The progressive and iterative approximation (PIA) method is an efficient and intuitive method for data fitting. However, in the classical PIA method, the number of the control points is equal to that of the data points. It is not feasible when the number of data points is very large. In this paper, we develop a new progressive and iterative approximation for least square fitting (LSPIA). LSPIA constructs a series of fitting curves (surfaces) by adjusting the control points iteratively, and the limit curve (surface) is the least square fitting result to the given data points. In each iteration, the difference vector for each control point is a weighted sum of some difference vectors between the data points and their corresponding points on the fitting curve (surface). Moreover, we present a simple method to compute the practical weight whose corresponding convergence rate is comparable to that of the theoretical best weight. The advantages of LSPIA are two-fold. First, with LSPIA, a very large data set can be fitted efficiently and robustly. Second, in the incremental data fitting procedure with LSPIA, a new round of iterations can be started from the fitting result of the last round of iterations, thus saving great amount of computation. Lots of empirical examples illustrated in this paper show the efficiency and effectiveness of LSPIA.

Progressive and iterative approximation for least squares B-spline curve and surface fitting

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

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline's control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.

B-spline surface fitting by iterative geometric interpolation/approximation algorithms

Technical Section: An extended iterative format for the progressive-iteration approximation

Local progressive-iterative approximation format for blending curves and patches

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C^2 continuity We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method Examples are provided to demonstrate the effectiveness of the proposed algorithms

Point-tangent/point-normal B-spline curve interpolation by geometric algorithms

When an interpolation point is interpolated using a curve/curved surface, represented by control points, such as a B-Spline curve/curved surface or a subdivision surface, interpolation processing is performed in such a way that a moving vector for moving each control point is calculated using position information on interpolation points constituting the point group, as well as vector information on the unit direction vector that is set for each interpolation point, and the control point is moved in the movement direction and for the movement amount of this moving vector. By, repeating the generation step of a curve/curved surface that interpolates the interpolation point and the movement step of the control point, the position of a new control point is found that fits the curve/curved surface to the positions and the unit direction vectors of the point group.

Shu-ichi Gofuku

Papers

Point-tangent/point-normal B-spline curve interpolation by geometric algorithms

Interpolation processing method and interpolation processor