Implicit Progressive-Iterative Approximation for Curve and Surface Reconstruction

Hongwei Lin; Yusuf Fatihu Hamza; Zihao Li

REVIEW 4 major objections 6 minor 35 references

Implicit Progressive-Iterative Approximation for Curve and Surface Reconstruction

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read I-PIA reconstructs implicit curves and surfaces by iterating B-spline coefficients from zero, converging to the minimum-norm least-squares fit and eliminating spurious sheets without explicit regularization.

desk verdict A correct but standard LSPIA convergence proof applied to implicit B-spline fitting; the 'no spurious sheets' claim is empirically supported but not proven, and Property 2 has a size error. read the letter →

arxiv 1909.00551 v1 pith:AWBTZIGX submitted 2019-09-02 math.NA cs.GRcs.NA

Yusuf Fatihu Hamza , Hongwei Lin , Zihao Li This is my paper · ORCID

classification math.NAcs.GRcs.NA MSC 65D1765D1065F2065F10

keywords ImplicitcurveandsurfacefittingProgressive-iterativeapproximationB-splineZero-levelsetsSpurioussheetsMoore-Penrosepseudoinverse

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

The paper's aim is to remove the spurious sheets that appear in implicit curve and surface reconstruction without paying the usual cost of adding explicit regularization. It does this by adapting the progressive-iterative approximation idea to implicit fitting: starting from zero control coefficients, the iteration converges to the minimum-Euclidean-norm solution of the normal equations. That minimum-norm solution acts as a built-in regularizer, so extra zero-level sets do not appear during iteration. The authors demonstrate on 2D and 3D models that the resulting reconstructions are robust to inaccurate distance fields, holes, non-uniform sampling, noise, and open surfaces, and that timings improve by one to three orders of magnitude over a total-variation-regularized baseline.

What carries the argument

The machinery is the stationary linear iteration $C^{{(α+1)}}$ = (I - μ B^T B) $C^{{(α)}}$ + μ B^T b, where B is the collocation matrix of tensor-product B-spline basis functions evaluated at the data points plus offset points generated by displacing each data point along its normal. The weight μ is chosen so that 0 < μ < 2/λ_max(B^T B), with a practical choice μ = 2 / ||B^T B||_∞. Because B^T B is positive semi-definite and singular (more coefficients than constraints), the iteration is a Richardson iteration on the normal equations; the eigen-decomposition of B^T B makes the error contract in the range of B^T B while the null-space component stays fixed. Starting from C^(0) = 0 therefore selects the projection onto the range, i.e., the minimum-norm least-squares coefficient vector.

What would settle it

Take a point cloud with two well-separated closed curve components, run I-PIA to convergence, and count the connected components of the zero level set of the reconstructed function; if any component appears away from the input data, the claim that I-PIA produces no extra zero-level sets is false.

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Extended reading notes

Core claim

The central claim is Theorem 3.4: with zero initial coefficients C^(0)=0, the I-PIA iteration converges to (B^T B)^+ B^T b, the Moore-Penrose pseudoinverse solution of the singular least-squares system B^T B X = B^T b. Since this is the solution of minimum Euclidean norm among all solutions of the constrained minimization min ||X||_E subject to B^T B X = B^T b, the iteration solves the normally underdetermined implicit-fitting problem with an implicit regularization: it selects the smallest coefficient vector that fits the data and offset points. On the paper's account this is why no extra zero-level sets (spurious sheets) arise in the reconstruction, and why no explicit minimization or regularization step is needed; the regularizer is a byproduct of starting at zero and iterating.

Load-bearing premise

The paper assumes, rather than proves, that the smallest-coefficient least-squares B-spline fit to the data and offset points does not contain extra loops or sheets; this is shown only by examples, not proved.

Editorial extensions

If this is right

Each I-PIA iteration is a matrix-vector product, so no linear system is solved during reconstruction; this is the source of the reported one-to-three-orders-of-magnitude speedup over regularized solvers.
Starting from zero control coefficients selects the minimum-norm B-spline coefficient vector among all fits, which is the regularization that suppresses spurious sheets.
Since no explicit regularization term is added, the same iteration applies to both implicit curves (bivariate B-splines) and implicit surfaces (trivariate B-splines).
The method inherits robustness to inaccurate distance fields, holes, non-uniform sampling, and noise because the offset-point construction and the min-norm selection do not depend on precise distance values.
Convergence is guaranteed for any weight in (0, 2/λ_max(B^T B)), so the iteration can be tuned for speed without changing the limit.

Reading between the lines

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

Beyond the paper, a direct test of the central practical claim would be to count the connected components of the zero level set on adversarial point clouds; if any spurious component appears away from the data, the advertised advantage would fail.
The min-norm limit depends on the offset distance σ and the sign pattern assigned to offset points; varying σ likely trades fidelity against sheet suppression, a trade the paper does not quantify.
The convergence proof only uses the eigen-decomposition of B^T B, so the same iteration could be accelerated with standard stationary-iteration accelerations (e.g., Chebyshev or conjugate-gradient-like schemes) that the paper does not explore.
Because the argument does not rely on B-spline-specific properties beyond a positive semi-definite Gram matrix, the same zero-start iteration may extend to other implicit bases, such as T-splines or radial basis functions, though the paper only treats B-splines.

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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

First thing to know: this is a modest, mostly sound application of an existing iteration to a new problem class. The paper transfers LSPIA to implicit B-spline reconstruction, proves convergence to the minimum-norm least-squares fit, and demonstrates big speedups over one TV-regularized baseline. The proof is correct as far as it goes, but it does not do what the authors claim it does: it says nothing about the number or location of zero-level components, so the 'no spurious sheets' headline is not supported by the theorem. It rests entirely on the examples.

What is genuinely new: PIA has not been used for implicit reconstruction before, and framing the minimum-norm solution as an implicit regularization is a reasonable idea. The experiments are fairly extensive—holes, noise, non-uniform sampling, open surfaces—and the reported speedups (one to three orders of magnitude over Liu et al.) are plausible because the iteration avoids solving linear systems. If the speedups reproduce, that is useful for practitioners.

Soft spots, in order of seriousness. First, the central practical claim is over-stated. Theorem 3.4 only establishes convergence to (B^T B)^+ B^T b. Minimizing coefficient norm does not control the geometry of the zero set; a tensor-product B-spline can still oscillate and produce extra branches. The paper shows only selected models with no failure cases and no topological counts. Second, Property 2 is factually wrong about the size of B: with offset points the collocation matrix is 2n × m, not n × m. The singularity argument then only holds when m > 2n; the case n < m ≤ 2n is not covered, which is exactly where the paper needs the argument to be cleanest. Third, the practical choice μ = 2/||B^T B||_∞ is taken from prior work and not shown to satisfy the strict inequality μ < 2/λ_max(B^T B). This is likely fine numerically but the paper leaves a gap in its own convergence condition. These are addressable weaknesses, not fatal ones. The iteration is coherent, the theorem is right under its stated hypothesis, and the examples are consistent with the claim, even if not proof.

Who gets value: researchers in geometric modeling and CAD who want a fast implicit fitting baseline and a clear statement of the minimum-norm convergence property. The paper deserves a serious referee, but the authors should be asked to fix Property 2, temper the no-spurious-sheets claim, add a failure analysis or at least a topological count on a test suite, and release code/data. I'd send it to review.

Referee Report

4 major / 6 minor

Summary. The paper proposes an implicit progressive-iterative approximation (I-PIA) method for reconstructing implicit curves and surfaces from unorganized point clouds with associated normals. The method appends offset points to the data, sets the initial tensor-product B-spline coefficients to zero, and iterates C^{(α+1)} = (I − μB^T B)C^{(α)} + μB^T b, where B is the collocation matrix over the data-plus-offset points and b encodes zero function values at the data points and ε at the offset points. The authors prove that for zero initial coefficients the iteration converges to (B^T B)^+ B^T b, the minimum-Euclidean-norm solution of the normal equations B^T B X = B^T b, and they interpret this as solving a constrained minimization problem with a natural regularization. The paper further claims that I-PIA eliminates extra zero-level sets ('spurious sheets') without explicit minimization and reports speedups of one to three orders of magnitude over a total-variation method on a wide range of curve and surface examples.

Significance. The idea of adapting progressive-iterative approximation to implicit fitting is novel, and the matrix formulation is clean. The convergence theorem (Theorem 3.4) is essentially a Richardson-iteration argument and is correct provided the step-size condition is satisfied; the paper also gives a self-contained derivation rather than citing the result. The experimental coverage is broad, including holes, noise, non-uniform sampling, porous surfaces, and fine details, and the reported speedups are substantial. However, the headline geometric claim that I-PIA produces no spurious sheets is not a consequence of the convergence theorem, which says nothing about the topology of the zero level set. That claim currently rests only on selected favorable examples, and the convergence analysis contains several gaps that need to be closed. If the geometric claim can be supported by a rigorous argument or a much more systematic empirical study, and the technical issues identified below are fixed, this would be a valuable contribution to implicit reconstruction.

major comments (4)

[Section 3.3, Theorem 3.4; Section 1, contribution bullet 2] Theorem 3.4 establishes only that the coefficient vector C^{(α)} converges to (B^T B)^+ B^T b, the minimum-Euclidean-norm solution of the normal equations. It contains no statement about the number or location of the connected components of the zero set {f^{(α)} = 0}, either in the limit or at any finite iteration. The paper's headline claim that 'No extra zero-level sets exist in the iteration of I-PIA' is therefore not a consequence of the proved result; minimizing the coefficient norm does not by itself imply any geometric regularity of the tensor-product B-spline function. The only support offered is a favorable selection of examples (Figures 1, 2, 4-8) with no negative cases and no counts of zero-level components. Because spurious-sheet elimination is the paper's main advertised advantage, this gap is load-bearing and needs either a theorem or a far more systematic empirical study with explicit topological counts.
[Section 3.3, Property 2; Eq. (12), Eq. (17)] Property 2 states that B is n × m and concludes that B^T B is singular from m > n. But B is defined as a collocation matrix on the 2n points (n data points plus n offset points), so its size is 2n × m. The claimed singularity does not follow from m > n. For the planar examples in Table 1, 2n exceeds m (e.g., Flower has 1018 rows versus 900 columns), so B^T B may well be nonsingular. The convergence theorem itself does not require singularity, but the motivation for the constrained minimum-norm problem (20) and the interpretation of the limit as the solution of a 'regularized' problem rest on this property. The property and the surrounding discussion must be corrected.
[Remarks 3.1 and 3.2] The practical step size μ = 2/||B^T B||_∞ is asserted to satisfy the convergence condition 0 < μ < 2/λ_max(B^T B), but no proof or reference is given for the strict inequality. For the symmetric positive semidefinite matrix B^T B, λ_max ≤ ||B^T B||_∞, so the chosen μ is only known to satisfy μ ≥ 2/λ_max when the bound is attained; the strict inequality used in Property 3 and in the proof of Theorem 3.4 may fail. The manuscript needs either a proof that ||B^T B||_∞ strictly exceeds λ_max for the specific B-spline collocation matrices used here, or a different step-size selection that is provably convergent.
[Sections 3.1 and 4.2] The algorithm is specified with exactly one offset point per data point (p_l = p_i + σ n_i, l = n+i, i = 1,...,n) and a right-hand side b of length 2n with entries 0 at the data points and +ε at the offset points. However, the experiments in Section 4.2 describe both inside and outside offset sets with signed values (−0.1, −0.2, 0.2, 0.1), and Figure 3 shows four offset layers. This inconsistency between the described method and the implementation used for the reported results affects reproducibility and makes it unclear which variant is being evaluated. The authors should specify the exact offset configuration used in each experiment and align the algorithm description with it.

minor comments (6)

[Abstract and Section 1] 'we proposed' should be 'we propose', and the sentence 'extra zero-level sets or spurious sheets arise in the reconstruction process makes' is ungrammatical; please revise for clarity.
[Eq. (17)] The displayed collocation matrix is internally inconsistent: the second column is B1(x1)B1(y1)B_{N_w}(z1) in the first row but B1(x2)B1(y2)B2(z2) in the second row, and the lexicographic ordering of the basis functions should be stated carefully.
[Section 3.2, definition of δ_l] In the surface case, the difference δ_l for l = n+i is defined using f^{(α)}(x_r, y_r, z_r), but the index r is not tied to i; this should be corrected to f^{(α)}(x_i, y_i, z_i).
[Table 2] The 'Maximum Error' column reports errors for I-PIA only; to support the time comparison, the errors of the comparison method [1] should be reported as well, or a statement should clarify that the accuracy of the two methods is comparable.
[Remark 3.1] The formula C = ||B^T B||_∞ = max_{ij} Σ_k B_i(x_k)B_j(y_k) is not the standard definition of the induced ∞-norm and is ambiguous; the maximum should be over one index only, with summation over the other.
[Figure 7 caption] 'porous surfac' is a typo for 'porous surface'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorem is proven from the iteration itself; the unproved zero-level-set claim is an evidentiary gap, not a circular reduction.

full rationale

The central derivation chain is self-contained. Theorem 3.4 proves convergence of iteration (18) to (B^T B)^+B^T b: the proof uses the eigen-decomposition of B^T B, the spectral bound on mu, and C(0)=0; no target result is inserted as an assumption. The limit (B^T B)^+B^T b is the minimum-norm solution of the normal equations by definition of the Moore-Penrose inverse, so the identification with problem (20) is a derivation, not a restatement. The only use of prior work by the authors is the practical choice of mu in Remarks 3.1 and 3.2 (referring to [8]) and background PIA literature; this choice is not load-bearing for the convergence theorem, which only requires 0<mu<2/lambda_max. The paper's advertised 'no extra zero-level sets' property is asserted in Section 1 and Section 4.1 and illustrated on examples, but it is not a consequence of Theorem 3.4; however, lack of proof is a correctness risk, not circularity, because no fitted parameter or self-citation is used to force that conclusion. The dimension inconsistency in Property 2 (B is 2n x m, not n x m) similarly affects a side justification, not the main reduction. No circular step satisfying the required 'specific reduction' standard was found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The algorithm's convergence proof uses standard linear algebra. The practical effectiveness claims add user-chosen parameters: offset distance, offset value, grid resolution, and step size. The no-spurious-sheets guarantee is an unproven domain assumption, and the rank argument in Property 2 miscounts the rows of the collocation matrix.

free parameters (4)

Offset distance sigma = Not fixed; e.g., 0.02, 0.05, 0.1 in Fig. 3; unspecified for most models
Offset points are generated as p_i + sigma n_i (Section 3.1); sigma controls the location of auxiliary constraints and affects the reconstructed zero set.
Offset value epsilon = Not fixed; examples use random values in [±0.5-sigma, ±0.5+sigma] and values like ±0.1, ±0.2
The target implicit value at offset points (Section 3.1); it sets the scale of f and is chosen by hand in experiments.
Step size mu = 2 / ||B^T B||_infinity
Chosen to ensure convergence (Remark 3.1); not fitted to data but is a user-selected hyperparameter of the iteration.
Grid resolution (knot counts) = e.g., 30x30, 50x50x50, 70x70x70, 190x190x190 in Tables 1-2
The B-spline grid size is chosen per model and strongly influences reconstruction detail and runtime.

assumptions (4)

standard math Symmetric positive semidefinite matrices admit eigen/SVD decomposition; Moore-Penrose inverse properties; spectral-radius convergence of the iteration.
Used in the proof of Theorem 3.4; standard linear algebra facts.
domain assumption The minimum-norm least-squares solution has no spurious zero-level sets for the tested inputs.
The central practical claim in Section 1 and Section 4.1; not proven, only illustrated on examples.
domain assumption Oriented unit normals are available for every data point, and one layer of offset points along the normals is enough to avoid the trivial zero solution.
Section 3.1; the algorithm depends on this input and offset construction.
ad hoc to paper The collocation matrix B has n rows and the number of control coefficients m exceeds the number of data points, so B^T B is singular (Property 2).
Used to justify the minimum-norm regularization story. The matrix B in Eqs. (12) and (17) has 2n rows (data plus offset points), so m > n does not imply singularity; the statement is incorrect or at least requires 2n < m.

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Cite this review

Pith. "Pith review of Implicit Progressive-Iterative Approximation for Curve and Surface Reconstruction." pith.science (2026). https://pith.science/paper/AWBTZIGX

@misc{pith2026190900551,
  author       = {Pith},
  title        = {Pith review of: Implicit Progressive-Iterative Approximation for Curve and Surface Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWBTZIGX}},
  note         = {Machine review of arXiv:1909.00551}
}

read the original abstract

Implicit curve and surface reconstruction attracts the attention of many researchers and gains a wide range of applications, due to its ability to describe objects with complicated geometry and topology. However, extra zero-level sets or spurious sheets arise in the reconstruction process makes the reconstruction result challenging to be interpreted and damage the final result. In this paper, we proposed an implicit curve and surface reconstruction method based on the progressive-iterative approximation method, named implicit progressive-iterative approximation (I-PIA). The proposed method elegantly eliminates the spurious sheets naturally without requiring any explicit minimization procedure, thus reducing the computational cost greatly and providing high-quality reconstruction results. Numerical examples are provided to demonstrate the efficiency and effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 1909.00551 by the authors.

**Figure 1.** Iterations in the reconstruction of 2D data sets: First row, flower model, and second row, Coons curve model. Blue points are the given data sets, and the red line is the reconstructed curve. From left to right: the 1st, 5th, 10th, 15th iteration steps. 4. Experiments and discussion Several experiments have been carried out to evaluate the performance of I-PIA. All the experiments are performed in MATLAB on a PC wi… view at source ↗

**Figure 2.** Iterations in the reconstruction of 3D data sets, Torus and double-Torus. The red points are the given data sets. From left to right: the 1st, 5th, 10th, 15th iteration steps. the reconstruction procedure. To show the effectiveness of I-PIA in the reconstruction of implicit curves and surfaces without the appearance of extra level sets, we test our algorithm on planar curves and 3D surfaces. The initial control coef… view at source ↗

**Figure 3.** Reconstruction of 2D data sets with inaccurate distance fields. First row: (a) the input data of dolphin (blue) and offset points (outside offset in green and inside offset in magenta). The preassigned function values selected as uniformly distributed random number in [±0.5 − σ, ±0.5 + σ] for the outside and inside offsets, respectively, (b)–(e): σ = 0, 0.02, 0.05, 0.1. Second row: (f) and (h) the input data of butt… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Holes and gaps filling. (a,d) elephant; (b,e) bunny; (c,f) fertility. First row: reconstructed surfaces. Second row: the reconstructed surfaces with the data points superimposed. right part of the bunny model is down-sampled, and 90% of the original data points are rem…

**Figure 5.** Figure 5: Surface reconstructions of the bunny model from different point clouds. (a) The original point cloud of the bunny model. (b) 90% of the right part of the bunny model are removed. (c) The original point cloud is disturbed by noise. (d)–(f) are the reconstructed surfaces…

**Figure 6.** Figure 6: Surface reconstructed from point clouds with di [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: Surface reconstruction of open surface (porous surfac). (a) The point cloud sampled from the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: Surface reconstruction of Buddha and BU with sectional enlargements. I-PIA [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Y . Liu, Y . Song, Z. Yang, J. Deng, Implicit surface reconstruction with total variation regularization, Computer Aided Geometric Design 52 (2017) 135–153. 17

work page 2017

[2] [2]

J ¨uttler, A

B. J ¨uttler, A. Felis, Least-squares ﬁtting of algebraic spline surfaces, Advances in Computational Mathematics 17 (1-2) (2002) 135–152

work page 2002

[3] [3]

Rouhani, A

M. Rouhani, A. D. Sappa, Implicit B-spline ﬁtting using the 3L algorithm, in: 2011 18th IEEE International Conference on Image Processing, IEEE, 2011, pp. 893–896

work page 2011

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M. Rouhani, A. D. Sappa, E. Boyer, Implicit B-spline surface reconstruction, IEEE Transactions on Image Pro- cessing 24 (1) (2014) 22–32

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