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<title>Progressive-iterative approximation method</title>

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Progressive-iterative approximation method

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In mathematics, the progressive-iterative approximation method is an <a href="Iterative_method" title="Iterative method">iterative method</a> of <a href="Curve_fitting" title="Curve fitting">data fitting</a> with geometric meanings.<a href="#cite_note-:0-1">[1]</a> Given a set of data points to be fitted, the method obtains a series of fitting curves (or surfaces) by iteratively updating the control points, and the <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> curve (surface) can <a href="Interpolation" title="Interpolation">interpolate</a> or <a href="Approximation_theory" title="Approximation theory">approximate</a> the given data points.<a href="#cite_note-:9-2">[2]</a> It avoids solving a linear system of equations directly and allows flexibility in adding constraints during the iterative process.<a href="#cite_note-:1-3">[3]</a> Therefore, it has been widely used in geometric design and related fields.<a href="#cite_note-:9-2">[2]</a>

The study of the iterative method with geometric meaning can be traced back to the work of scholars such as Dongxu Qi and <a href="Carl_R._de_Boor" title="Carl R. de Boor">Carl de Boor</a> in the 1970s.<a href="#cite_note-:10-4">[4]</a><a href="#cite_note-:11-5">[5]</a> In 1975, Qi et al. developed and proved the "profit and loss" algorithm for uniform cubic <a href="B-spline" title="B-spline">B-spline</a> curves,<a href="#cite_note-:10-4">[4]</a> and in 1979, de Boor independently proposed this algorithm.<a href="#cite_note-:11-5">[5]</a> In 2004, Hongwei Lin and coauthors proved that non-uniform cubic B-spline curves and surfaces have the "profit and loss" property.<a href="#cite_note-:1-3">[3]</a> 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).<a href="#cite_note-:0-1">[1]</a> In 2007, Maekawa et al. changed the algebraic distance in PIA to geometric distance and named it geometric interpolation (GI).<a href="#cite_note-6">[6]</a> In 2008, Cheng et al. extended it to <a href="Subdivision_surface" title="Subdivision surface">subdivision surfaces</a> and named the method progressive interpolation (PI).<a href="#cite_note-7">[7]</a> Since the iteration steps of the PIA, GI, and PI algorithms are similar and all have geometric meanings, they are collectively referred to as geometric iterative methods (GIM).<a href="#cite_note-:9-2">[2]</a>

PIA is now extended to several common curves and surfaces in the <a href="Geometric_design" title="Geometric design">geometric design</a> field,<a href="#cite_note-:13-8">[8]</a> including <a href="Non-uniform_rational_B-spline" title="Non-uniform rational B-spline">NURBS</a> curves and surfaces,<a href="#cite_note-:2-9">[9]</a> <a href="T-spline" title="T-spline">T-spline</a> surfaces,<a href="#cite_note-:3-10">[10]</a> and <a href="Implicit_curve" title="Implicit curve">implicit curves</a> and <a href="Implicit_surface" title="Implicit surface">surfaces</a>.<a href="#cite_note-:5-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Iteration_methods">Iteration methods</h2></div>

Generally, progressive-iterative approximation (PIA) can be divided into interpolation and approximation schemes.<a href="#cite_note-:9-2">[2]</a> In interpolation algorithms, the number of control points is equal to that of the data points; in approximation algorithms, the number of control points can be less than that of the data points. Specifically, there are some representative iteration methods—such as local-PIA,<a href="#cite_note-:6-12">[12]</a> implicit-PIA,<a href="#cite_note-:5-11">[11]</a> fairing-PIA,<a href="#cite_note-:7-13">[13]</a> and isogeometric least-squares progressive-iterative approximation (IG-LSPIA)<a href="#cite_note-:8-14">[14]</a>—that are specialized for solving the <a href="Isogeometric_analysis" title="Isogeometric analysis">isogeometric analysis</a> problem.<a href="#cite_note-:14-15">[15]</a>

<div class="mw-heading mw-heading3"><h3 id="Interpolation_scheme:_PIA">Interpolation scheme: PIA</h3></div>

In interpolation algorithms of PIA,<a href="#cite_note-:0-1">[1]</a><a href="#cite_note-:1-3">[3]</a><a href="#cite_note-:2-9">[9]</a><a href="#cite_note-16">[16]</a> every data point is used as a control point. To facilitate the description of the PIA iteration format for different forms of curves and surfaces, the following formula is uniformly used:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (\mathbf {t} )=\sum _{i=1}^{n}\mathbf {P} _{i}B_{i}(\mathbf {t} ).}</annotation>

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For example:

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</math><img src="./e13ee826ae98c55bf0e9ba93387199a7eccdd07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.675ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (\mathbf {t} )}" loading="lazy"> is a B-spline curve, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} }</annotation>

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</math><img src="./16ff63cc74a931900e79b3caacbae3fa8cc66845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} }" loading="lazy"> is a scalar, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle B_{i}(t)}</annotation>

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</math><img src="./13af4e4f3148782cf2bed68c7aaae3176939616a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.213ex; height:2.843ex;" alt="{\displaystyle B_{i}(t)}" loading="lazy"> is a B-spline basis function, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{i}}</annotation>

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</math><img src="./ebfbf0261f6dcefcfb6cdc7922af1c84f3c9d218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.626ex; height:2.509ex;" alt="{\displaystyle \mathbf {P} _{i}}" loading="lazy"> denotes the control point;<a href="#cite_note-:13-8">[8]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (\mathbf {t} )}</annotation>

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</math><img src="./e13ee826ae98c55bf0e9ba93387199a7eccdd07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.675ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (\mathbf {t} )}" loading="lazy"> is a B-spline patch with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{u}\times n_{v}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{u}\times n_{v}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} =(u,v)}</annotation>

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</math><img src="./868aac434ec43f9b191dfa0b346ecb24058c7664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.438ex; height:2.843ex;" alt="{\displaystyle \mathbf {t} =(u,v)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}(\mathbf {t} )=N_{i}(u)N_{i}(v)}">

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<annotation encoding="application/x-tex">{\displaystyle B_{i}(\mathbf {t} )=N_{i}(u)N_{i}(v)}</annotation>

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</math><img src="./9a109d40aa44f58c0646779997cf307fe128e89b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.918ex; height:2.843ex;" alt="{\displaystyle B_{i}(\mathbf {t} )=N_{i}(u)N_{i}(v)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{i}(u)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(u)}</annotation>

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</math><img src="./4de05db0191b67766ca33b85a40d5d07f9778427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.805ex; height:2.843ex;" alt="{\displaystyle N_{i}(u)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{i}(v)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(v)}</annotation>

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</math><img src="./ead6efd74586b28f280af865bea40c6a41eae2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.603ex; height:2.843ex;" alt="{\displaystyle N_{i}(v)}" loading="lazy"> are B-spline basis functions;<a href="#cite_note-:13-8">[8]</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (\mathbf {t} )}</annotation>

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</math><img src="./e13ee826ae98c55bf0e9ba93387199a7eccdd07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.675ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (\mathbf {t} )}" loading="lazy"> is a trivariate B-spline solid with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{u}\times n_{v}\times n_{w}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{u}\times n_{v}\times n_{w}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {t} =(u,v,w)}</annotation>

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</math><img src="./752bc9e2c643bc470b21a666e3a38544b86d2979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.136ex; height:2.843ex;" alt="{\displaystyle \mathbf {t} =(u,v,w)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}(\mathbf {t} )=N_{i}(u)N_{i}(v)N_{i}(w)}">

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<annotation encoding="application/x-tex">{\displaystyle B_{i}(\mathbf {t} )=N_{i}(u)N_{i}(v)N_{i}(w)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(u)}</annotation>

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</math><img src="./4de05db0191b67766ca33b85a40d5d07f9778427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.805ex; height:2.843ex;" alt="{\displaystyle N_{i}(u)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{i}(v)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(v)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(w)}</annotation>

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Additionally, this can be applied to NURBS curves and surfaces, T-spline surfaces, and triangular Bernstein–Bézier surfaces.<a href="#cite_note-:12-18">[18]</a>

Given an ordered data set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle i=1,2,\cdots ,n}</annotation>

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</math><img src="./14823e065543af3a10af6878d54744b6bad6fe01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.833ex; height:2.509ex;" alt="{\displaystyle i=1,2,\cdots ,n}" loading="lazy">, the initial fitting curve is:<a href="#cite_note-:0-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(0)}(t)=\sum _{i=1}^{n}\mathbf {P} _{i}^{(0)}B_{i}(t)}</annotation>

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where the initial control points of the initial fitting curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{i}^{(0)}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{i}^{(0)}}</annotation>

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</math><img src="./6facf65d5f40ede9303c015e42fb2667c988bd93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.16ex; height:3.676ex;" alt="{\displaystyle \mathbf {P} _{i}^{(0)}}" loading="lazy"> can be randomly selected. Suppose that after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">th fitting curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{(k)}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(k)}(t)}</annotation>

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</math><img src="./26b7f26b4f3fe073da885dafb2b5a1f3bf25565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.844ex; height:3.343ex;" alt="{\displaystyle \mathbf {P} ^{(k)}(t)}" loading="lazy"> is generated by

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{(k)}(t)=\sum _{i=1}^{n}\mathbf {P} _{i}^{(k)}B_{i}(t).}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(k)}(t)=\sum _{i=1}^{n}\mathbf {P} _{i}^{(k)}B_{i}(t).}</annotation>

</semantics>

</math><img src="./5b4862c5d72c759acc87bf06016f93910e6f6508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.739ex; height:6.843ex;" alt="{\displaystyle \mathbf {P} ^{(k)}(t)=\sum _{i=1}^{n}\mathbf {P} _{i}^{(k)}B_{i}(t).}" loading="lazy"></td> <td></td> <td class="nowrap">1</td></tr></tbody></table>

To construct the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

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</mrow>

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</msubsup>

<msub>

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</msup>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Delta } _{i}^{(k)}=\mathbf {Q} _{i}-\mathbf {P} ^{(k)}(t_{i}),\quad i=1,2,\cdots ,n}</annotation>

</semantics>

</math>

and use them to update the control points by

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{i}^{(k+1)}=\mathbf {P} _{i}^{(k)}+\mathbf {\Delta } _{i}^{(k)}}</annotation>

</semantics>

</math>

which leads to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(k+1)}(t)=\sum _{i=1}^{n}\mathbf {P} _{i}^{(k+1)}B_{i}(t).}</annotation>

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</math>

In this way, we obtain a sequence of curves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {P} ^{(\alpha )}(t),\alpha =0,1,2,\cdots }">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {P} ^{(\alpha )}(t),\alpha =0,1,2,\cdots }</annotation>

</semantics>

</math><img src="./fe6f8ee905d9dbd96d363dbf9cb15d6fd0dafb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.972ex; height:3.343ex;" alt="{\textstyle \mathbf {P} ^{(\alpha )}(t),\alpha =0,1,2,\cdots }" loading="lazy">, which converges to a limit curve that interpolates the give data points,<a href="#cite_note-:0-1">[1]</a><a href="#cite_note-:2-9">[9]</a> i.e.,

</mrow>

</munder>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \lim \limits _{\alpha \rightarrow \infty }\mathbf {P} ^{(\alpha )}(t_{i})=\mathbf {Q} _{i},\quad i=1,2,\cdots ,n.}</annotation>

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</math>

<div class="mw-heading mw-heading3"><h3 id="Approximation_scheme:_LSPIA">Approximation scheme: LSPIA</h3></div>

For the B-spline curve and surface fitting problem, Deng and Lin proposed a least-squares progressive–iterative approximation (LSPIA),<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:4-19">[19]</a> 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.<a href="#cite_note-:3-10">[10]</a>

Assume there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> data points and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> control points, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\leq m}</annotation>

</semantics>

</math><img src="./1be80a749be067190db51891139a12d4886fc054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle n\leq m}" loading="lazy">. Start with equation (<a href="#math_1">1</a>), which gives the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(k)}(t)=\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t).}</annotation>

</semantics>

</math>

To generate the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

</semantics>

</math><img src="./3f9f13644a6be482d7ddb19a6e0c706564773085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}" loading="lazy">th fitting curve, first compute the difference vectors for the data points<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:4-19">[19]</a>

</mrow>

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<msub>

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</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\delta }}_{i}^{(k)}=\mathbf {Q} _{i}-\mathbf {P} ^{(k)}(t_{i}),\quad i=1,2,\cdots ,m}</annotation>

</semantics>

</math>

and then the difference vectors for the control points

</mrow>

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</mrow>

<mrow>

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</munder>

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</msub>

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<msub>

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</msub>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Delta } _{j}^{(k)}={\frac {\sum _{i\in I_{j}}{c_{i}B_{j}(t_{i}){\boldsymbol {\delta }}_{i}^{(k)}}}{\sum _{i\in I_{j}}c_{i}B_{j}(t_{i})}},\quad j=1,2,\cdots ,n}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{j}}</annotation>

</semantics>

</math><img src="./b7cb04680c2246f78082e43ec912f72afe7af266.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.933ex; height:2.843ex;" alt="{\displaystyle I_{j}}" loading="lazy"> is the index set of the data points in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">th group, whose parameters fall in the local support of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">th basis function, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{j}(t_{i})\neq 0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{j}(t_{i})\neq 0}</annotation>

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</math><img src="./79f92ea12f8a7f569e559187859e27342da702bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.383ex; height:3.009ex;" alt="{\displaystyle B_{j}(t_{i})\neq 0}" loading="lazy">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>

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</math><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"> are weights that guarantee the convergence of the algorithm, usually taken as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}=1,i\in I_{j}}">

<msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{i}=1,i\in I_{j}}</annotation>

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</math><img src="./bdd78184126581fc36212d57a582f03b566b56db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.677ex; height:2.843ex;" alt="{\displaystyle c_{i}=1,i\in I_{j}}" loading="lazy">.

Finally, the control points of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

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</math><img src="./3f9f13644a6be482d7ddb19a6e0c706564773085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}" loading="lazy">th curve are updated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{j}^{(k+1)}=\mathbf {P} _{j}^{(k)}+\mathbf {\Delta } _{j}^{(k)},}">

</mrow>

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</msubsup>

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</mrow>

</msubsup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}^{(k+1)}=\mathbf {P} _{j}^{(k)}+\mathbf {\Delta } _{j}^{(k)},}</annotation>

</semantics>

</math><img src="./92160d4e59f9c19bd609d76c5c8a5acb9871bda3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.67ex; height:4.009ex;" alt="{\displaystyle \mathbf {P} _{j}^{(k+1)}=\mathbf {P} _{j}^{(k)}+\mathbf {\Delta } _{j}^{(k)},}" loading="lazy"> leading to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k+1)}</annotation>

</semantics>

</math><img src="./3f9f13644a6be482d7ddb19a6e0c706564773085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k+1)}" loading="lazy">th fitting curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{(k+1)}(t)}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(k+1)}(t)}</annotation>

</semantics>

</math><img src="./42af4a431fcef2377ef0a40a675266a6418786a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.944ex; height:3.343ex;" alt="{\displaystyle \mathbf {P} ^{(k+1)}(t)}" loading="lazy">. In this way, we obtain a sequence of curve, and the limit curve converges to the <a href="Least_squares" title="Least squares">least-squares fitting</a> result to the given data points.<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:4-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Local-PIA">Local-PIA</h3></div>

In the local-PIA method,<a href="#cite_note-:6-12">[12]</a> the control points are divided into active and fixed control points, whose subscripts are denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle I=\left\{i_{1},i_{2},\cdots ,i_{I}\right\}}">

<mrow>

<mrow>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle I=\left\{i_{1},i_{2},\cdots ,i_{I}\right\}}</annotation>

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</math><img src="./daa22752abc7b84d66afb4eac2d1702039b9e359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.384ex; height:2.843ex;" alt="{\textstyle I=\left\{i_{1},i_{2},\cdots ,i_{I}\right\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle J=\left\{j_{1},j_{2},\cdots ,j_{J}\right\}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</msub>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle J=\left\{j_{1},j_{2},\cdots ,j_{J}\right\}}</annotation>

</semantics>

</math><img src="./863bc57d6e90df80af09f405fe5b8b3080d5db00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.362ex; height:2.843ex;" alt="{\textstyle J=\left\{j_{1},j_{2},\cdots ,j_{J}\right\}}" loading="lazy">, respectively. Assume that, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle k}</annotation>

</semantics>

</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy">th fitting curve is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {P} ^{(k)}(t)=\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t)}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {P} ^{(k)}(t)=\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t)}</annotation>

</semantics>

</math><img src="./d006eb7f1fb2b7719156e70485fc1630d4533237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.311ex; height:4.009ex;" alt="{\textstyle \mathbf {P} ^{(k)}(t)=\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t)}" loading="lazy">, where the fixed control points satisfy

</mrow>

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</msubsup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}^{(k)}=\mathbf {P} _{j}^{(0)},\quad j\in J,\quad k=0,1,2,\cdots .}</annotation>

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</math>

Then, on the one hand, the iterative formula of the difference vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {\Delta } _{h}^{(k+1)}}">

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {\Delta } _{h}^{(k+1)}}</annotation>

</semantics>

</math><img src="./fb92844d2ec9a7c36da5855f039065ab91538dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.695ex; height:3.676ex;" alt="{\textstyle \mathbf {\Delta } _{h}^{(k+1)}}" loading="lazy"> corresponding to the fixed control points is

<mtr>

<mtd>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {\Delta } _{h}^{(k+1)}&=\mathbf {Q} _{h}-\sum _{j=1}^{n}\mathbf {P} _{j}^{(k+1)}B_{j}(t_{h})\\&=\mathbf {Q} _{h}-\sum _{j\in J}\mathbf {P} _{j}^{(k+1)}B_{j}(t_{h})-\sum _{i\in I}\left(\mathbf {P} _{i}^{(k)}+\mathbf {\Delta } _{i}^{(k)}\right)B_{i}(t_{h})\\&=\mathbf {Q} _{h}-\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t_{h})-\sum _{i\in I}\mathbf {\Delta } _{i}^{(k)}B_{i}(t_{h})\\&=\mathbf {\Delta } _{h}^{(k)}-\sum _{i\in I}\mathbf {\Delta } _{i}^{(k)}B_{i}(t_{h}),\quad h\in J.\end{aligned}}}</annotation>

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On the other hand, the iterative formula of the difference vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {D} _{l}^{(k+1)}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {\Delta } _{l}^{(k+1)}&=\mathbf {Q} _{l}-\sum _{j=1}^{n}\mathbf {P} _{j}^{(k+1)}B_{j}(t_{l})\\&=\mathbf {Q} _{l}-\sum _{j=1}^{n}\mathbf {P} _{j}^{(k)}B_{j}(t_{l})-\sum _{i\in I}\mathbf {\Delta } _{i}^{(k)}B_{i}(t_{l})\\&=\mathbf {\Delta } _{l}^{(k)}-\sum _{i\in I}\mathbf {\Delta } _{i}^{(k)}B_{i}(t_{l})\\&=-\mathbf {\Delta } _{i_{1}}^{(k)}B_{i_{1}}(t_{l})-\mathbf {\Delta } _{i_{2}}^{(k)}B_{i_{2}}(t_{l})-\cdots +\left(1-B_{l}(t_{l})\right)\mathbf {\Delta } _{l}^{(k)}-\cdots -\mathbf {\Delta } _{i_{I}}^{(k)}B_{i_{I}}(t_{l}),\quad l\in I.\end{aligned}}}</annotation>

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Arranging the above difference vectors into a one-dimensional sequence,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {D} ^{(k+1)}=\left[\mathbf {\Delta } _{j_{1}}^{(k+1)},\mathbf {\Delta } _{j_{2}}^{(k+1)},\cdots ,\mathbf {\Delta } _{j_{J}}^{(k+1)},\mathbf {\Delta } _{i_{1}}^{(k+1)},\mathbf {\Delta } _{i_{2}}^{(k+1)},\cdots ,\mathbf {\Delta } _{i_{I}}^{(k+1)}\right]^{T},\quad k=0,1,2,\cdots ,}</annotation>

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the local iteration format in matrix form is,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {D} ^{(k+1)}=\mathbf {T} \mathbf {D} ^{(k)},\quad k=0,1,2,\cdots ,}</annotation>

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<annotation encoding="application/x-tex">{\textstyle \mathbf {T} }</annotation>

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</math><img src="./9593d3daf4d4f5aebe663b792ea549da74c3a919.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\textstyle \mathbf {T} }" loading="lazy"> is the iteration matrix:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={\begin{bmatrix}\mathbf {E} _{J}&-\mathbf {B} _{1}\\0&\mathbf {E} _{I}-\mathbf {B} _{2}\end{bmatrix}},}</annotation>

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<annotation encoding="application/x-tex">{\textstyle \mathbf {E} _{J}}</annotation>

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<annotation encoding="application/x-tex">{\textstyle \mathbf {E} _{I}}</annotation>

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</math><img src="./b59531a1e414fc6aeca8780cdd1b4265dcdff09f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\textstyle \mathbf {E} _{I}}" loading="lazy"> are the identity matrices and

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{1}={\begin{bmatrix}B_{i_{1}}\left(t_{j_{1}}\right)&B_{i_{2}}\left(t_{j_{1}}\right)&\cdots &B_{i_{I}}\left(t_{j_{1}}\right)\\B_{i_{1}}\left(t_{j_{2}}\right)&B_{i_{2}}\left(t_{j_{2}}\right)&\cdots &B_{i_{I}}\left(t_{j_{2}}\right)\\\vdots &\vdots &\vdots &\vdots \\B_{i_{1}}\left(t_{j_{J}}\right)&B_{i_{2}}\left(t_{j_{J}}\right)&\cdots &B_{i_{I}}\left(t_{j_{J}}\right)\\\end{bmatrix}},\mathbf {B} _{2}={\begin{bmatrix}B_{i_{1}}\left(t_{i_{1}}\right)&B_{i_{2}}\left(t_{i_{1}}\right)&\cdots &B_{i_{I}}\left(t_{i_{1}}\right)\\B_{i_{1}}\left(t_{i_{2}}\right)&B_{i_{2}}\left(t_{i_{2}}\right)&\cdots &B_{i_{I}}\left(t_{i_{2}}\right)\\\vdots &\vdots &\vdots &\vdots \\B_{i_{1}}\left(t_{i_{I}}\right)&B_{i_{2}}\left(t_{i_{I}}\right)&\cdots &B_{i_{I}}\left(t_{i_{I}}\right)\\\end{bmatrix}}.}</annotation>

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</math>

The above local iteration format converges and can be extended to blending surfaces<a href="#cite_note-:6-12">[12]</a> and subdivision surfaces.<a href="#cite_note-20">[20]</a>

<div class="mw-heading mw-heading3"><h3 id="Implicit-PIA">Implicit-PIA</h3></div>

The PIA format for implicit curve and surface reconstruction is presented in the following.<a href="#cite_note-:5-11">[11]</a> Given an ordered point cloud <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\mathbf {Q} _{i}\right\}_{i=1}^{n}}">

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<annotation encoding="application/x-tex">{\textstyle \left\{\mathbf {Q} _{i}\right\}_{i=1}^{n}}</annotation>

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<annotation encoding="application/x-tex">{\textstyle \left\{\mathbf {n} _{i}\right\}_{i=1}^{n}}</annotation>

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</math><img src="./9f48f82f3d20cfa95b77c253e693eea620be9dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.51ex; height:3.176ex;" alt="{\textstyle \left\{\mathbf {n} _{i}\right\}_{i=1}^{n}}" loading="lazy"> on the data points, we want to reconstruct an implicit curve from the given <a href="Point_cloud" title="Point cloud">point cloud</a>. To avoid a trivial solution, some offset points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\mathbf {Q} _{l}\right\}_{l=n+1}^{2n}}">

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<annotation encoding="application/x-tex">{\textstyle \left\{\mathbf {Q} _{l}\right\}_{l=n+1}^{2n}}</annotation>

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</math><img src="./82bed08222abe660c02257a04f6e96d76b47d7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.143ex; height:3.509ex;" alt="{\textstyle \left\{\mathbf {Q} _{l}\right\}_{l=n+1}^{2n}}" loading="lazy"> are added to the point cloud.<a href="#cite_note-:5-11">[11]</a> They are offset by a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma }">

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<annotation encoding="application/x-tex">{\textstyle \sigma }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} _{l}=\mathbf {Q} _{i}+\sigma \mathbf {n} _{i},\quad l=n+i,\quad i=1,2,\cdots ,n.}</annotation>

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Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \epsilon }">

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<annotation encoding="application/x-tex">{\displaystyle f\left(\mathbf {Q} _{l}\right)=\epsilon ,\quad l=n+1,n+2,\cdots ,2n.}</annotation>

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Let the implicit curve after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }">

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<annotation encoding="application/x-tex">{\textstyle \alpha }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f^{(\alpha )}(x,y)=\sum _{i=1}^{N_{u}}\sum _{j=1}^{N_{v}}C_{ij}^{(\alpha )}B_{i}(x)B_{j}(y),}</annotation>

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Define the difference vector of data points as<a href="#cite_note-:5-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\delta }}_{k}^{(\alpha )}&=0-f^{(\alpha )}(x_{k},y_{k}),\quad k=1,2,\cdots ,n,\\{\boldsymbol {\delta }}_{l}^{(\alpha )}&=\epsilon -f^{(\alpha )}(x_{l},y_{l}),\quad l=n+1,n+2,\cdots ,2n.\end{aligned}}}</annotation>

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Next, calculate the difference vector of control coefficients

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Delta }}_{ij}^{(\alpha )}=\mu \sum _{k=1}^{2n}B_{i}(x_{k})B_{j}(y_{k}){\boldsymbol {\delta }}_{k}^{(\alpha )},\quad i=1,2,\cdots ,N_{u},\quad j=1,2,\cdots ,N_{v},}</annotation>

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</math><img src="./259577540a13444806174d5a1ae7662974f58085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\textstyle \mu }" loading="lazy"> is the convergence coefficient. As a result, the new control coefficients are

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<annotation encoding="application/x-tex">{\displaystyle C_{ij}^{(\alpha +1)}=C_{ij}^{(\alpha )}+{\boldsymbol {\Delta }}_{ij}^{(\alpha )},}</annotation>

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leading to the new algebraic B-spline curve

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<annotation encoding="application/x-tex">{\displaystyle f^{(\alpha +1)}(x,y)=\sum _{i=1}^{N_{u}}\sum _{j=1}^{N_{v}}C_{ij}^{(\alpha +1)}B_{i}(x)B_{j}(y).}</annotation>

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The above procedure is carried out iteratively to generate a sequence of algebraic B-spline functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{f^{(\alpha )}(x,y),\quad \alpha =0,1,2,\cdots \right\}}">

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<annotation encoding="application/x-tex">{\textstyle \left\{f^{(\alpha )}(x,y),\quad \alpha =0,1,2,\cdots \right\}}</annotation>

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</math><img src="./24cd0970c1c2ce9f98b826376a222ecb1d81ea46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.178ex; height:3.343ex;" alt="{\textstyle \left\{f^{(\alpha )}(x,y),\quad \alpha =0,1,2,\cdots \right\}}" loading="lazy">. The sequence converges to a minimization problem with constraints when the initial control coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{ij}^{(0)}=0}">

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</math><img src="./a5fcf7dd0a6dd2d1ef229c586dd6c71c5a651fa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.392ex; height:4.009ex;" alt="{\textstyle C_{ij}^{(0)}=0}" loading="lazy">.<a href="#cite_note-:5-11">[11]</a>

Assume that the implicit surface generated after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle f^{(\alpha )}(x,y,z)=\sum _{i=1}^{N_{u}}\sum _{j=1}^{N_{v}}\sum _{k=1}^{N_{w}}C_{ijk}^{(\alpha )}B_{i}(x)B_{j}(y)B_{k}(z),}</annotation>

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the iteration format is similar to that of the curve case.<a href="#cite_note-:5-11">[11]</a><a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="Fairing-PIA">Fairing-PIA</h3></div>

To develop fairing-PIA, we first define the functionals as follows:<a href="#cite_note-:7-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{r,j}(f)=\int _{t_{1}}^{t_{m}}B_{r,j}(t)fdt,\quad j=1,2,\cdots ,n,\quad r=1,2,3,}</annotation>

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<annotation encoding="application/x-tex">{\textstyle B_{r,j}(t)}</annotation>

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</math><img src="./e04c65a4dab04e1ba9fce9719e6e73276eba5b80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.522ex; height:3.009ex;" alt="{\textstyle B_{r,j}(t)}" loading="lazy"> represents the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}">

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</math><img src="./f2dfb06630b52c9e18fcc0a4688da10774206729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}" loading="lazy">th derivative of the basis function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B_{j}(t)}">

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<annotation encoding="application/x-tex">{\textstyle B_{j}(t)}</annotation>

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Let the curve after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{[k]}(t)=\sum _{j=1}^{n}B_{j}(t)\mathbf {P} _{j}^{[k]},\quad t\in [t_{1},t_{m}].}</annotation>

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To construct the new curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {P} ^{[k+1]}(t)}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {P} ^{[k+1]}(t)}</annotation>

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</math><img src="./5d055c4c77829e3728ca6c332d511984a8debeb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\textstyle (k+1)}" loading="lazy">st difference vectors for data points,<a href="#cite_note-:7-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} _{i}^{[k]}=\mathbf {Q} _{i}-\mathbf {P} ^{[k]}(t_{i}),\quad i=1,2,\cdots ,m.}</annotation>

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Then, the fitting difference vectors and the fairing vectors for control points are calculated by<a href="#cite_note-:7-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\delta }}_{j}^{[k]}&=\sum _{h\in I_{j}}B_{j}(t_{h})\mathbf {d} _{h}^{[k]},\quad j=1,2,\cdots ,n\\{\boldsymbol {\eta }}_{j}^{[k]}&=\sum _{l=1}^{n}{\mathcal {F}}_{r,l}\left(B_{r,j}(t)\right)\mathbf {P} _{l}^{[k]},\quad j=1,2,\cdots ,n\\\end{aligned}}}</annotation>

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Finally, the control points of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k+1)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{j}^{[k+1]}=\mathbf {P} _{j}^{[k]}+\mu _{j}\left[\left(1-\omega _{j}\right){\boldsymbol {\delta }}_{j}^{[k]}-\omega _{j}{\boldsymbol {\eta }}_{j}^{[k]}\right],\quad j=1,2,\cdots ,n,}</annotation>

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">th control point. The smoothing weights can be employed to adjust the smoothness individually, thus bringing great flexibility for smoothness.<a href="#cite_note-:7-13">[13]</a> The larger the smoothing weight is, the smoother the generated curve is. The new curve is obtained as follows

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In this way, we obtain a sequence of curves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\mathbf {P} ^{[k]}(t),\;k=1,2,3,\cdots \right\}}">

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</math><img src="./f54b773c855c3ef912d5deeb3ace18506a9551c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.881ex; height:4.843ex;" alt="{\textstyle \left\{\mathbf {P} ^{[k]}(t),\;k=1,2,3,\cdots \right\}}" loading="lazy">. The sequence converges to the solution of the conventional fairing method based on energy minimization when all smoothing weights are equal (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \omega _{j}=\omega }">

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</math><img src="./a324d700d54ff870e96ee95f8038fd22e2b07c00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.9ex; height:2.343ex;" alt="{\textstyle \omega _{j}=\omega }" loading="lazy">).<a href="#cite_note-:7-13">[13]</a> Similarly, the fairing-PIA can be extended to the surface case.

<div class="mw-heading mw-heading3"><h3 id="IG-LSPIA">IG-LSPIA</h3></div>

Isogeometric least-squares progressive-iterative approximation (IG-LSPIA).<a href="#cite_note-:8-14">[14]</a> Given a <a href="Boundary_value_problem" title="Boundary value problem">boundary value problem</a><a href="#cite_note-:14-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}{\mathcal {L}}u=f,&\quad {\text{in}}\;\Omega ,\\{\mathcal {G}}u=g,&\quad {\text{on}}\;\partial \Omega ,\end{aligned}}\right.}</annotation>

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<annotation encoding="application/x-tex">{\textstyle g}</annotation>

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</math><img src="./38dc9ad184fe5486391b456b9e68767ff77f3719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\textstyle g}" loading="lazy"> are the continuous functions. In the <a href="Isogeometric_analysis" title="Isogeometric analysis">isogeometric analysis method</a>, NURBS basis functions<a href="#cite_note-:13-8">[8]</a> are used as shape functions to solve the numerical solution of this boundary value problem.<a href="#cite_note-:14-15">[15]</a> The same basis functions are applied to represent the numerical solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u_{h}}">

<msub>

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</math><img src="./d7a091384c7b425b1353c8559eb6d7d420e78e37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\textstyle u_{h}}" loading="lazy"> and the geometric mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle G}">

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<annotation encoding="application/x-tex">{\textstyle G}</annotation>

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</math><img src="./febc2b9ff73bcca7b3fdb1432fddd1cdf3c8403c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\textstyle G}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u_{h}\left({\hat {\tau }}\right)&=\sum _{j=1}^{n}R_{j}({\hat {\tau }})u_{j},\\G({\hat {\tau }})&=\sum _{j=1}^{n}R_{j}({\hat {\tau }})P_{j},\end{aligned}}}</annotation>

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<annotation encoding="application/x-tex">{\textstyle R_{j}({\hat {\tau }})}</annotation>

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</math><img src="./a5ba753bf1d7ba4e4fdcf2b30e98101e52f35620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.851ex; height:3.009ex;" alt="{\textstyle R_{j}({\hat {\tau }})}" loading="lazy"> denotes the NURBS basis function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u_{j}}">

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<annotation encoding="application/x-tex">{\textstyle u_{j}}</annotation>

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</math><img src="./dd5dfece03b05e2eb6d9e8e2f0993cdbb3cae838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle u_{j}}" loading="lazy"> is the control coefficient. After substituting the collocation points<a href="#cite_note-:15-22">[22]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\tau }}_{i},i=1,2,...,{m}}">

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<annotation encoding="application/x-tex">{\textstyle {\hat {\tau }}_{i},i=1,2,...,{m}}</annotation>

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</math><img src="./8ccf55b0a2acfbf82ebf2b67a614a2c6caf90870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.671ex; height:2.509ex;" alt="{\textstyle {\hat {\tau }}_{i},i=1,2,...,{m}}" loading="lazy"> into the strong form of <a href="Partial_differential_equation" title="Partial differential equation">PDE</a>, we obtain a discretized problem<a href="#cite_note-:15-22">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}{\mathcal {L}}u_{h}({\hat {\tau }}_{i})=f(G({\hat {\tau }}_{i})),&\quad i\in {\mathcal {I_{L}}},\\{\mathcal {G}}u_{h}({\hat {\tau }}_{j})=g(G({\hat {\tau }}_{j})),&\quad j\in {\mathcal {I_{G}}},\end{aligned}}\right.}</annotation>

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {I_{L}}}}</annotation>

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</math><img src="./6f68ee883d3e1582446f5887678a19644ad44597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.069ex; width:2.702ex; height:2.509ex;" alt="{\textstyle {\mathcal {I_{L}}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {I_{G}}}}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {I_{G}}}}</annotation>

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</math><img src="./5b8f8f44d050b123785901f4a255d58ee63c4c2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.069ex; width:2.552ex; height:2.676ex;" alt="{\textstyle {\mathcal {I_{G}}}}" loading="lazy"> denote the subscripts of internal and boundary collocation points, respectively.

Arranging the control coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u_{j}}">

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</math><img src="./dd5dfece03b05e2eb6d9e8e2f0993cdbb3cae838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\textstyle u_{j}}" loading="lazy"> of the numerical solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle u_{h}({\hat {\tau }})}">

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<annotation encoding="application/x-tex">{\textstyle u_{h}({\hat {\tau }})}</annotation>

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</math><img src="./b5bc967a1857e0343f29fec5573022d14933bd0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.686ex; height:2.843ex;" alt="{\textstyle u_{h}({\hat {\tau }})}" loading="lazy"> into an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1}">

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</math><img src="./6706df9ed9f240d1a94545fb4e522bda168fe8fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 1}" loading="lazy">-dimensional column vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {U} =[u_{1},u_{2},...,u_{n}]^{T}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {U} =[u_{1},u_{2},...,u_{n}]^{T}}</annotation>

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</math><img src="./6595e18f6d17a58c974cdbfc1211557e76ca8e31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:3.009ex;" alt="{\textstyle \mathbf {U} =[u_{1},u_{2},...,u_{n}]^{T}}" loading="lazy">, the discretized problem can be reformulated in matrix form as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {AU} =\mathbf {b} }</annotation>

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<annotation encoding="application/x-tex">{\textstyle \mathbf {A} }</annotation>

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</math><img src="./b65702fed019f8853d55c303a73d6cbb1d7e0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\textstyle \mathbf {A} }" loading="lazy"> is the collocation matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {b} }">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {b} }</annotation>

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</math><img src="./83f84b3f70b8c2ab11ed37edba2ca75aec4d3a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\textstyle \mathbf {b} }" loading="lazy"> is the load vector.

Assume that the discretized load values are data points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{b_{i}\right\}_{i=1}^{m}}">

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<annotation encoding="application/x-tex">{\textstyle \left\{b_{i}\right\}_{i=1}^{m}}</annotation>

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</math><img src="./f4a228770afba3a076c880af74c8cd2feffbbe33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.022ex; height:3.176ex;" alt="{\textstyle \left\{b_{i}\right\}_{i=1}^{m}}" loading="lazy"> to be fitted. Given the initial guess of the control coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{u_{j}^{(0)}\right\}_{j=1}^{n},n<m}">

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<annotation encoding="application/x-tex">{\textstyle \left\{u_{j}^{(0)}\right\}_{j=1}^{n},n<m}</annotation>

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</math><img src="./0b455da4e415e04edb0f3c8219ac0c8aea42ef39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.342ex; height:5.176ex;" alt="{\textstyle \left\{u_{j}^{(0)}\right\}_{j=1}^{n},n<m}" loading="lazy">, we obtain an initial blending function<a href="#cite_note-:8-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle U^{(0)}({\hat {\tau }})=\sum _{j=1}^{n}A_{j}({\hat {\tau }})u_{j}^{(0)},\quad {\hat {\tau }}\in [{\hat {\tau }}_{1},{\hat {\tau }}_{m}],}</annotation>

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<annotation encoding="application/x-tex">{\textstyle A_{j}({\hat {\tau }})}</annotation>

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</math><img src="./ea1d0bf23441e16a351068561de0f46c6a122a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.83ex; height:3.009ex;" alt="{\textstyle A_{j}({\hat {\tau }})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j=1,2,\cdots ,n}">

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</math><img src="./43bf498c93ebc28603fe74066747909cd8e892a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:14.015ex; height:2.509ex;" alt="{\textstyle j=1,2,\cdots ,n}" loading="lazy">, represents the combination of different order derivatives of the NURBS basis functions determined using the operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}}">

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</math><img src="./9e97aae95c92641d7ecba829fa92d6fe23888465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\textstyle {\mathcal {L}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {G}}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{j}({\hat {\tau }})=\left\{{\begin{aligned}{\mathcal {L}}R_{j}({\hat {\tau }}),&\quad {\hat {\tau }}\ {\text{in}}\ \Omega _{p}^{in},\\{\mathcal {G}}R_{j}({\hat {\tau }}),&\quad {\hat {\tau }}\ {\text{in}}\ \Omega _{p}^{bd},\quad j=1,2,\cdots ,n,\end{aligned}}\right.}</annotation>

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<annotation encoding="application/x-tex">{\textstyle \Omega _{p}^{in}}</annotation>

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</math><img src="./4ca7355535f4dff4da9fb1fb5f2e76152fed367e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.464ex; height:3.176ex;" alt="{\textstyle \Omega _{p}^{in}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Omega _{p}^{bd}}">

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<annotation encoding="application/x-tex">{\textstyle \Omega _{p}^{bd}}</annotation>

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</math><img src="./1cdd62f06ab7d6afa180011e187a42e089f0b526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.475ex; height:3.176ex;" alt="{\textstyle \Omega _{p}^{bd}}" loading="lazy"> indicate the interior and boundary of the parameter domain, respectively. Each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A_{j}({\hat {\tau }})}">

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</math><img src="./ea1d0bf23441e16a351068561de0f46c6a122a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.83ex; height:3.009ex;" alt="{\textstyle A_{j}({\hat {\tau }})}" loading="lazy"> corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j}">

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</math><img src="./98c853a238654c3e0a13c0e7617ec5cb5352601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\textstyle j}" loading="lazy">th control coefficient. Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle J_{in}}">

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<annotation encoding="application/x-tex">{\textstyle J_{in}}</annotation>

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</math><img src="./63043dda4bcacefe7cf9ac030726ac9416991298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.076ex; height:2.509ex;" alt="{\textstyle J_{in}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle J_{bd}}">

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<annotation encoding="application/x-tex">{\textstyle J_{bd}}</annotation>

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</math><img src="./0d41b4218bbf1dc8fee1fdf3817a88cbc0e6e0e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.088ex; height:2.509ex;" alt="{\textstyle J_{bd}}" loading="lazy"> are the index sets of the internal and boundary control coefficients, respectively. <a href="Without_loss_of_generality" title="Without loss of generality">Without loss of generality</a>, we further assume that the boundary control coefficients have been obtained using strong or weak imposition and are fixed, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle u_{j}^{(k)}=u_{j}^{*},\quad j\in J_{bd},\quad k=0,1,2,\cdots .}</annotation>

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The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy">th blending function, generated after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k}">

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</math><img src="./0d5595fc0c47452f8fc2aa6e29c3611f047714b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\textstyle k}" loading="lazy">th iteration of IG-LSPIA,<a href="#cite_note-:8-14">[14]</a> is assumed to be as follows:

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<annotation encoding="application/x-tex">{\displaystyle U^{(k)}({\hat {\tau }})=\sum _{j=1}^{n}A_{j}({\hat {\tau }})u_{j}^{(k)},\quad {\hat {\tau }}\in [{\hat {\tau }}_{1},{\hat {\tau }}_{m}].}</annotation>

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Then, the difference vectors for collocation points (DCP) in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (k+1)}">

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<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

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</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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<msub>

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</munder>

<msub>

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</msub>

<msub>

<mover>

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<msub>

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</munder>

<msub>

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</msub>

<msub>

<mover>

</mover>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\delta }}_{i}^{(k)}&=b_{i}-\sum _{j=1}^{n}A_{j}({\hat {\tau }}_{i})u_{j}^{(k)}\\&=b_{i}-\sum _{j\in J_{bd}}A_{j}({\hat {\tau }}_{i})u_{j}^{(k)}-\sum _{j\in J_{in}}A_{j}({\hat {\tau }}_{i})u_{j}^{(k)},\quad i=1,2,...,m.\end{aligned}}}</annotation>

</semantics>

</math>

Moreover, group all load values whose parameters fall in the local support of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle j}</annotation>

</semantics>

</math><img src="./98c853a238654c3e0a13c0e7617ec5cb5352601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\textstyle j}" loading="lazy">th derivatives function, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A_{j}({\hat {\tau }}_{i})\neq 0}">

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A_{j}({\hat {\tau }}_{i})\neq 0}</annotation>

</semantics>

</math><img src="./ea5976520d4088c6334d7debdfdbc4ab2a35a316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.89ex; height:3.009ex;" alt="{\textstyle A_{j}({\hat {\tau }}_{i})\neq 0}" loading="lazy">, into the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle j}</annotation>

</semantics>

</math><img src="./98c853a238654c3e0a13c0e7617ec5cb5352601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\textstyle j}" loading="lazy">th group corresponding to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle j}</annotation>

</semantics>

</math><img src="./98c853a238654c3e0a13c0e7617ec5cb5352601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\textstyle j}" loading="lazy">th control coefficient, and denote the index set of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle j}</annotation>

</semantics>

</math><img src="./98c853a238654c3e0a13c0e7617ec5cb5352601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\textstyle j}" loading="lazy">th group of load values as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle I_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle I_{j}}</annotation>

</semantics>

</math><img src="./d58344801f199b23c0fd9017264b94d6703c70a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.933ex; height:2.843ex;" alt="{\textstyle I_{j}}" loading="lazy">. Lastly, the differences for control coefficients (DCC) can be constructed as follows:<a href="#cite_note-:8-14">[14]</a>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{j}^{(k)}=\mu \sum _{h\in I_{j}}A_{j}({\hat {\tau }}_{h}){\boldsymbol {\delta }}_{h}^{(k)},\quad j=1,2,...,n,}</annotation>

</semantics>

</math>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mu }</annotation>

</semantics>

</math><img src="./259577540a13444806174d5a1ae7662974f58085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\textstyle \mu }" loading="lazy"> is a normalization weight to guarantee the convergence of the algorithm.

Thus, the new control coefficients are updated via the following formula,

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{j}^{(k+1)}=u_{j}^{(k)}+d_{j}^{(k)},\quad j=1,2,...,n,}</annotation>

</semantics>

</math>

Consequently, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (k+1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (k+1)}</annotation>

</semantics>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U^{(k+1)}({\hat {\tau }})=\sum _{j=1}^{n}A_{j}({\hat {\tau }})u_{j}^{(k+1)}.}</annotation>

</semantics>

</math>

The above iteration process is performed until the desired fitting precision is reached and a sequence of blending functions is obtained

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{U^{(k)}({\hat {\tau }}),k=0,1,\dots \right\}.}</annotation>

</semantics>

</math>

The IG-LSPIA converges to the solution of a constrained least-squares collocation problem.<a href="#cite_note-:8-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="Proof_of_convergence">Proof of convergence</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Non-singular_case">Non-singular case</h3></div>

Let n be the number of control points and m be the number of data points.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n=m}</annotation>

</semantics>

</math><img src="./753c6840d55e2b1efd40c19d85cde16239ee4da5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\textstyle n=m}" loading="lazy">, the PIA iterative format in matrix form is

<mtr>

<mtd>

<msup>

</mrow>

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</mtd>

<mtd>

<msup>

</mrow>

</msup>

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<msup>

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<mtr>

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</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

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</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {P^{(\alpha +1)}} &=\mathbf {P^{(\alpha )}} +\mathbf {\Delta } ^{(\alpha )}\\&=\mathbf {P} ^{(\alpha )}+\mathbf {Q} -\mathbf {B} \mathbf {P} ^{(\alpha )}\\&=\left(\mathbf {I} -\mathbf {B} \right)\mathbf {P} ^{(\alpha )}+\mathbf {Q} \end{aligned}}}</annotation>

</semantics>

</math>

where

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msup>

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

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<msub>

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<msub>

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</mtr>

<mtr>

<mtd>

<msup>

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</msup>

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</mtd>

<mtd>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

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</mrow>

</mrow>

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</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

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</mrow>

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</mtd>

<mtd>

<msup>

<mrow>

<mrow>

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</msubsup>

</mrow>

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</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {Q} &=\left[\mathbf {Q} _{1},\mathbf {Q} _{2},\cdots ,\mathbf {Q} _{m}\right]^{T}\\\mathbf {P^{(\alpha )}} &=\left[\mathbf {P} _{1}^{(\alpha )},\mathbf {P} _{2}^{(\alpha )},\cdots ,\mathbf {P} _{n}^{(\alpha )}\right]^{T}\\\mathbf {\Delta } ^{(\alpha )}&=\left[\mathbf {\Delta } _{1}^{(\alpha )},\mathbf {\Delta } _{2}^{(\alpha )},\cdots ,\mathbf {\Delta } _{n}^{(\alpha )}\right]^{T}\\\mathbf {B} &={\begin{bmatrix}B_{1}(t_{1})&B_{2}(t_{1})&\cdots &B_{n}(t_{1})\\B_{1}(t_{2})&B_{2}(t_{2})&\cdots &B_{n}(t_{2})\\\vdots &\vdots &\ddots &\vdots \\B_{1}(t_{m})&B_{2}(t_{m})&\cdots &B_{n}(t_{m})\\\end{bmatrix}}.\end{aligned}}}</annotation>

</semantics>

</math>

The convergence of the PIA is related to the properties of the collocation matrix. If the <a href="Spectral_radius" title="Spectral radius">spectral radius</a> of the iteration matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} -\mathbf {B} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {I} -\mathbf {B} }</annotation>

</semantics>

</math><img src="./7448e7b3abc96b3be2e4311db050a110f406b254.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.755ex; height:2.343ex;" alt="{\displaystyle \mathbf {I} -\mathbf {B} }" loading="lazy"> is less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, then the PIA is convergent. It has been shown that the PIA methods are convergent for Bézier curves and surfaces, B-spline curves and surfaces, NURBS curves and surfaces, triangular Bernstein–Bézier surfaces, and subdivision surfaces (Loop, Catmull-Clark, Doo-Sabin).<a href="#cite_note-:9-2">[2]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n<m}</annotation>

</semantics>

</math><img src="./ff172acaeb3c6ecc3d6c1fead506a9b6c1b7ee70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\textstyle n<m}" loading="lazy">, the LSPIA in matrix form is<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:4-19">[19]</a>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {P^{(\alpha +1)}} &=\mathbf {P^{(\alpha )}} +\mu \mathbf {B} ^{T}\mathbf {\Delta } ^{(\alpha )}\\&=\mathbf {P} ^{(\alpha )}+\mu \mathbf {B} ^{T}\left(\mathbf {Q} -\mathbf {B} \mathbf {P} ^{(\alpha )}\right)\\&=\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)\mathbf {P} ^{(\alpha )}+\mu \mathbf {B} ^{T}\mathbf {Q} .\end{aligned}}}</annotation>

</semantics>

</math>

When the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {B} ^{T}\mathbf {B} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {B} ^{T}\mathbf {B} }</annotation>

</semantics>

</math><img src="./fcd356340ab1f3a16eb028a28131d3629aabbd4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.191ex; height:2.676ex;" alt="{\textstyle \mathbf {B} ^{T}\mathbf {B} }" loading="lazy"> is <a href="Invertible_matrix" title="Invertible matrix">nonsingular</a>, the following results can be obtained:<a href="#cite_note-23">[23]</a>

/* start https://en.wikipedia.org/ */

.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}

/* end https://en.wikipedia.org/ */

</style><div class="math_theorem" style="">

Lemma—If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}</annotation>

</semantics>

</math><img src="./d70264e7840f417985b4fd77534a2fd7084485ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.387ex; height:4.009ex;" alt="{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}" loading="lazy"> , where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \lambda _{0}}</annotation>

</semantics>

</math><img src="./deed531c8e299fbe2e4faf72d198dd0f60661a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\textstyle \lambda _{0}}" loading="lazy"> is the largest <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalue</a> of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {B} ^{T}\mathbf {B} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {B} ^{T}\mathbf {B} }</annotation>

</semantics>

</math><img src="./fcd356340ab1f3a16eb028a28131d3629aabbd4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.191ex; height:2.676ex;" alt="{\textstyle \mathbf {B} ^{T}\mathbf {B} }" loading="lazy">, then the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu \mathbf {B} ^{T}\mathbf {B} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mu \mathbf {B} ^{T}\mathbf {B} }</annotation>

</semantics>

</math><img src="./36e51dead8a45d686bc9b90184ae07a15408cbf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.593ex; height:3.176ex;" alt="{\textstyle \mu \mathbf {B} ^{T}\mathbf {B} }" loading="lazy"> are real numbers and satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<\lambda (\mu \mathbf {B} ^{T}\mathbf {B} )<2}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 0<\lambda (\mu \mathbf {B} ^{T}\mathbf {B} )<2}</annotation>

</semantics>

</math><img src="./6a5ebcae1ea42049655917ba45408bf830184ba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.279ex; height:3.176ex;" alt="{\textstyle 0<\lambda (\mu \mathbf {B} ^{T}\mathbf {B} )<2}" loading="lazy">.

</div>

Proof Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {B} ^{T}\mathbf {B} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {B} ^{T}\mathbf {B} }</annotation>

</semantics>

</math><img src="./fcd356340ab1f3a16eb028a28131d3629aabbd4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.191ex; height:2.676ex;" alt="{\textstyle \mathbf {B} ^{T}\mathbf {B} }" loading="lazy"> is nonsingular, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mu >0}</annotation>

</semantics>

</math><img src="./35bcf41c6e8a0aa7a3d1d507d9bd7f69093a6c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\textstyle \mu >0}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda (\mu \mathbf {B} ^{T}\mathbf {B} )>0}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \lambda (\mu \mathbf {B} ^{T}\mathbf {B} )>0}</annotation>

</semantics>

</math><img src="./f64d7e5e05e6177b97b624d20b362d305c478eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.018ex; height:3.176ex;" alt="{\textstyle \lambda (\mu \mathbf {B} ^{T}\mathbf {B} )>0}" loading="lazy">. Moreover,

<msup>

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<mfrac>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (\mu \mathbf {B} ^{T}\mathbf {B} )=\mu \lambda (\mathbf {B} ^{T}\mathbf {B} )<2{\frac {\lambda (\mathbf {B} ^{T}\mathbf {B} )}{\lambda _{0}}}<2.}</annotation>

</semantics>

</math>

In summary, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<\lambda (\mu \mathbf {B} ^{T}\mathbf {B} )<2}">

<msup>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 0<\lambda (\mu \mathbf {B} ^{T}\mathbf {B} )<2}</annotation>

</semantics>

Theorem—If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}</annotation>

</semantics>

</math><img src="./d70264e7840f417985b4fd77534a2fd7084485ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:11.387ex; height:4.009ex;" alt="{\textstyle 0<\mu <{\frac {2}{\lambda _{0}}}}" loading="lazy"> , LSPIA is convergent, and converges to the least-squares fitting result to the given data points.<a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:4-19">[19]</a>

</div>

Proof From the matrix form of iterative format, we obtain the following:

<mtr>

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</mtd>

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<mtd>

<msup>

<mrow>

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</mrow>

</msup>

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</munderover>

<mrow>

<mrow>

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<msup>

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</msup>

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</mrow>

<msup>

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</mtd>

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</munderover>

<msup>

<mrow>

<mrow>

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</msup>

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</mtable>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {P^{(\alpha +1)}} &=\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)\mathbf {P} ^{(\alpha )}+\mu \mathbf {B} ^{T}\mathbf {Q} ,\\&=\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)\left[\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)\mathbf {P} ^{(\alpha -1)}+\mu \mathbf {B} ^{T}\mathbf {Q} \right]+\mu \mathbf {B} ^{T}\mathbf {Q} ,\\&=\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)^{2}\mathbf {P} ^{(\alpha -1)}+\sum _{i=0}^{1}\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)\mu \mathbf {B} ^{T}\mathbf {Q} ,\\&=\cdots \\&=\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)^{\alpha +1}\mathbf {P} ^{(0)}+\sum _{i=0}^{\alpha }\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)^{\alpha }\mu \mathbf {B} ^{T}\mathbf {Q} .\\\end{aligned}}}</annotation>

</semantics>

</math>

According to above Lemma, the spectral radius of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu \mathbf {B} ^{T}\mathbf {B} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mu \mathbf {B} ^{T}\mathbf {B} }</annotation>

</semantics>

<mrow>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\rho \left({\mu \mathbf {B} ^{T}\mathbf {B} }\right)<2}</annotation>

</semantics>

</math>

and thus the spectral radius of the iteration matrix satisfies

<mrow>

</mrow>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\rho \left({\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} }\right)<1.}</annotation>

</semantics>

</math>

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha \rightarrow \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \alpha \rightarrow \infty }</annotation>

</semantics>

</math><img src="./4a8878aa46863804800d1660574f4b5c188d6e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.425ex; height:1.843ex;" alt="{\textstyle \alpha \rightarrow \infty }" loading="lazy">

<msup>

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</munderover>

<msup>

<mrow>

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<msup>

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</msup>

</mrow>

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</mrow>

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<mfrac>

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<mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)^{\infty }=0,\ \sum _{i=0}^{\infty }\left(\mathbf {I} -\mu \mathbf {B} ^{T}\mathbf {B} \right)^{\alpha }={\frac {1}{\mu }}\left(\mathbf {B} ^{T}\mathbf {B} \right)^{-1}.}</annotation>

</semantics>

</math>

As a result,

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{(\infty )}=\left(\mathbf {B} ^{T}\mathbf {B} \right)^{-1}\mathbf {B} ^{T}\mathbf {Q} ,}</annotation>

</semantics>

</math>

i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {B} ^{T}\mathbf {B} \mathbf {P} ^{(\infty )}=\mathbf {B} ^{T}\mathbf {Q} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {B} ^{T}\mathbf {B} \mathbf {P} ^{(\infty )}=\mathbf {B} ^{T}\mathbf {Q} }</annotation>

</semantics>

</math><img src="./b7f737847f4061bd49acd5f6ebcd2db02b600e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.569ex; height:3.176ex;" alt="{\textstyle \mathbf {B} ^{T}\mathbf {B} \mathbf {P} ^{(\infty )}=\mathbf {B} ^{T}\mathbf {Q} }" loading="lazy">, which is equivalent to the normal equation of the fitting problem. Hence, the LSPIA algorithm converges to the least squares result for a given sequence of points.

<div class="mw-heading mw-heading3"><h3 id="Singular_case">Singular case</h3></div>

Lin et al. showed that LSPIA converges even when the iteration matrix is singular.<a href="#cite_note-:12-18">[18]</a>

<div class="mw-heading mw-heading2"><h2 id="Acceleration_algorithms_and_others">Acceleration algorithms and others</h2></div>

<ul><li>Precondition: Liu et al. proposed a preconditioned PIA for Bézier surfaces via the diagonally compensated reduction method, effectively improving the accuracy and efficiency of the classical algorithm.<a href="#cite_note-24">[24]</a></li>

<li>Iteration matrix inverse approximation: Sajavičius improved the LSPIA based on the matrix approximate inverse method. In each iteration step, the approximate inverse of the coefficient matrix of the least-squares fitting problem is first computed and then used as the weight to adjust the control points.<a href="#cite_note-25">[25]</a></li>

<li>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.<a href="#cite_note-26">[26]</a> Moreover, Zhang et al. proposed a weighted local PIA format for tensor Bézier surfaces.<a href="#cite_note-27">[27]</a> Li et al. assigned initial weights to each data point, and the weights of the interpolated points are determined adaptively during the iterative process.<a href="#cite_note-28">[28]</a></li>

<li>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.<a href="#cite_note-29">[29]</a></li>

<li>Stochastic descent strategy: Rios and Jüttle explored the relationship between LSPIA and <a href="Gradient_descent" title="Gradient descent">gradient descent</a> method and proposed a stochastic LSPIA algorithm with parameter correction.<a href="#cite_note-30">[30]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

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.

<div class="mw-heading mw-heading3"><h3 id="Data_fitting">Data fitting</h3></div>

<ul><li>Adaptive data fitting: The control points are divided into active control points and fixed control points. In each round of iteration, if the fitting error of a data point reaches a given precision, its corresponding control point is fixed and not updated. This iterative process is repeated until all control points are fixed. The algorithm performs well on large-scale data fitting by adaptively reducing the number of active control points.<a href="#cite_note-31">[31]</a></li>

<li>Large-scale data fitting: By combining T-spline with PIA, an incremental fitting algorithm suitable for fitting large-scale data sets is proposed. During the incremental iteration, each new round of iterations reuses information from the last round of iterations to save computation. While the convergence speed of the traditional point-by-point iterative algorithm decreases as the number of control points increases, in PIA the computation of each iteration step is unrelated to the number of control points; this gives PIA a powerful capability for data fitting.<a href="#cite_note-:3-10">[10]</a></li>

<li>Local fitting: Based on the local property of PIA, a series of local PIA formats have been proposed.<a href="#cite_note-:6-12">[12]</a><a href="#cite_note-32">[32]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Implicit_reconstruction">Implicit reconstruction</h3></div>

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.<a href="#cite_note-:5-11">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="Offset_curve_approximation">Offset curve approximation</h3></div>

Firstly, the data points are sampled on the original curve. Then, the initial polynomial approximation curve or rational approximation curve of the offset curve is generated from these sampled points. Finally, the offset curve is approximated iteratively using the PIA method.<a href="#cite_note-33">[33]</a>

<div class="mw-heading mw-heading3"><h3 id="Mesh_generation">Mesh generation</h3></div>

Given a triangular mesh model as input, the algorithm first constructs the initial hexahedral mesh, then extracts the quadrilateral mesh of the surface as the initial boundary mesh. During the iterations, the movement of each mesh vertex is constrained to ensure the validity of the mesh. Finally, the hexahedral model is fitted to the given input model. The algorithm can guarantee the validity of the generated hexahedral mesh, i.e., the Jacobi value at each mesh vertex is greater than zero.<a href="#cite_note-34">[34]</a>

<div class="mw-heading mw-heading3"><h3 id="Data_compression">Data compression</h3></div>

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. This method can well preserve the neighborhood information of pixels.<a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading3"><h3 id="Fairing_curve_and_surface_generation">Fairing curve and surface generation</h3></div>

Given a data point set, we first define the fairing functional, and calculate the fitting difference vector and the fairing vector of the control point; then, adjust the control points with fairing weights. According to the above steps, the fairing curve and surface can be generated iteratively. Due to the sufficient fairing parameters, the method can achieve global or local fairing. It is also flexible to adjust knot vectors, fairing weights, or data parameterization after each round of iteration. The traditional energy-minimization method is a special case of this method, i.e., when the smooth weights are all the same.<a href="#cite_note-:7-13">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Isogeometric_analysis">Isogeometric analysis</h3></div>

The discretized load values are regarded as the set of data points, and the combination of the basis functions and their derivative functions is used as the blending function for fitting. 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. In addition, the average iteration time per step is only related to the number of data points (i.e., collocation points) and unrelated to the number of control coefficients.<a href="#cite_note-:8-14">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a href="https://creativecommons.org/licenses/by/4.0/" title="creativecommons:by/4.0/" class="external"></a> This article incorporates <a rel="nofollow" class="external text" href="https://www.mdpi.com/2227-7390/11/4/898">text</a> by zju_cagd available under the <a href="https://creativecommons.org/licenses/by/4.0/" class="extiw external" title="creativecommons:by/4.0/">CC BY 4.0</a> license.</li></ul>

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