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The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

Free-form surfaces have been widely used in aerospace, automotive and other fields. Due to its complex geometry, free-form surface inspection is generally conducted by touch-trigger or measuring probe-based Coordinate Measurement Machines or On-machine Measurement. Sampling strategy plays a decisive role in improving both measurement accuracy and efficiency, which is determined by sample size and distribution of sample points. However, it is difficult to simultaneously take the surface curvature, sampling density and approximation error into account, considering the complexity of surface geometry. In this paper, triangle mesh simplification is innovatively adopted in sampling planning to achieve multi-geometric constraints. As triangle mesh has outstanding advantages in representing the surface features, strong stability and is easy to modify its structure, free-form surface is converted to a dense triangle mesh. Triangle mesh simplification is implemented by iteratively contracting triangle edges. An improved quadric error metric is established to decide contraction order and optimal target vertices under discrete curvature constraint. Sampling density is controlled by limiting the triangle edge length. Detailed adaptive sampling algorithm under multi-geometric constraints is then developed. Both simulation and experiment are conducted to validate feasibility and robustness of the proposed method. The results are compared with uniform sampling and existing adaptive sampling strategy to show that the proposed method can prominently reduce sampling error when sample size is small.

Background: The supporting quadric method(SQM) is a versatile method for designing freeform optics for desired irradiance redistribution, but the time of solution optimization increases rapidly with the refinement of the mapping grid. Objective: As the complexity of light distribution is getting higher and higher, time consumed will also increase exponentially. This paper proposes an idea of applying the deep neural network method to optical design. Methods: In this article, we established a special corresponding relationship and prepared a dataset, which underwent deep network learning and training. Finally, a hybrid design method of deep learning and optical design was realized and verified. Results: Compared with the traditional method, this method is more efficient. Here, we used a deep neural network(DNN) to accelerate the freeform optical design. After the DNN was trained by a sample set consisting of a uniform pattern and eight different Chinese characters represented by an array with 11 × 11, it can generate a character's reflector within few milliseconds. Conclusion: As proof of this new method, a character pattern reflector was manufactured and tested, and the experimental irradiance distribution was found close to the expectation, which means that the neural network has the excellent capability to memorize all of the learned characters. SQM combined with DNN has the potential to establish a particular “optical font library” and even offers a promising path for rapid freeform optical design to realize the function of “optical typography”.

In this note, we extend work of Farkas and Rim��nyi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $��(\log(N)^{-0.087})$ for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$., Revised according to referee comments, to appear in Experimental Mathematics

In this paper, a new refined quasi-3-dimensional logarithmic shear deformation theory (RQ-3DLSDT) and an advanced moving Kriging interpolation (AMKI) based-meshless method is combined for the first time to study the static bending, free vibration, and compressive buckling analysis of isotropic and sandwich functionally graded plates laid on elastic foundations. The RQ-3DLSDT considering the effect of thickness stretching based only on four-unknown variables to determine the displacement field leading to reduce computational efforts. The weak forms of the equilibrium equation are discretized and numerically solved by the AMKI mesh-free method employed a proposed quadric correlation function for getting stable solutions.

Orthoptic curves for the conics are well known. It is the Monge's circle for ellipse and hyperbola, and for parabola it is its directrix. These conics are level sets of quadratic functions in the plane. We consider level sets of quadratic functions in higher dimension, known as quadric hypersurfaces. For these hypersurfaces we present and study their orthoptic sets, which extend the idea of orthoptic curves for conics.

This paper is concerned with the problem of multi-view 3D reconstruction with an un-calibrated micro-lens array based light field camera. To acquire 3D Euclidean reconstruction, existing approaches commonly apply the calibration with a checkerboard and motion estimation from static scenes in two steps. Self-calibration is the process of simultaneously estimating intrinsic and extrinsic parameters directly from un-calibrated light fields without the help of a checkerboard. While the self-calibration technique for conventional (pinhole) camera is well understood, how to extend it to light field camera remains a challenging task. This is primarily due to the ultra-small baseline of the light field camera. We propose an effective self-calibration method for a light field camera for automatic metric reconstruction without a laborious pre-calibration process. In contrast to conventional self-calibration, we show how such a self-calibration method can be made numerically stable, by exploiting the regularity and measurement redundancies unique for the light field camera. The proposed method is built upon the derivation of a novel ray-space homography constraint (RSHC) using Plucker parameterization as well as a ray-space infinity homography (RSIH). We also propose a new concept of “rays of the absolute conic (RAC)” defined as a special quadric in 5D projective space $${\mathbb {P}}^5$$ . A set of new equations are established and solved for self-calibration and 3D metric reconstruction specifically designed for a light field camera . We validate the efficacy of the proposed method on both synthetic and real light fields, and have obtained superior results in both accuracy and robustness.

A Minkowski plane is Euclidean if and only if at least one ellipse is a quadric. We discuss the higher dimensional consequences too.

Signal modeling is an important technique in many engineering applications. This paper is concerned about signal modeling problem for the sine multi-frequency signals or periodic signals. In terms of different characteristics between the signal output and the signal parameters, a separable modeling scheme is presented for estimating the signal parameters. In order to seize the real-time information of the signals to be modeled, a sliding measurement window is designed for using the observations dynamically and implementing accurate parameter estimates. Because the amplitude parameters are linear with respect to the signal output and the angular frequency parameters are nonlinear with respect to the signal output, the signal parameters are separated into a linear parameter set and a nonlinear parameter set. Based on these separable parameter sets, a nonlinear optimization problem is converted into a combination of the optimization quadric and the nonlinear optimization. Then, a separable multi-innovation Newton iterative signal modeling method is derived and implemented to estimate sine multi-frequency signals and periodic signals. The simulation results are found to be effective of modeling dynamic signals. For the reason that the proposed method is based on dynamic sliding measurement window, it can be used for online estimation applications.

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

Herein, we present a novel approach for monocular dual quadric initialization that combines three-dimensional (3D) map points with two-dimensional (2D) object detection for forward-translating camera movements. The traditional approach using 2D detection bounding boxes in multiple views fails in straight vehicle motion scenarios as object observation is limited to few frames. Although single image initialization is possible when multiple constraints are introduced, such initialization is based on strong assumptions. In this letter, we incorporate constraints from 3D map points with single-view 2D object detection to robustly initialize the dual quadric. Constraints from 3D map points are converted to planar constraints from their convex hull. Together with the projective planar constraints from bounding boxes, the proposed method can infer accurate dual quadric parameters. Further, comparison studies with the state of the art (SOTA) show that the proposed approach achieves the same accuracy of center localization but outperforms the existing methods in shape estimation and success ratio of initialization. The proposed method dose not rely on assumptions of dimension and pose of 3D objects; hence, it is more generic and accurate. Based on the KITTI raw dataset, the initialization success ratio is up to 97.7% with an average position error of 1.58 m, and 2D IoU of 80% when the number of map points per object accumulates to 60. When applied to the TUM RGB-D dataset, the proposed approach yields an initialization success ratio of 92.7% when the number of map points per object accumulates to 30, revealing a 16.2% increment compared with the SOTA using an RGB-D camera. Finally, we integrate the initialization method into a simultaneous localization and mapping system.

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality. Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank. We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components., Final version, to appear in Moscow Mathematical Journal

In this paper two new tensor fields on real hypersurfaces in complex quadric are introduced. Real hypersurfaces on which the derivatives of the tensor fields with respect to the Levi-Civita connection and the k-th generalized Tanaka–Webster connection of them coincide are studied leading to new classification results on real hypersurfaces in a complex quadric.

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ F q with q odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson’s classification (for q odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) , q odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ π , and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

3D object detection is an essential component of scene perception and motion prediction in autonomous driving. Previous methods represent objects as the truncated signed distance fields (3D bounding box), which can only provide the geometric constraints of point-to-line. In this work, we define the object as a more compact representation, quadric (ellipsoid) in a 3D scene and a conic (ellipse) in an image, which can provide stronger geometric constraints of surface-to-curve. Specifically, we estimate a ellipsoid from a conic fitted by a 2D bounding box to obtain 3D object localization and occupancy. We further to formulate this constraint relation as a nonlinear optimization problem in dual space, which enables us to easy recover stable and accurate 3D object parameters by adding only three additional direction-aware branches to the existing 2D detection networks. In addition, we decouple the dimensions of object and update the length and orientation of objects in our iterative algorithm when the estimations from the 2D detection networks have different deviations. The final detection results can be obtained after passing through our geometry-related refinement network. We evaluate our method on the KITTI object detection benchmark and achieve the best performance among published monocular competitors.

The μ-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ-basis from an implicit form. In this paper, we derive the explicit forms of μ-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate μ-basis” as well as the error estimation., Agencia Estatal de Investigación

The purpose of this paper is to find pointwise estimates on the integral kernel for the $$\Box _b$$ Green operator, N, for a particular quadric submanifold of codimension 2 in $${\mathbb {C}}^4$$ . This particular kernel has singularities away from the diagonal. Furthermore, the estimates exhibited by this kernel, both on and off the diagonal, follow no known paradigm.