On a progressive and iterative approximation method with memory for least square fitting

Hui-Di Wang; Zheng-Da Huang

REVIEW 2 major objections 5 minor 36 references

On a progressive and iterative approximation method with memory for least square fitting

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read An iteration with memory converges to least-squares curve fits at a rate provably faster than LSPIA, even for rank-deficient collocation matrices.

desk verdict The curve-case acceleration result is genuine and the convergence proof checks out; the surface extension in Theorem 7 is asserted without proof and should be supplied or softened before publication. read the letter →

arxiv 1908.06417 v2 pith:7J64LVFJ submitted 2019-08-18 math.NA cs.NA

Zheng-Da Huang , Hui-Di Wang This is my paper · ORCID

classification math.NAcs.NA MSC 65D1765F1041A1515B48

keywords progressiveanditerativeapproximationleastsquaresfittingnormalizedtotallypositivebasisconvergenceratespectralradiusrank-deficientcollocationmatrixB-splinecurvetensorproductsurface

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces MLSPIA, a progressive and iterative approximation method for least-squares fitting that stores the previous correction and uses three weights when updating control points. Its central claim is that, for any normalized totally positive basis, the iterated curves converge to the least-squares fit even when the collocation matrix has deficient column rank, provided the weights satisfy an explicit interval condition. At the paper's chosen optimal weights, the asymptotic contraction rate is $(\sigma_1-\sigma_r)/(\sigma_1+\sigma_r)$, strictly smaller than the optimal rate $(\sigma_1^2-\sigma_r^2)/(\sigma_1^2+\sigma_r^2)$ of the earlier LSPIA method whenever $\sigma_1>\sigma_r$. Because the method never assembles or solves a linear system directly, a better contraction rate translates directly into fewer iterations on very large point sets. A tensor-product surface version is stated with the same kind of rate, with the proof deferred as similar.

What carries the argument

The load-bearing object is the iteration matrix $H_{\omega,\gamma,\upsilon}$ that governs the coupled update of the auxiliary weighted sums $\Lambda^k$ and the control points $P^k$; the paper works in any normalized totally positive (NTP) basis, meaning basis functions that are nonnegative, sum to one, and have totally positive collocation matrices. An orthogonal change of variables built from the SVD of the collocation matrix $B$ block-diagonalizes $H_{\omega,\gamma,\upsilon}$, leaving eigenvalues $1-\omega$ together with the roots of a quadratic equation $\lambda^2+[\gamma\upsilon\sigma_i^2-(2-\omega)]\lambda+\sigma_i^2\upsilon(\omega-\gamma)+1-\omega=0$ for each positive singular value $\sigma_i$. A root-bound lemma converts those quadratics into the explicit weight condition (11), and at the paper's optimal weights every active eigenvalue has modulus $(\sigma_1-\sigma_r)/(\sigma_1+\sigma_r)$, the contraction factor that determines the convergence rate.

What would settle it

Construct a small tensor-product example, say with $B_1$ rank-deficient and $B_2$ a single-column positive basis, compute the spectral radius of the iteration matrix $H_{\omega^*,\gamma^*,\upsilon^*}$ formed from $B_1\otimes B_2$, and compare it with $(\sigma_1\mu_1-\sigma_r\mu_s)/(\sigma_1\mu_1+\sigma_r\mu_s)$; a mismatch in the asymptotic error contraction would show the omitted surface proof does not follow from the stated assumptions.

Watch

Extended reading notes

Core claim

The discovery, on the paper's own terms, is that the least-squares progressive iteration can be accelerated by adding memory: the increment to each control point at step $k$ blends the previous increment with two successive residual-weighted sums, controlled by weights $\omega$, $\gamma$, and $\upsilon$. The paper proves the resulting curves converge to a solution of the least-squares normal equations $B^T B X = B^T Q$ whenever (11) holds, regardless of whether $B$ has full column rank. The proof rewrites the iteration in coupled variables $(\Lambda^k, P^k)$, block-diagonalizes the iteration matrix by an orthogonal transform, and shows each singular value $\sigma_i$ contributes a quadratic eigenvalue equation; the root bound of Lemma 4 converts those equations into the explicit condition (11). At the weights $\omega^*=\gamma^*=4\sigma_1\sigma_r/(\sigma_1+\sigma_r)^2$, $\upsilon^*=1/(\sigma_1\sigma_r)$, the spectral radius is $(\sigma_1-\sigma_r)/(\sigma_1+\sigma_r)$, which is less than the fastest LSPIA rate $(\sigma_1^2-\sigma_r^2)/(\sigma_1^2+\sigma_r^2)$ whenever $\sigma_1>\sigma_r$. Numerical examples with cubic B-spline curves and a tensor-product cubic B-spline surface confirm that fewer iterations are needed.

Load-bearing premise

The surface result (Theorem 7) is stated without proof and rests on the unstated assumption that the tensor-product iteration behaves exactly like the curve iteration, with singular values of the two factors multiplying pairwise; if that spectral factorization fails for a rank-deficient factor, the stated surface rate does not follow from the paper's arguments.

Editorial extensions

If this is right

For very large point sets, MLSPIA reaches a fixed fitting tolerance in fewer iterations than LSPIA, because its spectral radius is provably smaller at the recommended weights.
The convergence guarantee covers rank-deficient collocation matrices, so redundant or nearly dependent basis functions no longer make the iterative fit fail or require a separate least-squares solver.
The added memory costs only two extra additions and three extra scalar multiplications per control point per step, so the faster convergence is obtained at a small per-iteration premium.
If Theorem 7's tensor-product claim is accepted, surface fitting inherits the same speed-up, with the extreme singular products $\sigma_1\mu_1$ and $\sigma_r\mu_s$ replacing $\sigma_1$ and $\sigma_r$ in the rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not analyze how estimation error in knowing $\sigma_1$ and $\sigma_r$ affects the promised rate; a practical version would need cheap approximations of these extremes, and the sensitivity of the spectral radius to those estimates is an open question.
The two-step memory is a linear acceleration; storing more past corrections could in principle push the contraction factor further toward a value governed by the square root of the condition number, although the present paper gives no such analysis.
Because the argument relies only on the singular values of the collocation matrix, the same three-weight acceleration should transfer to other fitting settings with totally positive bases, such as generalized B-spline or subdivision-surface fitting, whenever the same spectral structure can be verified.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core claim is real: the MLSPIA recurrence with two prior-step memory terms and three weights is a legitimate extension of LSPIA, and the paper proves that for any NTP basis, with a rank-deficient collocation matrix, the iteration converges to a least-squares solution. I checked the eigenvalue analysis in Lemma 5 and the discriminant calculation in Theorem 6; they are correct. The rate comparison is also fair: the optimal MLSPIA rate is strictly better than the LSPIA rate when sigma1 > sigma_r, and the numerical experiments confirm the improvement on independent data sets. The paper is honest about what it does not do: Remark 2 admits that no general weight optimization is attempted, and Remark 3 notes that shape issues are out of scope.

The main problem is Theorem 7. The surface extension, which is part of the abstract's promise, is stated and then dismissed with "The proof to the Theorem 7 is omitted since it is similar to the case for Theorems 2 and 6." That is not adequate. The surface iteration is not literally the same as the curve iteration: it acts through the Kronecker product B1⊗B2, and one needs to show that the SVD decouples the iteration matrix the way it does in the curve case, and that the extremal singular values are products sigma1*mu1 and sigma_r*mu_s. These are standard facts, but they are not written down, and if the null-space block structure did not carry over, the stated surface rate would not follow. The authors could fix this by including the full proof, or by restating the surface claim as conjectural and moving the surface rate out of the abstract.

Two smaller issues. First, there is a placeholder citation "[ ? ]" in Section 2 where a T-spline development is mentioned. That needs to be resolved before any final version. Second, the abstract says the method converges to "the" least-squares curve, but in the rank-deficient case the proof establishes convergence to one of possibly several least-squares solutions. The wording should be adjusted.

None of these problems undermine the curve-case result, which is solid and practically useful. The citation pattern looks normal, and the numerical comparisons are reproducible in spirit, including the use of the same data sets as the original LSPIA paper. The paper does not oversell its incremental contribution; it is a targeted speedup, not a new paradigm.

For a CAGD audience, this is worth engaging with. I would send it to peer review, expecting a conditional acceptance: the surface proof must be supplied or the claims appropriately narrowed. The placeholder and the abstract wording are minor and easily fixed.

Referee Report

2 major / 5 minor

Summary. The paper proposes MLSPIA, a progressive and iterative approximation method with memory for least-squares fitting. The iteration (1)-(3) uses three weights to update control points and accumulated difference terms. For curves, the authors prove (Theorem 2) that for any normalized totally positive basis, even when the collocation matrix B has deficient column rank, the iteration converges to a least-squares fit provided the weights satisfy condition (11). They then derive a spectral-radius estimate at a particular weight choice (Theorem 6) and show that this rate is strictly smaller than the optimal LSPIA rate from [7] whenever the relevant extremal singular values differ. A tensor-product surface extension is stated as Theorem 7, and five numerical examples, including one surface example, illustrate the acceleration. The proof of Theorem 7 is explicitly omitted, although the abstract, the conclusions, and Example 5 rely on the surface result.

Significance. If the curve-case theorems are correct, the paper makes a useful contribution: it extends LSPIA convergence to rank-deficient collocation matrices, provides a constructive weight choice with a rigorously faster theoretical rate, and the numerical examples confirm the acceleration with negligible final deviations. The eigenvalue analysis in Lemma 3 and the discriminant computation leading to (29) are sound, and the treatment of the rank-deficient null-space block in Theorem 2 is careful. The omitted surface proof is the main weakness: because the surface theorem is a central advertised claim, the manuscript is not yet complete. A full proof of Theorem 7 is required before the surface claims can be accepted; the curve part alone would be a modest but valid contribution.

major comments (2)

[Section 5, Theorem 7] The proof of the surface extension is explicitly omitted with the sentence 'The proof to the Theorem 7 is omitted since it is similar to the case for Theorems 2 and 6.' This is a load-bearing gap. The surface iteration (33)-(35) acts through the Kronecker product B1⊗B2, and the proof must establish the singular-value factorization of B1⊗B2, the corresponding block structure of the iteration matrix (a 2rs-dimensional coupled block, an (m1m2-rs)-dimensional diagonal block, and an (n1n2-rs)-dimensional identity block), and the rate formula ρ = (σ1μ1 - σrμs)/(σ1μ1 + σrμs). These are standard facts, but they are not written down, and the abstract, the conclusions, and Example 5 all rely on Theorem 7. Please supply a complete proof, or explicitly restrict the paper's central claims to curves.
[Theorem 6(2) and its proof] The statement 'When the NTP basis is linearly independent, B is of full column rank' is not true in general. A normalized totally positive basis can be linearly independent as a set of functions while its collocation matrix at the chosen parameters is rank-deficient; for example, a B-spline basis with no data parameter in the support of one basis function yields a zero column. The comparison with the LSPIA rate from [7] is valid under the assumption that B has full column rank. Please restate item 2 in those terms and remove the false implication.

minor comments (5)

[Equations (11) and (36)] The inequalities in (11) and (36) appear to be missing division signs: the lower bound should read ω - ω/(σ1^2 υ) and the upper bound ω/2 - (ω-2)/(σ1^2 υ); as typeset, the expressions are dimensionally inconsistent. The same correction applies to (36) with σ1μ1 in place of σ1.
[Figure 5 and Section 6.3] The caption of Figure 5 says '(b) Step 1000', while the text says the second and third panels are at steps 200 and 362, and Table 2 reports 362 iterations. Please reconcile these numbers.
[Notation, Table 2] The symbols ν* and υ* are used interchangeably for the third weight in Table 2 and the surrounding text; please use a single notation throughout.
[Section 2 and general text] There are several typographical errors, including 'woks', 'dada set', and 'expect' for 'except', and the running header 'Journal of LaTeX Templates' should be removed.
[Reference [10]] Reference [10] is cited as an arXiv preprint; if a published version exists, please cite the published source.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the convergence and rate theorems are derived from the iteration matrix, and the comparison target is an external benchmark, not the paper's own fitted output.

full rationale

The central derivation is self-contained. Theorem 2 obtains convergence of (Λ^k, P^k) from the contraction of H_{ω,γ,υ}, whose eigenvalues are characterized in Lemma 3 via the SVD of B; condition (11) is derived, not assumed, and the fixed point is shown to solve B^T B P = B^T Q. Theorem 6 compares the spectral radius at the analytically chosen weights ω*=γ*=4σ1σr/(σ1+σr)^2, υ*=1/(σ1σr) with the LSPIA rate (σ1^2−σr^2)/(σ1^2+σr^2) taken from Deng and Lin [7], an external benchmark whose authors do not overlap with the present paper. The weights depend only on singular values of the collocation matrix B, not on the data Q, so the claimed rate is a theorem rather than a fitted prediction. I also flag one non-circular gap: in Section 5, Theorem 7 (surface case) states `The proof to the Theorem 7 is omitted since it is similar to the case for Theorems 2 and 6.` The surface rate ρ=(σ1μ1−σrμs)/(σ1μ1+σrμs) and convergence condition (36) are therefore asserted without the needed Kronecker-product/SVD block-decoupling argument; this is a completeness and correctness risk, but it is an omitted proof, not an equation that reduces to its input by construction. No circularity score penalty is appropriate.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard linear algebra (SVD, spectral radius), the classical quadratic root criterion, and the cited LSPIA rate formula. The three method weights are analytically chosen from singular values, not fitted to data. No new physical or mathematical entities are introduced; the surface extension is asserted without proof and relies on the Kronecker-product singular value fact.

free parameters (3)

weight omega = omega* = 4 sigma1 sigma_r / (sigma1 + sigma_r)^2
The MLSPIA recurrence (3) has three tuning weights. Theorem 2 gives a convergence region (11); Theorem 6 proposes a closed-form default based on extreme singular values of B. The default is a formula, not a fit to data, but the method's behavior depends on it.
weight gamma = gamma* = 4 sigma1 sigma_r / (sigma1 + sigma_r)^2
Same as omega; in the proposed default gamma* equals omega*.
weight upsilon = upsilon* = 1 / (sigma1 sigma_r)
Third weight, set by the reciprocal product of extreme singular values in Theorem 6.

assumptions (6)

standard math The collocation matrix B is real and admits an SVD as in (12); the convergence analysis relies on the spectral decomposition of B^T B and B B^T.
Theorem 2 uses SVD to block-diagonalize the iteration matrix (14)-(15).
domain assumption The bases B_i(t) form a normalized totally positive basis, giving nonnegative entries and partition of unity, which is the application setting but not used in the linear-algebra proof.
The method is framed for NTP bases in CAGD; the proof of convergence only uses SVD, so the NTP property is a domain assumption for the intended use.
standard math Lemma 4 (both roots of a real quadratic have modulus less than 1 iff |c| < 1 and |b| < 1 + c) from [35] is used to characterize spectral radius.
Lemma 5 relies on this classical stability criterion for second-order recurrences.
domain assumption The fastest LSPIA convergence rate is (sigma1^2 - sigma_r^2) / (sigma1^2 + sigma_r^2), as proved in the cited paper [7].
Theorem 6 compares against this benchmark without reproving it; the comparison's validity inherits from [7].
standard math For the surface case, singular values of B1 tensor B2 are products sigma_i mu_j (undisplayed background fact).
Theorem 7 asserts the surface rate without proof; this Kronecker-product fact is the missing justification.
domain assumption The parameterization uses increasing sequences and the Schoenberg-Whitney condition to build cubic B-spline bases.
Numerical examples use normalized accumulated chord parameterization; the theory itself only needs increasing knots.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On a progressive and iterative approximation method with memory for least square fitting." pith.science (2026). https://pith.science/paper/7J64LVFJ

@misc{pith2026190806417,
  author       = {Pith},
  title        = {Pith review of: On a progressive and iterative approximation method with memory for least square fitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J64LVFJ}},
  note         = {Machine review of arXiv:1908.06417}
}

read the original abstract

In this paper, we present a progressive and iterative approximation method with memory for least square fitting(MLSPIA). It adjusts the control points and the weighted sums iteratively to construct a series of fitting curves (surfaces) with three weights. For any normalized totally positive basis even when the collocation matrix is of deficient column rank, we obtain a condition to guarantee that these curves (surfaces) converge to the least square fitting curve (surface) to the given data points. It is proved that the theoretical convergence rate of the method is faster than the one of the progressive and iterative approximation method for least square fitting (LSPIA) in [Deng C-Y, Lin H-W. Progressive and iterative approximation for least squares B-spline curve and surface fitting. Computer-Aided Design 2014;47:32-44] under the same assumption. Examples verify this phenomenon.

Figures

Figures reproduced from arXiv: 1908.06417 by the authors.

**Figure 2.** A point set of 305 points is fitted by a cubic B-spline curve with 30 control points. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** A point set of 501 points is fitted by a cubic B-spline curve with 50 control points. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: A point set of 269 points is fitted by a cubic B-spline curve with 35 control points. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: A point set of 81 × 81 points is fitted by a cubic B-spline tensor product surface with 35 × 35 control points.. It can be seen that the fittings at the left side in [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Curvature combs for Examples 1-4 and zebra maps for Example 5: For Example 1, [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[7]

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

Deng C-Y, Lin H-W. Progressive and iterative approximation for least squares B-spline curve and surface ﬁtting. Comput-Aided Des 2014;47:32– 44

work page 2014
[1]

The NURBS book

Piegl L, Tiller W. The NURBS book. 2nd ed.; New York, USA: Springer- Verlag; 1997

work page 1997
[2]

Handbook of computer aided geometric design

Farin G, Hoschek J, Kim M-S. Handbook of computer aided geometric design. 1st ed.; North-Holland; 2002

work page 2002
[3]

Total least squares ﬁtting of bezier and b-spline curves to ordered data

Borges CF, Pastva T. Total least squares ﬁtting of bezier and b-spline curves to ordered data. Comput-Aided Des 2002;19:275–89

work page 2002
[4]

Fitting b-spline curves to point clouds by curvature-based squared distance minimization

Wang W, Pottmann H, Liu Y . Fitting b-spline curves to point clouds by curvature-based squared distance minimization. ACM Transactions on Graphics 2006;25:214–38

work page 2006
[5]

B-spline surface ﬁtting by it- erative geometric interpolation/approximation algorithms

Kineri Y, Wang M, Lin H, Maekawa T . B-spline surface ﬁtting by it- erative geometric interpolation/approximation algorithms. Comput-Aided Des 2012;44(7):697–708

work page 2012
[6]

An eﬃcient method for ﬁtting large data sets using T-splines

Lin H-W, Zhang Z-Y. An eﬃcient method for ﬁtting large data sets using T-splines. SIAM J Sci Comput 2013;35(6):A3052–68. 22

work page 2013
[8]

Elitist clonal selection algorithm for optimal choice of free knots in b-spline data ﬁtting

Galveza A, Iglesiasa A, Avilaa A, Oteroc C, Ariasc R, Manchadoca C. Elitist clonal selection algorithm for optimal choice of free knots in b-spline data ﬁtting. Applied Soft Computing 2015;26:90–106

work page 2015

Show all 36 references

[9]

Least square geometric iterative ﬁtting method for generalized B-spline curves with two diﬀerent kinds of weights

Zhang L, Ge X-Y, Tan J-Q. Least square geometric iterative ﬁtting method for generalized B-spline curves with two diﬀerent kinds of weights. Vis Comput 2016;32(9):1109–20

2016
[10]

The convergence of least-squares progressive iterative approximation with singular iterative matrix

Lin H, Cao Q, Zhang X . The convergence of least-squares progressive iterative approximation with singular iterative matrix. arXiv 2017;URL: arXiv:1707.09109

2017 arXiv
[11]

Shape modeling based on specifying the initial b-spline curve and scaled bfgs optimization method

Ebrahimi A, Loghmani BG. Shape modeling based on specifying the initial b-spline curve and scaled bfgs optimization method. Multimed Tools Appl 2018;77:30331–51

2018
[12]

Survey on geometric iterative methods and their applications

Lin H, Maekawa T, Deng C. Survey on geometric iterative methods and their applications. Comput-Aided Des 2018;95:40–51

2018
[13]

Progressive iterative approximation for regularized least square bivariate B-spline surface ﬁtting

Liu M-Z, Li B-J, Guo Q-J, Zhu C-G, Hu P, Shao Y-H. Progressive iterative approximation for regularized least square bivariate B-spline surface ﬁtting. J Comput Appl Math 2018;327:175–87

2018
[14]

Parameterizing and extending trimmed regions for tensor-product surface ﬁtting

Vaitkus M, Varady T. Parameterizing and extending trimmed regions for tensor-product surface ﬁtting. Comput-Aided Des 2018;104:125–40

2018
[15]

Totally positive bases and progressive iteration approximation

Lin H-W, Bao H-J, Wang G-J . Totally positive bases and progressive iteration approximation. Comput Math Appl 2005;50(3-4):575–86

2005
[16]

The convergence of the geometric interpolation algorithm

Lin H-W . The convergence of the geometric interpolation algorithm. Comput-Aided Des 2010;42(6):505–8

2010
[17]

Interpolation by geometric algo- rithm

Maekawa T, Matsumoto Y, Namiki K. Interpolation by geometric algo- rithm. Comput-Aided Des 2007;39(4):313–23

2007
[18]

How does Agee’s smoothing method work? Proceedings of the 1979 army numerical analysis and computers conference, ARO report 1979;79-3:299–302

De Boor C. How does Agee’s smoothing method work? Proceedings of the 1979 army numerical analysis and computers conference, ARO report 1979;79-3:299–302

1979
[19]

The method of numeric polish in curve ﬁtting

Qi D, Tian Z, Zhang Y, Feng J. The method of numeric polish in curve ﬁtting. Acta Mathematica Sinica 1975;18:173–84

1975
[20]

A method of designing free surfaces by computer display (1st report)

Yamaguchi F. A method of designing free surfaces by computer display (1st report). Precision Machinery 1977;43(2):168–73

1977
[21]

Convexity of rational curves and total positivity

Carnicer JM, Garc´ ıa-Esnaola M, Pe˜ na JM. Convexity of rational curves and total positivity. J Comput Appl Math 1996;71(2):365–82. 23

1996
[22]

A shape preserving representation with a evaluation algorithm of linear complexity

Delgado J, Pe˜ na, JM. A shape preserving representation with a evaluation algorithm of linear complexity. Comput Aided Geom Design 2003;20(1):1– 10

2003
[23]

Constructing iterative non-uniform B- spline curve and surface to ﬁt data points

Lin H-W, Wang G-J, Dong C-S. Constructing iterative non-uniform B- spline curve and surface to ﬁt data points. Sci China Ser F 2004;47(3):315– 31

2004
[24]

Weighted progressive iteration approximation and convergence analysis

Lu L-Z. Weighted progressive iteration approximation and convergence analysis. Comput Aided Geom Design 2010;27(2):129–37

2010
[25]

An extended iterative format for the progressive- iteration approximation

Lin H-W, Zhang Z-Y. An extended iterative format for the progressive- iteration approximation. Comput Graph 2011;35(5):967–75

2011
[26]

An iterative algorithm of NURBS interpolation and approximation

Shi L-M, Wang R-H. An iterative algorithm of NURBS interpolation and approximation. J Math Res Exposition 2006;26(4):735–43

2006
[27]

Local progressive-iterative approximation format for blending curves and patches

Lin H-W. Local progressive-iterative approximation format for blending curves and patches. Comput Aided Geom Design 2010;27(4):322–39

2010
[28]

Progressive iterative approximation for triangular B´ ezier surfaces

Chen J, Wang G-J . Progressive iterative approximation for triangular B´ ezier surfaces. Comput-Aided Des 2011;43(8):889–95

2011
[29]

An iterative algorithm for polynomial approximation of rational triangular B´ ezier surfaces

Hu Q-Q. An iterative algorithm for polynomial approximation of rational triangular B´ ezier surfaces. Appl Math Comput 2013;219(17):9308–16

2013
[30]

Loop subdivision surface based progressive interpolation

Cheng F-H, Fan F-T, Lai S-H, Huang C-L, Wang J-X, Yong J-H. Loop subdivision surface based progressive interpolation. J Comput Sci Tech 2009;24(1):39–46

2009
[31]

Weighted progressive interpolation of Loop subdivi- sion surfaces

Deng C-Y, Ma W-Y. Weighted progressive interpolation of Loop subdivi- sion surfaces. Comput-Aided Des 2012;44(5):424–31

2012
[32]

Progressive interpolation based on Catmull-Clark subdivision surfaces

Chen Z-X, Luo X-N, Tan L, Ye B-H, Chen J-P. Progressive interpolation based on Catmull-Clark subdivision surfaces. Computer Graphics Forum 2008;27(7):1823–7

2008
[33]

Subdivision based interpolation with shape control

Fan F-T, Cheng F-H, Lai S-H. Subdivision based interpolation with shape control. Computer-Aided Design and Applications 2008;5(1-4):539–47

2008
[34]

Least squares cubic spline approximation I-Fixed knots

De Boor C, Rice JR. Least squares cubic spline approximation I-Fixed knots. Computer Sciences, Purdue University; 1968

1968
[35]

Iterative solution of large linear systems

Young DM. Iterative solution of large linear systems. New York-London: Academic press; 1971

1971
[36]

On polya frequency functions

Sch¨ oenberg IJ, Whitney A. On polya frequency functions. Trans Amer Math Soc 1953;74:246–59. 24

1953

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

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

Deng C-Y, Lin H-W. Progressive and iterative approximation for least squares B-spline curve and surface ﬁtting. Comput-Aided Des 2014;47:32– 44

work page 2014

[2] [1]

The NURBS book

Piegl L, Tiller W. The NURBS book. 2nd ed.; New York, USA: Springer- Verlag; 1997

work page 1997

[3] [2]

Handbook of computer aided geometric design

Farin G, Hoschek J, Kim M-S. Handbook of computer aided geometric design. 1st ed.; North-Holland; 2002

work page 2002

[4] [3]

Total least squares ﬁtting of bezier and b-spline curves to ordered data

Borges CF, Pastva T. Total least squares ﬁtting of bezier and b-spline curves to ordered data. Comput-Aided Des 2002;19:275–89

work page 2002

[5] [4]

Fitting b-spline curves to point clouds by curvature-based squared distance minimization

Wang W, Pottmann H, Liu Y . Fitting b-spline curves to point clouds by curvature-based squared distance minimization. ACM Transactions on Graphics 2006;25:214–38

work page 2006

[6] [5]

B-spline surface ﬁtting by it- erative geometric interpolation/approximation algorithms

Kineri Y, Wang M, Lin H, Maekawa T . B-spline surface ﬁtting by it- erative geometric interpolation/approximation algorithms. Comput-Aided Des 2012;44(7):697–708

work page 2012

[7] [6]

An eﬃcient method for ﬁtting large data sets using T-splines

Lin H-W, Zhang Z-Y. An eﬃcient method for ﬁtting large data sets using T-splines. SIAM J Sci Comput 2013;35(6):A3052–68. 22

work page 2013

[8] [8]

Elitist clonal selection algorithm for optimal choice of free knots in b-spline data ﬁtting

Galveza A, Iglesiasa A, Avilaa A, Oteroc C, Ariasc R, Manchadoca C. Elitist clonal selection algorithm for optimal choice of free knots in b-spline data ﬁtting. Applied Soft Computing 2015;26:90–106

work page 2015