Three-dimensional Convex Research Articles

Vision-based localisation of mature apples in tree images using convexity

This paper details a procedure for detecting apples in tree images using shape analysis are presented. The core of the procedure consists of a so-termed convexity test that identifies edges that could correspond to three-dimensional convex objects of a given size range from a much larger set of edges. This is achieved by analysing a number of intensity profiles that originate at each edge and determining whether they have a shape that is suitable with a 3D convex object of the correct size. We show that contrarily to the prevailing opinion, the intensity functions of three-dimensional convex objects are not necessarily convex, which led us to developing models for describing such profiles. The simplest suitable model includes four parameters that can be easily estimated by a standard least square constrained optimization procedure. After merging the selected edges that fall on circles, a second analysis is performed to remove false positive detections and eliminate multiple detections of apples. The procedure was demonstrated on 51 grey-level images that were recorded in a Golden Delicious apple variety orchard under natural light conditions. On average, together with preliminary pre-processing operations, the convexity test removed 99.8% of the edges initially identified by Canny filter. Analysis based on the remaining edges led to correct detection of 94% of the apples visible in the images. Fourteen percent of the identified objects were “false positive” detections, mainly due to leaves or parts of leaves that generated convex surfaces very similar to apples, or by leaves that lay on apples and created misleading edges.

Analytic formulas for complete hyperbolic affine spheres

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions into three-dimensional real space. The generic member of the family of immersions is given by elliptic integrals.

Rapid Characterization of Vegetation Structure with a Microsoft Kinect Sensor

The importance of vegetation structure and biomass in controlling land-atmosphere exchange is widely recognized, but measurements of canopy structure are challenging, time consuming, and often rely on destructive methods. The Microsoft Kinect is an infrared sensor designed for video gaming that outputs synchronized color and depth images and that has the potential to allow rapid characterization of vegetation structure. We compared depth images from a Kinect sensor with manual measurements of plant structure and size for two species growing in a California grassland. The depth images agreed well with the horizontal and vertical measurements of plant size made manually. Similarly, the plant volumes calculated with a three-dimensional convex hulls approach was well related to plant biomass. The Kinect showed some limitations for ecological observation associated with a short measurement range and daytime light contamination. Nonetheless, the Kinect's light weight, fast acquisition time, low power requirement, and cost make it a promising tool for rapid field surveys of canopy structure, especially in small-statured vegetation.

On convex envelopes for bivariate functions over polytopes

In this paper we discuss convex envelopes for bivariate functions, satisfying suitable assumptions, over polytopes. We first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two-dimensional polytope through the solution of a three-dimensional convex subproblem with continuously differentiable constraint functions. Then, for quadratic functions as well as for some polynomial and rational ones, again satisfying suitable assumptions, we show how the same computations can be carried out through the solution of a single semidefinite problem.

Size and shape estimation of 3‐D convex objects from their 2‐D projections: application to crystallization processes

The aim of this paper is to present a new projective stereological method that enables to estimate the size and the shape of a three-dimensional convex object from measurements only made on its two-dimensional projections. To do so, some geometrical and morphometrical measurements on the projected shadows of the three-dimensional convex object are done and the value of the three-dimensional convex object size and shape parameters are retrieved using the maximum likelihood estimation method. The proposed method is then applied to estimate three-dimensional particle size distributions during crystallization processes.

Flat tori in three-dimensional space and convex integration

It is well-known that the curvature tensor is an isometric invariant of C(2) Riemannian manifolds. This invariant is at the origin of the rigidity observed in Riemannian geometry. In the mid 1950s, Nash amazed the world mathematical community by showing that this rigidity breaks down in regularity C(1). This unexpected flexibility has many paradoxical consequences, one of them is the existence of C(1) isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and 1980s, M. Gromov, revisiting Nash's results introduced convex integration theory offering a general framework to solve this type of geometric problems. In this research, we convert convex integration theory into an algorithm that produces isometric maps of flat tori. We provide an implementation of a convex integration process leading to images of an embedding of a flat torus. The resulting surface reveals a C(1) fractal structure: Although the tangent plane is defined everywhere, the normal vector exhibits a fractal behavior. Isometric embeddings of flat tori may thus appear as a geometric occurrence of a structure that is simultaneously C(1) and fractal. Beyond these results, our implementation demonstrates that convex integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.

A polyhedral beam-tracing method for modeling ultrawideband indoor radio propagation

A three-dimensional convex polyhedral beam-tracing method for modeling ultrawideband indoor radio channels is presented.The model enables a fast and unambiguous generation of all propagation paths between the transmitter and the receiver over an 850-MHz bandwidth. To validate the model, its results are compared with measurements. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:904–909, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26670

Meissner’s Mysterious Bodies

Certain three-dimensional convex bodies have a counterintuitive property; they are of constant width. In this particular respect they resemble a sphere without being one. Discovered a century ago, Meissner’s bodies have often been conjectured to minimize volume among bodies of given constant width. However, this conjecture is still open. We draw attention to this challenging and beautiful open problem by presenting some of its history and its recent development.

Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Let $ \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) $ be the tautological vector bundle over the Grassmann manifold of 2-planes in $ {\mathbb{R}^3} $ , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of $ {\mathbb{R}^3} $ . A field of convex figures is given in γ 2 3 if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ 2 3 contains a figure K containing a centrally symmetric convex figure of area $ \left( {4 + 16\sqrt {2} } \right) $ S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area $ \left( {12\sqrt {2} - 8} \right) $ S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume $ \left( {36\sqrt {2} - 24} \right) $ V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.

Quantum sign permutation polytopes

Convex polytopes are convex hulls of point sets in the n-dimensional space that generalize two-dimensional convex polygons and three-dimensional convex polyhedra. We concentrate on the class of n-dimensional polytopes in called sign permutation polytopes. We characterize sign permutation polytopes before relating their construction to constructions over the space of quantum density matrices. Finally, we consider the problem of state identification and show how sign permutation polytopes may be useful in addressing issues of robustness.

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

We propose strongly consistent algorithms for reconstructing the characteristic function $1_K$ of an unknown convex body $K$ in $\mathbb {R}^n$ from possibly noisy measurements of the modulus of its Fourier transform $\widehat {1_K}$. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing $K$ from noisy measurements of its covariogram, the function giving the volume of the intersection of $K$ with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use $O(k^n)$ noisy covariogram measurements to construct a convex polytope $P_k$ that approximates $K$ or its reflection $-K$ in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between $P_k$ and $\pm K$ tends to zero as $k$ tends to infinity.

Semidefinite programming for optimizing convex bodies under width constraints

We consider the problem of minimizing a functional (such as the area, perimeter and surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér–Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of constant-width planar bodies, rotors and space bodies of revolution are revisited. The approach seems promising to investigate more difficult optimization problems in the class of three-dimensional convex bodies.

A Brunn–Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R 3 , as an application we get the Brunn–Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R 3 .

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Approximation of three-dimensional convex bodies by affine-regular prisms

Let K ⊂ ℝ3 be a convex body of unit volume. It is proved that K contains an affine-regular pentagonal prism of volume $$ {{4\left( {5 - 2\sqrt 5 } \right)} \mathord{\left/{\vphantom {{4\left( {5 - 2\sqrt 5 } \right)} 9}} \right.} 9} $$ (which is greater than 0.2346) and an affine-regular pentagonal antiprism of volume $$ {{4\left( {3\sqrt 5 - 5} \right)} \mathord{\left/{\vphantom {{4\left( {3\sqrt 5 - 5} \right)} {27}}} \right.} {27}} $$ (which is greater than 0,253). Furthermore, K is contained in an affine-regular pentagonal prism of volume $$ 6\left( {3 - \sqrt 5 } \right) $$ (which is less than 4.5836), and in an affine-regular heptagonal prism of volume 21(2 cos π/7 − 1)/4 (which is less than 4.2102). If K is a tetrahedron, then the latter estimate is sharp. Bibliography: 8 titles.

Polygons inscribed in a closed curve and a three-dimensional convex body

Here are samples of results obtained in the paper. Let γ be a centrally symmetric closed curve in ℝ n that does not contain its center of symmetry, O. Then γ is circumscribed about a square (with center O), as well as about a rhombus (also with center O) whose vertices split γ into parts of equal length. If n is odd, then there is a centrally symmetric equilateral 2n-link polyline inscribed in γ and lying in a hyperplane. Let K ⊂ ℝ3 be a convex body, and let x ∈ (0; 1). Then K is circumscribed about an affine-regular pentagonal prism P such that the ratio of the lateral edge l of P to the longest chord of K parallel to l is equal to x. Bibliography: 7 titles.

Fully discrete potential-based finite element methods for a transient eddy current problem

Fully discrete potential-based finite element methods called $${\mathbf{A}-\phi}$$methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using $${\mathbf{A}-\phi}$$methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7.

A packing algorithm for three-dimensional convex particles

Simulation of granular particles is an important tool in many fields. However, simulation of particles of complex shapes remains largely out of reach even in two-dimension. One of the major hurdles is the difficulty in representing particles in an efficient, flexible, and accurate manner. By representing particles as convex polyhedrons which are themselves the intersection of a set of half spaces, we develop a method that allows one to efficiently carry out key operations, including particle–particle and particle–container wall overlapping detection, precise identification of the overlapping region, particle shifting, particle rotation, and others. The simulation of packing 1,000 particles into a container takes only a few minutes with this approach. We further demonstrate the potential of this approach with a simulation that re-generates the “Brazil nut” phenomenon by mixing and shaking particles of two different sizes.

3차원 Acceleration Convex Polytope를 기반으로 한 로봇 손의 안정한 파지 분석

This article describes the analysis of stable grasping for multi-fingered robot. An analysis method of stable grasping, which is based on the three-dimensional acceleration convex polytope, is proposed. This method is derived from combining dynamic equations governing object motion and robot motion, force relationship and acceleration relationship between robot fingers and object`s gravity center through contact condition, and constraint equations for satisfying no-slip conditions at every contact points. After mapping no-slip condition to torque space, we derived intersected region of given torque bounds and the mapped region in torque space so that the intersected region in torque space guarantees no excessive torque as well as no-slip at the contact points. The intersected region in torque space is mapped to an acceleration convex polytope corresponding to the maximum acceleration boundaries which can be exerted by the robot fingers under the given individual bounds of each joints torque and without causing slip at the contacts. As will be shown through the analysis and examples, the stable grasping depends on the joint driving torque limits, the posture and the mass of robot fingers, the configuration and the mass of an object, the grasp position, the friction coefficients between the object surface and finger end-effectors.

The covariogram determines three-dimensional convex polytopes

The cross covariogram g K , L of two convex sets K , L ⊂ R n is the function which associates to each x ∈ R n the volume of the intersection of K with L + x . The problem of determining the sets from this function is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that g K , K determines three-dimensional convex polytopes K and that g K , L determines both K and L when K and L are convex polyhedral cones satisfying certain assumptions. These results settle a conjecture of G. Matheron in the class of convex polytopes. Further results regard the known counterexamples in dimension n ⩾ 4 .