National flags are very important symbols of countries.They represent the countries’s authority and dignity of a country.However,there are some occasions where the flags were produced incorrectly(or as frauds)but sti...National flags are very important symbols of countries.They represent the countries’s authority and dignity of a country.However,there are some occasions where the flags were produced incorrectly(or as frauds)but still hung officially without a formal inspection.In this paper,we propose a photogrammetric inspection method for hanging national flags of China,for which only one single image is required to perform the inspection.The proposed method allows automatic estimation of the relative positions and orientation of the pentagrams,so exposure of inappropriate flags can be identified avoided.The method invokes a novel 2D geometric model of a pentagram(five-pointed star)to constrain an adjustment to estimate the camera’s exterior orientation parameters based on a single image of a statically hung flag.Conventional error parameters such as the radial distortion parameters are integrated into the pentagram model to form a calibration process to reduce the 3D reconstruction errors.Once the camera and the distortion parameters are estimated,the relative positions,orientations,and dimensions of all the five pentagrams can be readily computed with independent pentagram fitting so the flag quality can be verified using the national standard.More than 20 different hanging flags were captured to verify the proposed method.The results indicate that the method is flexible and accurate,with an accuracy of 1.1mm for the position/dimension,and 0.2°for the orientation on average.Since the method is based on the proposed geometric model of the pentagram,it can be readily adapted to form another system to verify other countries’national flags containing more than one pentagram.展开更多
Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present wor...Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “<em>The point and the straight line are one and the same</em>”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: <em>ab</em> = <em>ba</em> and (<em>ab</em>)(<em>ac</em>) = <em>a</em>, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.展开更多
基金Science and Technology Program of Guangzhou,China(202102080287)。
文摘National flags are very important symbols of countries.They represent the countries’s authority and dignity of a country.However,there are some occasions where the flags were produced incorrectly(or as frauds)but still hung officially without a formal inspection.In this paper,we propose a photogrammetric inspection method for hanging national flags of China,for which only one single image is required to perform the inspection.The proposed method allows automatic estimation of the relative positions and orientation of the pentagrams,so exposure of inappropriate flags can be identified avoided.The method invokes a novel 2D geometric model of a pentagram(five-pointed star)to constrain an adjustment to estimate the camera’s exterior orientation parameters based on a single image of a statically hung flag.Conventional error parameters such as the radial distortion parameters are integrated into the pentagram model to form a calibration process to reduce the 3D reconstruction errors.Once the camera and the distortion parameters are estimated,the relative positions,orientations,and dimensions of all the five pentagrams can be readily computed with independent pentagram fitting so the flag quality can be verified using the national standard.More than 20 different hanging flags were captured to verify the proposed method.The results indicate that the method is flexible and accurate,with an accuracy of 1.1mm for the position/dimension,and 0.2°for the orientation on average.Since the method is based on the proposed geometric model of the pentagram,it can be readily adapted to form another system to verify other countries’national flags containing more than one pentagram.
文摘Numerous authors studied polarities in incidence structures or algebrization of projective geometry <a href="#1">[1]</a> <a href="#2">[2]</a>. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “<em>The point and the straight line are one and the same</em>”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: <em>ab</em> = <em>ba</em> and (<em>ab</em>)(<em>ac</em>) = <em>a</em>, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.
