B-Spline Surfaces with Free Parameters for the Interpolation of Curve Networks

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor product B-spline surfaces and bivariate B-spline surfaces on criss-cross triangulations. Finally, we present graphical results.

On Quadratic Interpolation of Image Cross-Correlation for Subpixel Motion Extraction.

Digital image correlation techniques are well known for motion extraction from video images. Following a two-stage approach, the pixel-level displacement is first estimated by maximizing the cross-correlation between two images, then the estimation is refined in the vicinity of the cross-correlation peak. Among existing subpixel refinement methods, quadratic surface fitting (QSF) provides good performances in terms of accuracy and computational burden. It estimates subpixel displacement by interpolating cross-correlation values with a quadratic surface. The purpose of this paper is to analytically investigate the QSF method. By means of counterexamples, it is first shown in this paper that, contrary to a widespread intuition, the quadratic surface fitted to the pixel-level cross-correlation values in the neighborhood of the cross-correlation peak does not always have a maximum. The main contribution of this paper then consists in establishing the mathematical conditions ensuring the existence of a maximum of this fitted quadratic surface, based on a rigorous analysis. Algorithm modifications for handling the failure cases of the QSF method are also proposed in this paper, in order to consolidate it for subpixel motion extraction. Experimental results based on two typical types of images are also reported.

A Modified Local Distance-weighted (MLD) Method of Interpolation and Its Numerical Performances for Large Scattered Datasets

The purpose of this study was to propose a new interpolation scheme that was designed to remedy the shortcomings encountered in two popular interpolation methods; the triangle-based blending (TBB) method and the inverse distance weighted (IDW) method. At the same time, the proposed method combines their desirable aspects, which are the local nature and non-use of quadratic surface construction, making it comparatively less time-consuming and more independent of the global effect. Because of these properties, the new scheme was named the ‘Modified Local Distance-Weighted (MLD)’ method, and it was tested in detail with different sizes of datasets. The datasets involved in this investigation were of two types; uniformly and non-uniformly distributed. The performances were carefully monitored and assessed via several criteria; accuracy, sensitivity to parameters, CPU-time, storage requirement and ease of implementation. For comparative purposes, three alternative interpolation methods were simultaneously carried out, i.e. TBB, IDW and Radial Basis Function-Based (RBF) method. The investigation clearly revealed promising aspects of the proposed scheme where a good quality of results was anticipated, and CPU-time and storage were seen to have been significantly reduced. The research strongly indicates the benefits of the proposed method for larger sized datasets for real practical scientific and engineering uses in the future.

Numerical simulations for incompressible turbulence cavitation flows with tangent of hyperbola interface capturing (THINC) scheme

In this work, we developed high-fidelity numerical solvers for turbulent cavitation flows and presented numerical simulations of cavitation and supercavitation behind an axisymmetric projectile and a conical cavitator. The proposed numerical solver is based on the homogeneous equilibrium model where the mass transfer rate between vapor and liquid is dependent on a volume of fraction (VOF) function. As a result, the numerical results of cavitation turbulent flow are heavily affected by the accuracy of the VOF evolution prediction. Thus, the proposed solver employs the Tangent of Hyperbola for INterface Capture method with Quadratic surface representation and Gaussian Quadrature scheme to accurately resolve the interfacial structures of cavitation bubbles. To effectively model the turbulent effects, the solver utilizes two approaches, respectively, i.e., the Smagorinsky model of large eddy simulation and the shear stress transport model of the Reynolds-averaged Navier–Stokes equation. The homogeneous turbulence equation systems are then discretized by the linear second-order schemes in space and time. The phase transition including both evaporation and condensation is described by the finite-rate mass transfer models such as Schnerr–Sauer and Kunz models. Numerical simulations and comparison studies are performed with the proposed solvers. Compared with previous simulation works, the current simulation results of cavity shape patterns and related hydrodynamic characteristics are in better agreement with experimental data and analytical theories, as well as reference solutions. These results validate that the proposed solver can produce high-fidelity predictions for solving the flow structures of cavitation and reentrant jet in the turbulent cavitation simulations.

APPLICATION OF BOX-WILSON CENTRAL COMPOSITE EXPERIMENTAL DESIGN TO OPTIMIZE THE PROTEOLYTIC ACTIVITY IN MILK BY LACTOCOCCUS LACTIS LCL STRAIN

In this work, the individual and interactive effects of temperature and pH on the proteolytic activity of Lactococcus lactis LCL strain in milk were studied by quadratic Response Surface Methodology. The effects of the selected variables have been analysed using a Central Composite Design and the maximum response (proteolysis rate) under the optimum values of variables and of a satisfactory fit model was realized. The mathematical relationship of the proteolytic activity on the two significant independent variables can be approximated by a nonlinear polynomial model. Predicted values were found to be in a good agreement with experimental values (R2 = 0.937) demonstrating the good significance of the model. The result of optimization predicted by the model has shown that the maximal result for the proteolysis rate revolved around of 43.7 ± 1.4 % at the optimal condition (30°C and pH 7).

Study on the degradation mechanism of 2-amino-4-acetaminoanisole from wastewater by nano-Fe3O4-catalyzed Fenton system

2-Amino-4-acetaminoanisole (AMA) is an intermediate product in the synthesis of many commercial dyes, and its wide application has led to the generation of a series of AMA dye wastewater. Discharge of untreated AMA dyed wastewater could bring environmental concerns. The present study featured H2O2 Fenton system to degrade 2-amino-4-acetaminoanisole from wastewater using nano-Fe3O4 catalyst prepared via the co-precipitation method. Furthermore, the Box-Behnken design (BBD) response surface method was used to investigate the individual effects of Fe3O4 dosage, H2O2 dosage, initial pH, and reaction time on AMA removal, while in the interaction study, the Design-Expert 10.0 software was applied to obtain a quadratic response surface model. Results indicated that the catalytic effect of nano-Fe3O4 showed better degradation performance as compared to FeSO4 Fenton system. The order of the influence of the selected independent variables on the response value is as follows: nano-Fe3O4 dosage > H2O2 dosage > reaction time > pH. As for 3.04 × 105μg/L of AMA dye wastewater, the optimal reaction conditions considered in this study are 1.70g/L of nano-Fe3O4 dosage, 53.52mmol/L of H2O2 dosage, pH 5.14, and 388.97min as system reaction time. Furthermore, HPLC-MS was employed to analyze the degradation mechanism of AMA and the reaction intermediate products. Possible degradation pathways of AMA by radicals were addressed. Findings of this research provide fundamental theory and guide practical AMA treatment during wastewater treatment.

Calibration of line-structured light vision sensor based on free-placed single cylindrical target

This paper proposes a novel calibration method for the Line-structured Light Vision Sensor (L-LVS) based on a straight cylindrical target that can be free placed. The cylindrical target is placed at least once in the measurement space. The camera captures the image of the cylindrical target and the light stripe. Based on the geometric properties of the cylinder and vanishing points, we calculate the 3D centers of the cylindrical end circles and reconstruct the cylindrical quadric surface. Then, multiple 3D points on the cylindrical surface are obtained according to the perspective projection relationship, which finally allows for the reconstruction of the light plane. Both simulation and real experiments show that the proposed method is effective. In the real experiments, the measurement accuracy of a L-LVS based on the proposed calibration method achieves 0.056 mm, when the measuring distance is about 400 mm. Due to the small-size and pose-unconstrainted target, our calibration method is superior to be applied in the confined measurement space in the wild.

A Novel Chinese Points of Interest Classification Method Based on Weighted Quadratic Surface Support Vector Machine

A Novel Chinese Points of Interest Classification Method Based on Weighted Quadratic Surface Support Vector Machine

Quadrics for Structuring Invariant Space-Time Wave Packets

We provide a general approach for structuring invariant 3D+1 optical wave packets in both bulk and structured dispersive media, through a simple engineering of phase-matched space-time frequencies on quadric surfaces.