Progressive-iterative approximation method
[2] It avoids solving a linear system of equations directly and allows flexibility in adding constraints during the iterative process.
[2] The study of the iterative method with geometric meaning can be traced back to the work of scholars such as Dongxu Qi and Carl de Boor in the 1970s.
[5] In 2004, Hongwei Lin and coauthors proved that non-uniform cubic B-spline curves and surfaces have the "profit and loss" property.
[3] Later, in 2005, Lin et al. proved that the curves and surfaces with normalized and totally positive basis all have this property and named it progressive iterative approximation (PIA).
[6] In 2008, Cheng et al. extended it to subdivision surfaces and named the method progressive interpolation (PI).
Specifically, there are some representative iteration methods—such as local-PIA,[12] implicit-PIA,[11] fairing-PIA,[13] and isogeometric least-squares progressive-iterative approximation (IG-LSPIA)[14]—that are specialized for solving the isogeometric analysis problem.
To facilitate the description of the PIA iteration format for different forms of curves and surfaces, the following formula is uniformly used:
, which converges to a limit curve that interpolates the give data points,[1][9] i.e.,
For the B-spline curve and surface fitting problem, Deng and Lin proposed a least-squares progressive–iterative approximation (LSPIA),[10][19] which allows the number of control points to be less than the number of the data points and is more suitable for large-scale data fitting problems.
th fitting curve, first compute the difference vectors for the data points[10][19]
[20] The PIA format for implicit curve and surface reconstruction is presented in the following.
The above procedure is carried out iteratively to generate a sequence of algebraic B-spline functions
The sequence converges to a minimization problem with constraints when the initial control coefficients
The sequence converges to the solution of the conventional fairing method based on energy minimization when all smoothing weights are equal (
, represents the combination of different order derivatives of the NURBS basis functions determined using the operators
Without loss of generality, we further assume that the boundary control coefficients have been obtained using strong or weak imposition and are fixed, i.e.,
Moreover, group all load values whose parameters fall in the local support of the
The above iteration process is performed until the desired fitting precision is reached and a sequence of blending functions is obtained
The IG-LSPIA converges to the solution of a constrained least-squares collocation problem.
Hence, the LSPIA algorithm converges to the least squares result for a given sequence of points.
Lin et al. showed that LSPIA converges even when the iteration matrix is singular.
[18] Since PIA has obvious geometric meaning, constraints can be easily integrated in the iterations.
Currently, PIA has been widely applied in many fields, such as data fitting, reverse engineering, geometric design, mesh generation, data compression, fairing curve and surface generation, and isogeometric analysis.
For implicit curve and surface reconstruction, PIA avoids the additional zero level set and regularization term, which greatly improves the speed of the reconstruction algorithm.
Finally, the offset curve is approximated iteratively using the PIA method.
[34] First, the image data are converted into a one-dimensional sequence by Hilbert scan.
Then, these data points are fitted by LSPIA to generate a Hilbert curve.
Finally, the Hilbert curve is sampled, and the compressed image can be reconstructed.
It is also flexible to adjust knot vectors, fairing weights, or data parameterization after each round of iteration.
The method automatically adjusts the degrees of freedom of the numerical solution of the partial differential equation according to the fitting result of the blending function to the load values.
Top left: The data points and the initial control polygon (Here, the initial control points are taken as the data points). Top right: The initial curve and the difference vectors. Bottom left: New control polygon is generated by adding the difference vectors to the old control points. Bottom right: The new control polygon and new curve (in purple).
Top left: Data points (blue circles), initial control polygon (green lines) constructed from a subset of , and initial fitting curve . Top right: Difference vectors for data points and difference vectors for control points. Bottom: A new control polygon (purple lines) is generated by adding to the old control points; it then creates the next fitting curve (purple curve).
