An Efficient Geometric Method for 3D Reconstruction Based on Images

An Efficient Geometric Method for 3D Reconstruction Based on Images

Ying Zhu, Jingyue Jiang, Zhiling Tang, Fang Wang, Rong Wang, Si Zhong, Xiaonan Luo
Copyright: © 2018 |Pages: 13
DOI: 10.4018/IJGHPC.2018040103
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In this article, a new geometric method for 3D reconstruction based on images is presented. The main advantage of this method is low data storage requirement and high computation process efficiency. The key mathematical methods are curve and surface modeling technology. In order to implement geometric method for 3D reconstruction, a new surface is developed named as rational W-spline surface via tensor product, through the important properties of the basis function, the authors get the important properties of the rational surface. By giving the new control matrix and the change of the parameters based on the images, 3D reconstruction effect can be obtained. This surface includes non-rational surface and rational surface, and their tensor product forms are adopted. In order to express this algorithm intuitively, the bilinear W-spline surface is realized by the experiment. The experimental results showed that the geometric method can be used to the 3D reconstruction.
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1. Introduction

In computer-aided design and digital geometry processing, it involves a lot of data, such as image data and geometric data. In order to represent and process this data, the challenging research issues include how to decrease the storage requirement and how to design the highly performance computation algorithm. In the application of surface reconstruction, some important research results are reported, such as big data computing and simulation modeling (Chen, Wang, Sun & Lu, 2016), shape analysis, animation, medical phenomena (Zhang, Hoffman & Reinhardt, 2006), image processing, image representation and so on. The aim of surface reconstruction is essentially to construct a mathematical model to represent a surface based on a set of data points. The advantage of using geometric method is low data storage requirement and high computation process efficiency.

Researchers made tremendous effort on reconstruction of 3D shapes and some achieved significant results. Spline curves and surfaces are important geometric method in the CAD. When the curves and surfaces are created, they can be modified to satisfy the design requirement (Wu & Tao, 2009). By changing the control points of the curves via optimizations and constrained conditions, it is more convenient for designing and modifying curves and surfaces. Bézier presented a method to control the vertices of polygon, the designer can easily modify the shape of the curve by moving the control vertices to obtain the satisfying shape (Gálvez, Iglesias & Avila, 2013; Belbis, Garnier & Foufou, 2013).The representation of shapes using parametric curves and surfaces is widely used in domains that make use of computer graphics, such as industrial design, the animation industry, as well as for the analysis of biomedical images (Schmitter, Delgado-Gonzalo & Unser, 2016). DeBoor reported a standard algorithm about B-spline curves and surfaces (Rogina & Bosner, 2003). Bézier curve and B-spline curve are two important methods of curve modeling in CAGD (Guid, Kolmanic & Strnad, 2006; Prautzsch, Boehm, & Paluszny, 2002). B-splines have become the industrial standard for representing geometric information in the CAGD fields (Yamaura, Nanya, Imoto & Maekawa, 2015). Once objects are represented in a B-spline form, they can be used in a number of downstream applications (Patrikalakis & Maekawa, 2010).

As the well-known rational Bézier and B-spline curves and surfaces, the weights can also be used to modify their shapes (Schmitter, Delgado-Gonzalo & Unser, 2016; Zeng, 2009). The spline-based representation of parametric shapes proved to be a convenient choice about user interactivity in shape modeling due to the underlying control-point-based nature of spline functions. The blending functions of Bernstein polynomials with shifted knots (Khan, Lobiyal & Kilicman, 2015) were applied to construct the Bézier curves and surfaces. An analysis of formal CAD modeling strategies and best practices for history-based parametric design were presented (Camba, Contero & Company, 2016). Zieniuket et al. proposed an alternative approach to boundary shape representation for 3D boundary value problems based on parametric surface patches. Surface patches is used to describe a shape of 3D objects using a given set of control points and associated basis functions (Zieniuk & Szerszen, 2013). In the shape matching application, the structural and boundary information are combined and applied to establish regional correspondences in temporal pairs of mammograms (Jananet et al., 2015). A group testing based method is proposed for B-spline curve and surface fitting that approximates a model by processing few data points. This method can be employed without an initial estimation and find both knot vector and control points simultaneously by finding the most informative or salient points (Norouzzadeh Ravari & Taghirad, 2016).

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