Progressive-iterative approximation method

=== Implicit-PIA ===

=== Implicit-PIA ===

The PIA format for implicit curve and surface reconstruction is presented in the following.<ref name=":5" /> Given an ordered point cloud <math display="inline">\left\{\mathbf{Q}_i\right\}_{i=1}^n</math> and a unit normal vector <math display="inline">\left\{\mathbf{n}_i\right\}_{i=1}^n</math> on the data points, we want to reconstruct an implicit curve from the given point cloud. To avoid a trivial solution, some offset points <math display="inline">\left\{\mathbf{Q}_l\right\}_{l=n+1}^{2n}</math> are added to the point cloud.<ref name=":5" /> They are offset by a distance <math display="inline">\sigma</math> along the unit normal vector of each point

The PIA format for implicit curve and surface reconstruction is presented in the following.<ref name=":5" /> Given an ordered point cloud <math display="inline">\left\{\mathbf{Q}_i\right\}_{i=1}^n</math> and a unit normal vector <math display="inline">\left\{\mathbf{n}_i\right\}_{i=1}^n</math> on the data points, we want to reconstruct an implicit curve from the given [[point cloud]]. To avoid a trivial solution, some offset points <math display="inline">\left\{\mathbf{Q}_l\right\}_{l=n+1}^{2n}</math> are added to the point cloud.<ref name=":5" /> They are offset by a distance <math display="inline">\sigma</math> along the unit normal vector of each point

<math display="block">

<math display="block">

\mathbf{Q}_l=\mathbf{Q}_i+\sigma\mathbf{n}_i,\quad l=n+i,\quad i=1,2,\cdots,n.

\mathbf{Q}_l=\mathbf{Q}_i+\sigma\mathbf{n}_i,\quad l=n+i,\quad i=1,2,\cdots,n.

\right.

\right.

</math>

</math>

where <math display="inline">\Omega_p^{in}</math> and <math display="inline">\Omega_p^{bd}</math> indicate the interior and boundary of the parameter domain, respectively. Each <math display="inline">A_j(\hat\tau)</math> corresponds to the <math display="inline">j</math>th control coefficient. Assume that <math display="inline">J_{in}</math> and <math display="inline">J_{bd}</math> are the index sets of the internal and boundary control coefficients, respectively. 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.,

where <math display="inline">\Omega_p^{in}</math> and <math display="inline">\Omega_p^{bd}</math> indicate the interior and boundary of the parameter domain, respectively. Each <math display="inline">A_j(\hat\tau)</math> corresponds to the <math display="inline">j</math>th control coefficient. Assume that <math display="inline">J_{in}</math> and <math display="inline">J_{bd}</math> are the index sets of the internal and boundary control coefficients, respectively. [[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.,

<math display="block">

<math display="block">

u_{j}^{(k)}=u_{j}^{*},\quad j\in J_{bd},\quad k=0,1,2,\cdots.

u_{j}^{(k)}=u_{j}^{*},\quad j\in J_{bd},\quad k=0,1,2,\cdots.

* '''Optimal weight''': Lu initially presented a weighted progressive-iterative approximation (WPIA) that introduces the optimal weight of difference vectors for control points to accelerate the convergence.<ref>{{Cite journal |last=Lu |first=Lizheng |title=Weighted progressive iteration approximation and convergence analysis |journal=Computer Aided Geometric Design |date=2010 |volume=27 |issue=2 |pages=129–137 |doi=10.1016/j.cagd.2009.11.001 |issn=0167-8396}}</ref> Moreover, Zhang et al. proposed a weighted local PIA format for tensor Bézier surfaces.<ref>{{Cite journal |last=Zhang |first=Li |date=2014-05-01 |title=Weighted Local Progressive Iterative Approximation for Tensor-product B\'{e}zier Surfaces |journal=Journal of Information and Computational Science |volume=11 |issue=7 |pages=2117–2124 |doi=10.12733/jics20103359 |issn=1548-7741}}</ref> Li et al. assigned initial weights to each data point, and the weights of the interpolated points are determined adaptively during the iterative process.<ref>{{Cite journal |last1=Li |first1=Shasha |last2=Xu |first2=Huixia |last3=Deng |first3=Chongyang |year=2019 |title=Data-weighted least square progressive and iterative approximation and related B-Spline curve fitting |journal=Journal of Computer-Aided Design & Computer Graphics |volume=31 |issue=9 |pages=1574–1580}}</ref>

* '''Optimal weight''': Lu initially presented a weighted progressive-iterative approximation (WPIA) that introduces the optimal weight of difference vectors for control points to accelerate the convergence.<ref>{{Cite journal |last=Lu |first=Lizheng |title=Weighted progressive iteration approximation and convergence analysis |journal=Computer Aided Geometric Design |date=2010 |volume=27 |issue=2 |pages=129–137 |doi=10.1016/j.cagd.2009.11.001 |issn=0167-8396}}</ref> Moreover, Zhang et al. proposed a weighted local PIA format for tensor Bézier surfaces.<ref>{{Cite journal |last=Zhang |first=Li |date=2014-05-01 |title=Weighted Local Progressive Iterative Approximation for Tensor-product B\'{e}zier Surfaces |journal=Journal of Information and Computational Science |volume=11 |issue=7 |pages=2117–2124 |doi=10.12733/jics20103359 |issn=1548-7741}}</ref> Li et al. assigned initial weights to each data point, and the weights of the interpolated points are determined adaptively during the iterative process.<ref>{{Cite journal |last1=Li |first1=Shasha |last2=Xu |first2=Huixia |last3=Deng |first3=Chongyang |year=2019 |title=Data-weighted least square progressive and iterative approximation and related B-Spline curve fitting |journal=Journal of Computer-Aided Design & Computer Graphics |volume=31 |issue=9 |pages=1574–1580}}</ref>

* '''Acceleration with memory''': In 2020, Huang et al. proposed a PIA method with memory for least square fitting (MLSPIA), which has a similar format to the momentum method. MLSPIA generates a series of fitting curves with three weights by iteratively adjusting the control points. With appropriate parameter selection, these curves converge to the least squares fit results for a given data point and are more efficient than LSPIA.<ref>{{Cite journal |last1=Huang |first1=Zheng-Da |last2=Wang |first2=Hui-Di |date=2020 |title=On a progressive and iterative approximation method with memory for least square fitting |journal=Computer Aided Geometric Design |volume=82 |pages=101931 |arxiv=1908.06417 |doi=10.1016/j.cagd.2020.101931 |issn=0167-8396 |s2cid=201070122}}</ref>

* '''Acceleration with memory''': In 2020, Huang et al. proposed a PIA method with memory for least square fitting (MLSPIA), which has a similar format to the momentum method. MLSPIA generates a series of fitting curves with three weights by iteratively adjusting the control points. With appropriate parameter selection, these curves converge to the least squares fit results for a given data point and are more efficient than LSPIA.<ref>{{Cite journal |last1=Huang |first1=Zheng-Da |last2=Wang |first2=Hui-Di |date=2020 |title=On a progressive and iterative approximation method with memory for least square fitting |journal=Computer Aided Geometric Design |volume=82 |pages=101931 |arxiv=1908.06417 |doi=10.1016/j.cagd.2020.101931 |issn=0167-8396 |s2cid=201070122}}</ref>

* '''Stochastic descent strategy''': Rios and Jüttle explored the relationship between LSPIA and gradient descent method and proposed a stochastic LSPIA algorithm with parameter correction.<ref>{{Cite journal |last1=Rios |first1=Dany |last2=Jüttler |first2=Bert |date=2022 |title=LSPIA, (stochastic) gradient descent, and parameter correction |journal=Journal of Computational and Applied Mathematics |volume=406 |pages=113921 |doi=10.1016/j.cam.2021.113921 |issn=0377-0427 |s2cid=244018717}}</ref>

* '''Stochastic descent strategy''': Rios and Jüttle explored the relationship between LSPIA and [[gradient descent]] method and proposed a stochastic LSPIA algorithm with parameter correction.<ref>{{Cite journal |last1=Rios |first1=Dany |last2=Jüttler |first2=Bert |date=2022 |title=LSPIA, (stochastic) gradient descent, and parameter correction |journal=Journal of Computational and Applied Mathematics |volume=406 |pages=113921 |doi=10.1016/j.cam.2021.113921 |issn=0377-0427 |s2cid=244018717}}</ref>

== Applications ==

== Applications ==