Directional Smoothness

• Directional smoothness is defined as controlling local variations along specific directions using metrics like directional derivatives, curvature integrals, and Laplacian energies.
• It is applied across geometric modeling, image and surface smoothing, and point cloud processing to enhance fidelity and reduce artifacts.
• Algorithmic implementations leverage metrics such as total normal curvature and path-dependent optimization to improve convergence, feature preservation, and overall performance.

Directional smoothness denotes the property of a function, signal, shape, mapping, or model output in which local variations are preferentially controlled or measured along specific directions rather than being constrained uniformly. This concept finds rigorous expression in multiple domains including geometric modeling, functional analysis, optimization, point-cloud generation, and image/surface smoothing. Directional smoothness generalizes isotropic smoothness metrics by decomposing local behavior into direction-dependent terms—frequently via directional derivatives, energy integrals, curvature, or graph-based differences. Its algorithmic handling enables models and numerical methods to better preserve geometric fidelity, reduce artifacts, and realize trajectory-dependent convergence in optimization.

1. Mathematical Formulations of Directional Smoothness

Directional smoothness in geometrical settings is typically encoded by measuring the squared differences or derivatives along multiple directions at each point. For point clouds, given $X = \{x_1, ..., x_n\} \subset \mathbb{R}^3$, the local, point-wise directional smoothness at $x_i$ is expressed as

$$S_{\text{dir}}(x_i) = \sum_{j\in N(i)} w_{ij}\, \|x_j - x_i\|^2,$$

where $N(i)$ indexes the $K$-nearest neighbors of $x_i$, and $w_{ij}$ are adjacency weights, typically binary or Gaussian kernels. More generally, this coincides locally with the Laplacian energy restricted to $i$, which can be interpreted as penalizing discrete directional derivatives along the edges to $N(i)$ (Li et al., 2024).

In surface and image smoothing, directional smoothness is formulated via total normal curvature (TNC) regularization. Consider a surface $u: \Omega \subset \mathbb{R}^2 \to \mathbb{R}$, with normal curvature in direction $\theta$ given by

$$\kappa_n(\theta) = \frac{u_{xx}\cos^2\theta + 2u_{xy}\cos\theta\sin\theta + u_{yy}\sin^2\theta}{\sqrt{1 + |\nabla u|^2}\,\left[1+(\nabla u \cdot \mathbf{t})^2\right]},$$

where $\mathbf{t} = (\cos\theta, \sin\theta)$. The TNC integrates the absolute curvature in all directions: $$\kappa_T(x, y; u) = \int_0^{2\pi} |\kappa_n(\theta)|\, d\theta.$$ The associated variational energy combines this term with total variation and data fidelity components (Lu et al., 22 Dec 2025).

For maps between surfaces (e.g., triangle meshes), the continuum Dirichlet energy

$$E_{\text{Dir}}(f) = \frac{1}{2} \int_{M_1} \|\nabla f(x)\|^2\, d\mu_1(x)$$

and its discrete cotan-Laplacian form penalize pointwise directional derivative magnitudes and inform the smoothness of correspondence maps (Magnet et al., 2022).

In functional analysis, directional regularity (a smoothness notion) is detected by decay properties of the directional short-time Fourier transform (DSTFT): $$V_{g, u}f(x, \omega) = \int_{\mathbb{R}^n} f(t)\, \overline{g(u \cdot t - x)}\, e^{-2\pi i\, t\cdot\omega}\, dt,$$ with window $g$ and analysis direction $u$. Rapid decay in $\omega$ signals $C^\infty$-regularity along the corresponding direction (Atanasova et al., 2017).

2. Directional Smoothness in Algorithmic Frameworks

Differentiated implementation of directional smoothness depends on the application domain:

• Diffusion-based point cloud generation augments reverse SDE steps with graph Laplacian-derived smoothness gradients, computed at each denoising step on the Tweedie-estimated clean cloud, thereby penalizing local non-smoothness at each point in neighbor directions. The discrete update integrates the learned score function and the smoothness prior gradient, which is the negative Laplacian energy gradient. Pseudocode includes neighbor graph construction, Laplacian formation, gradient computation, and point updates per diffusion time step (Li et al., 2024).
• Surface/image smoothing using TNC regularization solves high-order PDEs whose steady-state minimizes curvature in all directions. A gradient flow formulation and operator splitting (e.g., Lie splitting) decompose the time evolution into curvature, TV, constraint, and fidelity substeps; most are closed-form or efficiently solvable (fixed-point iteration, ADMM, FFT for elliptic components) (Lu et al., 22 Dec 2025).
• Non-rigid shape matching incorporates the Dirichlet energy of pulled-back coordinates into the optimization pipeline. An alternating minimization with auxiliary variables permits efficient decoupling of the smoothness quadratic term and functional/spectral constraints. Each iteration comprises spectral least squares for functional maps, sparse linear solves for Dirichlet variables, and row-wise nearest-neighbor assignments for correspondence matrices (Magnet et al., 2022).
• Path-dependent optimization in gradient descent replaces the global $L$-smoothness constant with directional smoothness functions ($D(y,x)$, $A(y,x)$), enabling adaptive, trajectory-dependent step sizes and tighter convergence bounds. Point-wise smoothness is measurable as

$$D(y, x) := 2\frac{\|\nabla f(y) - \nabla f(x)\|}{\|y-x\|}.$$

Strongly adapted steps are found by root-finding involving $M(x^+, x)$, and Polyak or normalized schemes implicitly exploit path-wise smoothness without explicit estimation (Mishkin et al., 2024).

3. Metric-Based Evaluation and Empirical Validation

Directional smoothness is evaluated using metrics tailored to the domain:

• Point cloud generation: Relative Smoothness ($$RS = |E_{X\sim p_{\text{model}}}\operatorname{tr}(X^\top L X) - E_{X\sim p_{\text{data}}}\operatorname{tr}(X^\top L X)|$$), Minimum Matching Distance (MMD), Coverage (COV), and 1-NN Accuracy (1-NNA) are standard. Incorporating smoothness constraint yields RS reductions of 20–40% for typical shape categories. Visual inspection reveals smoother, less noisy surfaces (Li et al., 2024).
• Image/surface smoothing: Sharp edges and isotropy of output are validated empirically. Robustness to algorithmic and parameter choices is observed. Feature preservation under TNC is distinct compared to TV or mean-curvature flows—edges survive, and curved regions avoid staircasing or collapse (Lu et al., 22 Dec 2025).
• Shape matching: Smooth correspondence maps are obtained even under strong non-isometries, outperforming global spectral constraints in terms of local fidelity and continuity (Magnet et al., 2022).
• Gradient methods: Empirically, directional smoothness-based rates track actual optimization progress more tightly than classical $L$-based theory (one–two orders of magnitude improvement in bounds on non–strongly–convex logistic regression) (Mishkin et al., 2024).

4. Theoretical Foundations and Regularity Detection

Directional smoothness generalizes standard smoothness by enabling control and detection along arbitrary directions:

• The equivalence of decay in DSTFT and $C^\infty$-regularity along hyperplanes ($u \cdot t = x$) connects Fourier-analytic criteria to classical smoothness theorems. Directional wave-front sets classify singularities with directionality, independent of window choice (Atanasova et al., 2017).
• In geometry, the continuum formulation using directional derivatives (as in Dirichlet energy or curvature integrals) aligns with the second fundamental form and harmonic map theory, directly relating smoothness penalties to differential geometry fundamentals (Magnet et al., 2022, Lu et al., 22 Dec 2025).
• Graph-based energies (point clouds, meshes) decompose to sums over edge directions, making pointwise directional smoothness both computationally tractable and mathematically interpretable (Li et al., 2024, Magnet et al., 2022).

5. Broader Implications and Extensions

Directional smoothness priors and measurements have broad utility and present several extension paths:

• Generative modeling/point clouds: Smoothness energies can govern point-cloud completion, up-sampling, or segmentation; anisotropic modifications (directional weights tied to local normals) further enrich control over curvature and geometry (Li et al., 2024).
• Surface/image processing: TNC regularization balances sharp feature retention against overall isotropy and avoids artifacts common to simpler regularizers. Operator-splitting frameworks can be generalized to other nonlinear or high-order PDE constraints (Lu et al., 22 Dec 2025).
• Shape matching: Pointwise directional smoothness enables map refinement frameworks applicable to inter-category correspondences, subsuming various prior registration and map-smoothing methods (Magnet et al., 2022).
• Optimization theory: Path-wise directional smoothness delivers adaptive rates and step-size rules that are tighter and more reflective of encountered curvature than global worst-case constants. This approach is particularly effective in non-strongly-convex regimes and in trajectory-sensitive optimization settings (Mishkin et al., 2024).
• Further extensions may involve higher-order directional derivatives, discrete normal evolution for enhanced curvature control, and incorporation into geometric deep learning as a regularizer for embedding fidelity and manifold-aware representations.

6. Domain-Specific Challenges and Outlook

Implementing directional smoothness requires attention to algorithmic and domain-specific details:

• Graph and mesh-based energies necessitate careful construction of adjacency, weighting schemes, and Laplacians to accurately reflect underlying geometry without introducing numerical anisotropy (Li et al., 2024, Magnet et al., 2022).
• Penalizing curvature—especially total normal curvature—demands efficient handling of high-order derivatives and robust operator-splitting schemes to preserve isotropy and avoid sensitivity to time-differencing or parameter tuning (Lu et al., 22 Dec 2025).
• DSTFT-based smoothness regularity is invariant under the choice of compactly supported windows, but proper selection of neighborhoods and cones is crucial for precise singularity localization and regularity detection (Atanasova et al., 2017).
• Robust adaptivity in gradient-based optimization relies on accurate (yet efficiently computable) directional smoothness functions along the actual trajectory, and closes the gap between empirical behavior and theoretical guarantees (Mishkin et al., 2024).
• Across domains, the integration of directional smoothness must be tuned to the specific tradeoffs between fidelity to data, geometric regularity, and computational practicality.

In summary, directional smoothness provides a theoretically sound and practically effective means to control, assess, and optimize local variations along specific directions, yielding substantive improvements in generative fidelity, correspondence quality, regularity detection, and optimization adaptivity, with robust implementation strategies spanning combinatorial, variational, and analytic frameworks.