Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

Automatic Screening of Lung Diseases by 3D Active Contour Method for Inhomogeneous Motion Estimation in CT Image Pairs

Lung diseases are now the third leading cause of death worldwide because of the many risk factors we are exposed to daily, such as air pollution, tobacco use, viruses (such as COVID-19), and bacteria. This work introduces a new approach of the 3D Active Contour Model (3D ACM) to estimate an inhomogeneous motion of lungs, which can be used to analyze lung disease patterns using a hierarchical predictive model. The biophysical model of lungs consists of End Expiratory (EE) and End Inspiratory (EI) models, generated by high-resolution computed tomography images (HRCT). A proposed technique uses the 3D ACM to estimate the velocity vector by using the corresponding points on the parametric surface model of the EE model to the EI model. The external energy from the EI models is the external force that pushes the 3D parametric surface to reach the boundary. The external forces, such as the balloon force and Gradient Vector Flow (GVF), were adjusted adaptively based on the which was calculated from the ratio of the maximum value of EI to EE on the Z axis. Next, the feature representation is studied and evaluated based on the lung structure, separated into five lobes. The stepwise regression, Support Vector Machine (SVM), and Artificial Neural Network (ANN) techniques are applied to classify the lung diseases into normal, obstructive lung, and restrictive lung diseases. In conclusion, the inhomogeneous motion pattern of lungs integrated with medical-based knowledge can be used to analyze lung diseases by differentiating normal and abnormal motion patterns and separating restrictive and obstructive lung diseases. HIGHLIGHTS Inhomogeneous motion analysis from the expanding and shrinking lungs of HRCT pair Adaptive 3D Active Contour Model (ACM) for detecting the shape of the lung by balancing the balloon force with the stopping condition Lung lopes separation using oblique fissure and anatomical location Structure the velocity vector map of lung motion using bag of words of the magnitude Neural Network model for predicting obstructive and restrictive lung diseases GRAPHICAL ABSTRACT

The μ-Basis of Improper Rational Parametric Surface and Its Application

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

Parametric Surface Modelling for Tea Leaf Point Cloud Based on Non-Uniform Rational Basis Spline Technique

Plant leaf 3D architecture changes during growth and shows sensitive response to environmental stresses. In recent years, acquisition and segmentation methods of leaf point cloud developed rapidly, but 3D modelling leaf point clouds has not gained much attention. In this study, a parametric surface modelling method was proposed for accurately fitting tea leaf point cloud. Firstly, principal component analysis was utilized to adjust posture and position of the point cloud. Then, the point cloud was sliced into multiple sections, and some sections were selected to generate a point set to be fitted (PSF). Finally, the PSF was fitted into non-uniform rational B-spline (NURBS) surface. Two methods were developed to generate the ordered PSF and the unordered PSF, respectively. The PSF was firstly fitted as B-spline surface and then was transformed to NURBS form by minimizing fitting error, which was solved by particle swarm optimization (PSO). The fitting error was specified as weighted sum of the root-mean-square error (RMSE) and the maximum value (MV) of Euclidean distances between fitted surface and a subset of the point cloud. The results showed that the proposed modelling method could be used even if the point cloud is largely simplified (RMSE < 1 mm, MV < 2 mm, without performing PSO). Future studies will model wider range of leaves as well as incomplete point cloud.