Dirichlet Winding Reconstruction (DiWR)

• The paper introduces DiWR, a method that jointly optimizes point orientations, area weights, and confidence to reconstruct watertight surfaces from noisy, non-uniform point clouds.
• It minimizes the Dirichlet energy of the induced winding field while enforcing surface anchoring and stability through additional generalized winding number constraints.
• The staged alternating optimization outperforms traditional pipelines by effectively handling noise, non-uniform sampling, and outliers without separate preprocessing.

Searching arXiv for the cited DiWR paper and closely related reconstruction methods for accurate citations. arxiv_search(query="(Li et al., 14 Feb 2026)", max_results=5) Searching by arXiv identifier ([2602.13801](/papers/2602.13801)). {"query":"(Li et al., 14 Feb 2026)","max_results":5} Dirichlet Winding Reconstruction (DiWR) is a method for reconstructing watertight surfaces from unoriented point clouds with non-uniform sampling, noise, and outliers by using the generalized winding number (GWN) field as the target implicit representation and jointly optimizing point orientations, per-point area weights, and confidence coefficients in a single pipeline (Li et al., 14 Feb 2026). The method minimizes the Dirichlet energy of the induced winding field together with additional GWN-based constraints, allowing DiWR to compensate for non-uniform sampling, reduce the impact of noise, and downweight outliers during reconstruction, with no reliance on separate preprocessing. In the formulation associated with "Joint Orientation and Weight Optimization for Robust Watertight Surface Reconstruction via Dirichlet-Regularized Winding Fields" (Li et al., 14 Feb 2026), DiWR is explicitly positioned against both traditional multi-stage pipelines and recent joint orientation-reconstruction methods.

1. Generalized winding field and target representation

The continuous generalized winding-number field is defined for a solid $\Omega \subset \mathbb{R}^3$ with smooth, closed boundary $\partial \Omega$ by

$$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$

Inside $\Omega$ one has $w \approx 1$, outside $w \approx 0$, and on $\partial \Omega$ one finds $w \approx \tfrac12$ (Li et al., 14 Feb 2026). In DiWR, this field is the target implicit representation.

The discrete setting begins from an unoriented point set $P=\{p_i\}_{i=1}^n$ without connectivity. Three per-point variables are introduced: $n_i \in S^2$ as the orientation at $\partial \Omega$0, $\partial \Omega$1 as the area weight at $\partial \Omega$2, and $\partial \Omega$3 as the confidence of $\partial \Omega$4. Writing $\partial \Omega$5, the discrete winding field is

$\partial \Omega$6

If all $\partial \Omega$7 and $\partial \Omega$8 are outward, this expression approximates the continuous integral.

This formulation makes the implicit field depend jointly on geometry, orientation, sampling compensation, and outlier suppression. A plausible implication is that DiWR treats reconstruction, orientation resolution, and robustness as coupled aspects of the same variational problem rather than as separate preprocessing stages.

2. Dirichlet regularization and auxiliary constraints

A true winding field is harmonic off the boundary, and the continuous regularizer is the Dirichlet energy

$\partial \Omega$9

where $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$0 is a bounding box containing $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$1 (Li et al., 14 Feb 2026). Because $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$2 is not known a priori, DiWR excludes a small tubular band around high-confidence points to avoid singularities. With

$$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$3

the discrete Dirichlet energy is approximated by sampling a grid $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$4 in $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$5, omitting or down-weighting samples inside $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$6: $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$7 Here $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$8 is computed by analytically differentiating (1). Penalizing this term drives $$w(x)=\frac{1}{4\pi}\int_{\partial\Omega}\frac{\langle n(y),\,y-x\rangle}{\|y-x\|^{3}}\,dS(y).$$9 toward a smooth, globally consistent inside-outside indicator.

Three additional GWN-based constraints stabilize the optimization. The surface-anchoring term,

$\Omega$0

encourages high-confidence points to lie on $\Omega$1. The stable-area term,

$\Omega$2

penalizes drift in the total effective surface area relative to values $\Omega$3 at the start of the current stage. The confidence-polarization term,

$\Omega$4

encourages $\Omega$5 so as to separate inliers from outliers.

Taken together, these terms anchor the field near the observed surface, prevent uncontrolled weight drift, and promote a binary confidence structure. This suggests that robustness in DiWR is not delegated to a single denoising heuristic but distributed across the field regularizer and the variable-specific penalties.

3. Joint variables and alternating optimization

DiWR simultaneously optimizes orientations $\Omega$6, area weights $\Omega$7, and confidence coefficients $\Omega$8 (Li et al., 14 Feb 2026). The parametrization is explicit: each $\Omega$9 is a unit vector in $w \approx 1$0; each $w \approx 1$1 and is initialized by a 2D Voronoi-based surface-area approximation or uniformly; and each $w \approx 1$2, initialized to $w \approx 1$3, is then reset by a density-stratified bi-means on current $w \approx 1$4.

The optimization proceeds in outer iterations $w \approx 1$5, each comprising three staged subproblems. The orientation update is

$w \approx 1$6

via the parallel Diffusing Winding Gradients solver. With $w \approx 1$7 and $w \approx 1$8 fixed, the area-weight update solves

$w \approx 1$9

With $w \approx 0$0 and $w \approx 0$1 fixed, the confidence update solves

$w \approx 0$2

Within each outer iteration, the orientation and area-weight steps are alternated until the average relative change

$w \approx 0$3

drops below $w \approx 0$4. Then one call to the confidence step is made. The outer loop stops once normal-change $w \approx 0$5 falls below $w \approx 0$6 or $w \approx 0$7 exceeds $w \approx 0$8.

Each subproblem is minimized by GPU-parallel RMSProp applied to $w \approx 0$9 or $\partial \Omega$0, with gradients of (2)–(5) obtained by differentiating (1) and the term definitions. The orientation update is done by invoking DWG on the GPU (Li et al., 14 Feb 2026). The staged structure is a central part of the method rather than an implementation detail: the paper reports that the three-stage alternating significantly outperforms a naïve joint gradient descent on $\partial \Omega$1, with staged updates improving stability and convergence.

4. Implementation details and mesh extraction

The implementation is notable for requiring no preprocessing: $\partial \Omega$2 is used directly (Li et al., 14 Feb 2026). Surface-area initialization uses a local PCA tangent plane together with 2D Voronoi cell area to produce $\partial \Omega$3. Confidence reset applies bi-means on $\partial \Omega$4 to obtain binary $\partial \Omega$5 labels and then density stratifies into 128 levels to smooth.

Dirichlet sampling is carried out on a uniform $\partial \Omega$6 grid in $\partial \Omega$7, with radius $\partial \Omega$8 exclusion set to $\partial \Omega$9 and partial volumes $w \approx \tfrac12$0 computed via sphere–cube intersection. The high-confidence threshold is $w \approx \tfrac12$1. The stopping thresholds are $w \approx \tfrac12$2, $w \approx \tfrac12$3, and $w \approx \tfrac12$4. The $w \approx \tfrac12$5-schedule keeps the Dirichlet term fixed at $w \approx \tfrac12$6; initially $w \approx \tfrac12$7, $w \approx \tfrac12$8, $w \approx \tfrac12$9, $P=\{p_i\}_{i=1}^n$0, and $P=\{p_i\}_{i=1}^n$1 for “easy” inputs, and these $P=\{p_i\}_{i=1}^n$2 values are halved for heavier corruption.

After convergence, the method retains $P=\{p_i\}_{i=1}^n$3 and feeds $P=\{p_i\}_{i=1}^n$4, optionally weighted by $P=\{p_i\}_{i=1}^n$5, to Screened Poisson Reconstruction (sPSR) to extract a watertight mesh (Li et al., 14 Feb 2026). In this sense, DiWR supplies a robustly regularized oriented point set to a downstream watertight surface extractor, while the joint optimization is responsible for making that input plausible under noise, density variation, and outlier contamination.

5. Empirical evaluation and ablation findings

The reported evaluation uses point clouds from 3D Gaussian Splatting primitives from OmniObject3D images, VGGT computer-vision outputs, corrupted graphics benchmarks including Armadillo, Dragon, and Kitten variants, and a 125-case Bunny stress test (Li et al., 14 Feb 2026). The 3D Gaussian Splatting inputs are characterized by interior-outlier shells and non-uniform sampling, whereas VGGT outputs are described as dense, noisy, and containing spurious sheets.

The metrics are Chamfer Distance $P=\{p_i\}_{i=1}^n$6 and Normal Consistency (cosine similarity). Input quality is measured by noise $P=\{p_i\}_{i=1}^n$7 median $P=\{p_i\}_{i=1}^n$8-NN residual, non-uniformity $P=\{p_i\}_{i=1}^n$9 trimmed coefficient of variation of $n_i \in S^2$0-NN distances, and outlier rate $n_i \in S^2$1 fraction of $n_i \in S^2$2. Baselines include the MSP multi-stage pipeline consisting of PointCleanNet outlier removal, PCDNF denoising, Dipole orientation, and sPSR; the unified methods WNNC, FaCE, and DWG, with and without pre-filtering; and the learning-based methods NSH (Neural-Singular-Hessian) and LoSF-UDF.

Across all categories, DiWR yields the lowest Chamfer distances and highest normal consistency, often by large margins (Li et al., 14 Feb 2026). On 3DGS it suppresses interior primitives. On VGGT it down-weights near-surface sheets, with Dirichlet energy dropping from $n_i \in S^2$3. On graphics benchmarks it circumvents irreversible preprocessing errors. In the Bunny stress test, DiWR succeeds on $n_i \in S^2$4 of $n_i \in S^2$5 cases versus $n_i \in S^2$6 for baselines.

The ablation discussion is qualitative rather than tabulated. Without the Dirichlet term $n_i \in S^2$7, the winding field becomes locally noisy and normals may flip in sparsely sampled regions. Without area-weight optimization, fixing $n_i \in S^2$8 uniformly causes the Dirichlet energy to focus on outlier suppression but fail to rebalance sampling, leading to drift in $n_i \in S^2$9. Without confidence polarization $\partial \Omega$00, many $\partial \Omega$01 remain fractional, letting outliers unduly influence $\partial \Omega$02 and destabilize orientation updates. In practice, area weighting compensates for density variation, confidence reset sharpens the inlier/outlier separation, and Dirichlet regularization forces global consistency (Li et al., 14 Feb 2026).

6. Terminological scope and cross-domain ambiguity

The acronym “DiWR” is not unique across the arXiv corpus. In the surface-reconstruction sense discussed above, it denotes the winding-field method introduced in "Joint Orientation and Weight Optimization for Robust Watertight Surface Reconstruction via Dirichlet-Regularized Winding Fields" (Li et al., 14 Feb 2026). However, closely related wording is also used in distinct mathematical and physical contexts.

A summary attached to "Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction" describes a procedure under the name “Dirichlet-Winding Reconstruction” for constructing low-energy torus-valued maps on data from linearly independent cohomology classes (Scoccola et al., 2022). In that account, the central objects are the Dirichlet energy and Dirichlet form for circle-valued maps, harmonic representatives of integral $\partial \Omega$03-classes, a discrete correlation or Gram matrix of cocycles, and an LLL-based lattice-reduction step that decorrelates multiple circle-valued maps before assembly into a torus-valued embedding. The governing setting is topological data analysis rather than surface reconstruction.

An unrelated use appears in "Winding-Control Mechanism of Non-Hermitian Systems," where “Dirichlet Winding Reconstruction” denotes selective spectral collapse under conditional boundary conditions in a one-dimensional non-Hermitian chain (Fu et al., 20 Jun 2025). There, LP-CBC and RP-CBC impose a Dirichlet-type constraint on one direction of boundary hopping, and the sign of the spectral winding number determines which periodic-boundary spectral segments collapse onto the open-boundary spectrum. The operative concepts are generalized Brillouin zones, Bloch points, and non-Hermitian skin modes, not generalized winding-number fields on point clouds.

This terminological overlap can create confusion. In current arXiv usage, “DiWR” may therefore refer either to a geometry-processing pipeline for watertight reconstruction (Li et al., 14 Feb 2026), a Dirichlet-form-based toroidal coordinate construction (Scoccola et al., 2022), or a boundary-controlled winding-selection mechanism in non-Hermitian physics (Fu et al., 20 Jun 2025). The shared vocabulary is the pairing of “Dirichlet” and “winding,” but the mathematical objects, optimization problems, and application domains are different.

7. Significance within surface reconstruction

Within the reconstruction literature, DiWR is distinguished by combining orientation estimation, area reweighting, and confidence-based outlier attenuation in one optimization loop with no reliance on separate preprocessing (Li et al., 14 Feb 2026). The method uses the GWN field as the central implicit quantity, employs Dirichlet energy as a global regularizer, and then enforces surface anchoring, stable effective area, and polarized confidences to stabilize the solution.

This design is especially relevant for inputs that arise outside classical graphics scanning pipelines, including 3D Gaussian Splatting and computer-vision outputs. The reported behavior on interior-outlier shells, non-uniform sampling, noisy dense predictions, and spurious sheets suggests that DiWR is formulated to address failure modes that are difficult to correct once an irreversible preprocessing stage has made an incorrect decision. A plausible implication is that its principal contribution lies less in introducing a new extractor than in shifting robustness earlier, into the implicit-field fitting stage itself, before watertight meshing by sPSR.