Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, present the local properties and accelerating techniques, and show their convergence. Moreover, because it is easy to integrate geometric constraints in the iterative procedure, GIM has been widely applied in geometric design and related areas. We survey the successful applications of geometric iterative methods, including applications in geometric design, data fitting, reverse engineering, mesh and NURBS solid generation.

Survey on geometric iterative methods and their applications

Data fitting is an extensively employed modeling tool in geometric design. With the advent of the big data era, the data sets to be fitted are made larger and larger, leading to more and more leastsquares fitting systems with singular coefficient matrices. LSPIA (least-squares progressive iterative approximation) is an efficient iterative method for the least-squares fitting. However, the convergence of LSPIA for the singular least-squares fitting systems remains as an open problem. In this paper, the authors showed that LSPIA for the singular least-squares fitting systems is convergent. Moreover, in a special case, LSPIA converges to the Moore-Penrose (M-P) pseudo-inverse solution to the leastsquares fitting result of the data set. This property makes LSPIA, an iterative method with clear geometric meanings, robust in geometric modeling applications. In addition, the authors discussed some implementation detail of LSPIA, and presented an example to validate the convergence of LSPIA for the singular least-squares fitting systems.

The Convergence of Least-Squares Progressive Iterative Approximation for Singular Least-Squares Fitting System

For several years, researchers have focused on Reverse Engineering techniques, which generate the geometric model of an object from a point set, a mesh, a triangulation, etc. These data are...

Reconstruction of a CAD model from the deformed mesh using B-spline surfaces

A Sustainable Reverse Engineering Process

We introduce a framework for modeling of heterogeneous objects in terms of trivariate B-spline functions and a method for slicing them directly for additive manufacturing. We first fit volumetric attribute data associated with the geometry in terms of trivariate B-spline functions under the assumption that the geometric volume is already defined by the trivariate B-spline functions. Then, the B-spline volume and the associated attribute data are directly sliced without converting them to stereo-lithography format, resulting in a tool path with fewer errors. Furthermore, adaptive ray shooting is introduced in the slicing plane so that the zigzag tool path passes through all the tangential intersection points of the heterogeneous objects to represent all the feature points in the fabricated model. Complex examples illustrate the effectiveness of our method.

Adaptive direct slicing of volumetric attribute data represented by trivariate B-spline functions

Topologically robust B-spline surface reconstruction from point clouds using level set methods and iterative geometric fitting algorithms

A method for reconstructing a complete three-dimensional (3D) voxel model from multiple images captured during two circular motions of an object is proposed. The two circular motions include the object with a regular orientation and the corresponding reversed orientation. The shape-from-silhouette method, which constructs a voxelbased 3D model from the silhouettes of two-dimensional (2D) images, is used for the reconstruction. In general, a single camera focused on an object’s circular motion cannot capture the entire object using a single fixed orientation. Thus, it is required to invert the orientation of the object at least once to capture the entire object. In this paper, a novel method for integrating two incomplete reconstructed voxel models to produce a single complete voxel model is discussed. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.

https://www.jstage.jst.go.jp/article/jamdsm/7/3/7_362/_pdf

Reconstruction of Complete 3D Models by Voxel Integration

3308 Reverse Engineering of Complex 3D Objects by Voxel Integration

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Topologically robust B-spline surface reconstruction from point clouds using level set methods and iterative geometric fitting algorithms

Reconstruction of Complete 3D Models by Voxel Integration

3308 Reverse Engineering of Complex 3D Objects by Voxel Integration